phd thesis in applied mathematics

Recent PhD Theses - Applied Mathematics

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Department of Applied Mathematics University of Waterloo Waterloo, Ontario Canada N2L 3G1 Phone: 519-888-4567, ext. 32700 Fax: 519-746-4319

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Ph.D. in Applied Mathematics

See the catalog copy of the description of the Ph.D. in Applied Mathematics program.

1. Overview

A student in the Ph.D. in Applied Mathematics degree program must maintain satisfactory academic progress towards completion of the degree. Student satisfactory academic progress is primarily assessed by: (a) satisfactory coursework performance, (b) the Qualifying Examination, (c) the Dissertation Topic Approval Defense, and (d) the Dissertation Defense. Courses and the Qualifying Examination are used to ensure that the student has the breadth as well as the depth of knowledge needed for research success. The Dissertation Topic Approval Defense is used to ensure that the scope of dissertation research is important, that the plan is well thought out, and that the student has sufficient skills and thoughtfulness needed for success. The Dissertation Defense is used to assess the outcomes of the dissertation research, and whether or not the plan agreed upon by the Dissertation Committee has been appropriately followed.

The key requirements and milestones for the Ph.D. in Applied Mathematics degree are provided below. Failure to satisfy the requirements can result in suspension or dismissal from the program.

• Minimum Hours
• Interdisciplinary Minor
• Core Courses
• Additional “Core” Courses
• Qualifying Examination
• Dissertation Committee
• Dissertation Topic Approval Defense
• Dissertation Defense

2. Minimum Hours

To earn a Ph.D. in Applied Mathematics degree, a student must complete at least 56 approved post baccalaureate credit hours. This includes 2 hours of Responsible Conduct of Research (GRAD 8302), at least 18 hours of dissertation research and reading (MATH 8994), and the hours for the interdisciplinary minor. Graduation requirements mandate that students must achieve a minimum grade point average of 3.0 to graduate. Receiving more than two grades of C or a single grade of U in any graduate course will result in a suspension from the program.

A limited amount of transfer credit is allowed. In accordance with rules of the UNC Charlotte Graduate School, students are allowed to transfer up to 30 semester hours of graduate credit earned at UNC Charlotte or other recognized graduate programs. Only courses with grades A or B may be accepted for transfer credit. To receive transfer credit, students must file an online request (and submit all necessary documents including copies of transcripts and course syllabi if requesting to transfer non-UNC Charlotte courses).

File an online request to transfer post-Baccalaureate credits at http://gpetition.uncc.edu .

3. Interdisciplinary Minor

The interdisciplinary minor may be satisfied by 9 hours of graduate work outside the mathematics department, by 6 credit hours for a directed project in an area of application (MATH 8691/8692), or by a combination of external coursework and a directed project in an area of application totaling 9 credit hours.

It is expected that interdisciplinary minor courses shall in general be in STEM disciplines, but if there are applications in the student’s dissertation work towards the social sciences, courses in those fields are allowed too. The following is a non-exhaustive list of interdisciplinary minor courses allowed for several fields.

Physics: PHYS 5222, 5232, 5242, 5271, 6101 through 6201, 6203 through 6211, 6221 through 6271. A common example is PHYS 6210, but 5242 and 5271 would also be along the same lines.

Optics: OPTI 8101, 8102, 8104, 8105, 8211 with 8102, 8104, and 8211 being particularly relevant.

Molecular Biophysics: PHYS 6108/OPTI 8000, PHYS 6204, PHYS 6610 ( https://mbp.charlotte.edu/ )

Mechanical Engineering: MEGR 6116, 7113, 7164 for students who have specialized in math of fluids, while 6141, 6125, 7102, 7142, and 7143 for those specializing in continuum mechanics and elasticity.

Computer Science: ITCS 6111, 6114, 6150, 6153, 6155, 6165, 6170, 6171, 6220, 6226 with 6114 commonly taken.

Finance and Economics: Any of FINN or ECON courses listed under the MS Mathematical Finance program. Common examples include FINN 6203, 6210, 6211, and ECON 6206, 6113, 6219.

Mathematics Education: Any graduate level MAED courses such as MAED 6122, 6123, 6124.

4. Core Courses

All students in the Ph.D. in Applied Mathematics degree program must take the following courses, regardless of their intended area of study:

• GRAD 8302 Responsible Conduct of Research (2 hours, usually required to take within the first year in the program)
• MATH 8143 Real Analysis I (3 hours)
• MATH 8144 Real Analysis II (3 hours)
• MATH 8994 Doctoral Research and Reading (at least 18 hours)

Students whose intended area of study is statistics or mathematical finance are also required to take

• MATH 8120 Theory of Probability I (3 hours)

5. Additional “Core” Courses

The following courses, though not explicitly required, are strongly recommended for each area of study.

Statistics: STAT 5123, 5124, 5126, 5127, 6115, 8127, 8133, 8135, 8137, 8139, 8122, 8123, 8027 (at least once)

Computational Math: MATH 5165, 5171, 5172, 5173, 5174, 5176, 8172, 8176

PDE and Mathematical Physics: MATH 5173, 5174, 8172 

Probability: MATH 5128, 5129, 8120, 8125

Dynamical Systems: MATH 5173, 5174, 7275, 7276, 7277

Topology: MATH 5181, 8171, 8172 and independent study

Algebra: MATH 5163, 5164, 8163, 8164, and 8065 and/or independent study

Mathematical Finance: MATH 6202, 6203, 6204, 6205, 6206

6. Qualifying Examination

After being admitted to the Ph.D. program, a student is expected to take the qualifying examination within three semesters. This time limit may be extended up to two additional semesters in certain cases, depending on the background of the student and with program approval. The qualifying examination consists of two parts: the first part is a written examination based on Real Analysis I and II (MATH 8143/8144) or Theory of Probability I and Real Analysis I (MATH 8120/8143), the latter intended for a student with intended area of study in statistics or mathematical finance . The second part is a written examination based on two other courses chosen by the student to be specifically related to the student’s intended area of study and approved by the Graduate Coordinator. Typical choices for Part II are STAT 5126/5127, MATH 5173/5174, MATH 5172/5176, MATH 5163/5164, MATH 6205/6206, etc. The student may be allowed to retake a portion of the qualifying examination a second time if the student does not pass that portion on the first attempt within the guidelines of the Graduate School regulations pertaining to the qualifying examination and as overseen by the department Graduate Committee. A student who does not complete the qualifying examination as per the regulations of the Graduate School will be terminated from the Ph.D. program.

Complete and submit the following form after taking the Qualifying Examination. (Qualifying Exam Report Form) -> Graduate School form.

7. Dissertation Committee

After passing the Qualifying Examination, the student must set up a Dissertation Committee of at least four graduate faculty members, which must include at least three graduate faculty members from the Department of Mathematics and Statistics and one member appointed by the Graduate School. The committee is chaired by the student’s dissertation advisor. If the dissertation advisor is a graduate faculty member from an outside department or institution, a graduate faculty member from the Department of Mathematics and Statistics must be a co-chair of the committee. The Dissertation Committee must be approved by the Graduate Coordinator. After identifying and obtaining the signatures of the Dissertation Committee faculty, the Appointment of Doctoral Dissertation Committee Form must be sent to the Graduate School for the appointment of the Graduate Faculty Representative.

The Dissertation Committee should be appointed as soon as it is feasible, usually within a year after passing the Qualifying Examination.

Complete and submit the following form within a year of passing the Qualifying Examination. (Appointment of Doctoral Dissertation Committee Form) -> Graduate School form.

8. Dissertation Topic Approval Defense

Each student must present and orally defend a Ph.D. dissertation proposal after passing the Qualifying Examination and within ten semesters of entering the Program. The Dissertation Topic Approval Defense will be conducted by the student's Dissertation Committee, and will be open to faculty and students. The dissertation proposal must address a significant, original and substantive piece of research. The proposal must include sufficient preliminary data and a timeline such that the Dissertation Committee can assess its feasibility.

The student should provide copies of the written dissertation proposal to the Dissertation Committee at least two weeks prior to the oral defense. At the discretion of the Dissertation Committee, the defense may include questions that cover the student's program of study and background knowledge and techniques in the research area. The Dissertation Committee will unanimously grade the Dissertation Topic Approval Defense as pass/fail according to the corresponding rubrics. A student may retake the Dissertation Topic Approval Defense if he/she fails the first time. The second failed attempt will result in the termination of the student's enrollment in the Ph.D. program. It is expected that the student first take the proposal defense by the ninth semester after enrollment to provide time for a second try should the first one fail. A doctoral student advances to Ph.D. candidacy after the dissertation proposal has been successfully defended. Candidacy must be achieved at least six months before the degree is conferred (so if you plan to graduate in a spring semester with the commencement on May 14, then you would need to successfully defend your dissertation topic by November 13 the prior year.)

The student must follow the following procedure in order to defend the dissertation proposal.

• Communicate with the Dissertation Committee to set up a date/time for the oral defense, and reserve a defense room for at least two hours .
• Send an electronic or written copy of the dissertation proposal to each member of the Dissertation Committee at least two weeks prior to the oral defense.
• Inform the Graduate Coordinator the schedule at least one week prior to the oral defense.

Complete and submit the following form only after successfully passing the Dissertation Topic Approval Defense. (Petition for Topic Approval Form) -> Graduate School Form.

9. Dissertation

Each student must complete and defend a dissertation based on a research program approved by the student's dissertation advisor which results in a high quality, original and substantial piece of research. The student must orally present and successfully defend the dissertation before the student's doctoral dissertation committee in a defense that is open to the public. The Dissertation will be unanimously graded as pass/fail based on the corresponding rubrics by the Dissertation Committee and must be approved by the Dean of the Graduate School. Two attempts of the Dissertation Defense are permitted. The second failed attempt will result in the termination of the student's enrollment in the Ph.D. program.

The student must follow the following procedure in order to defend the dissertation.

• Communicate with the Dissertation Committee to set up a date/time for the public defense, and reserve a defense room for at least two hours with the help of the Graduate Coordinator.
• Send an electronic or written copy of the dissertation to each member of the Dissertation Committee at least three weeks prior to the public defense.
• Send an electronic copy of the dissertation in PDF as well as an abstract in a separate word file to the Graduate Coordinator at least two weeks prior to the public defense. The abstract is limited to 200 words, and does not have to be the same as the abstract included in the dissertation.
• Submit dissertation defense announcement to the general public at least 10 days prior to the scheduled defense date through https://graduateschool.charlotte.edu/current-students/graduation-clearance/submit-dissertation-defense-announcement.
• Prepare a presentation that should be at least 45 minutes long.

Complete and submit the following forms after defending your Dissertation. (Dissertation Report for Doctoral Candidates Form) -> Graduate School Form.

Also, submit the Dissertation Title Page with Original Committee Signatures.

In addition, submit ETD Signature Form with original committee and student signatures to the Graduate School within 24 hours after defense.

10. Graduation

Detailed information about graduation including the dissertation manual can be found on  the Graduate School's Graduation website .

Please pay attention to the various deadlines in the official UNC Charlotte academic calendar , in particular, the following deadlines if you are planning to graduate.

• Deadline for graduate students to apply for graduation
• Doctoral dissertation pre-defense formatting consultation deadline (may no longer be required)
• Doctoral dissertation defense deadline
• Doctoral dissertation post-defense formatting consultation deadline (may no longer be required)
• Last day to submit doctoral dissertations to Graduate School

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Mathematics PhD theses

A selection of Mathematics PhD thesis titles is listed below, some of which are available online:

2022   2021 2020 2019 2018 2017 2016 2015 2014 2013 2012 2011 2010 2009 2008 2007 2006 2005 2004 2003 2002 2001 2000 1999 1998 1997 1996 1995 1994 1993 1992 1991

Melanie Kobras –  Low order models of storm track variability

Ed Clark –  Vectorial Variational Problems in L∞ and Applications to Data Assimilation

Katerina Christou – Modelling PDEs in Population Dynamics using Fixed and Moving Meshes  

Chiara Cecilia Maiocchi –  Unstable Periodic Orbits: a language to interpret the complexity of chaotic systems

Samuel R Harrison – Stalactite Inspired Thin Film Flow

Elena Saggioro – Causal network approaches for the study of sub-seasonal to seasonal variability and predictability

Cathie A Wells – Reformulating aircraft routing algorithms to reduce fuel burn and thus CO 2 emissions  

Jennifer E. Israelsson –  The spatial statistical distribution for multiple rainfall intensities over Ghana

Giulia Carigi –  Ergodic properties and response theory for a stochastic two-layer model of geophysical fluid dynamics

André Macedo –  Local-global principles for norms

Tsz Yan Leung  –  Weather Predictability: Some Theoretical Considerations

Jehan Alswaihli –  Iteration of Inverse Problems and Data Assimilation Techniques for Neural Field Equations

Jemima M Tabeart –  On the treatment of correlated observation errors in data assimilation

Chris Davies –  Computer Simulation Studies of Dynamics and Self-Assembly Behaviour of Charged Polymer Systems

Birzhan Ayanbayev –  Some Problems in Vectorial Calculus of Variations in L∞

Penpark Sirimark –  Mathematical Modelling of Liquid Transport in Porous Materials at Low Levels of Saturation

Adam Barker –  Path Properties of Levy Processes

Hasen Mekki Öztürk –  Spectra of Indefinite Linear Operator Pencils

Carlo Cafaro –  Information gain that convective-scale models bring to probabilistic weather forecasts

Nicola Thorn –  The boundedness and spectral properties of multiplicative Toeplitz operators

James Jackaman  – Finite element methods as geometric structure preserving algorithms

Changqiong Wang - Applications of Monte Carlo Methods in Studying Polymer Dynamics

Jack Kirk - The molecular dynamics and rheology of polymer melts near the flat surface

Hussien Ali Hussien Abugirda - Linear and Nonlinear Non-Divergence Elliptic Systems of Partial Differential Equations

Andrew Gibbs - Numerical methods for high frequency scattering by multiple obstacles (PDF-2.63MB)

Mohammad Al Azah - Fast Evaluation of Special Functions by the Modified Trapezium Rule (PDF-913KB)

Katarzyna (Kasia) Kozlowska - Riemann-Hilbert Problems and their applications in mathematical physics (PDF-1.16MB)

Anna Watkins - A Moving Mesh Finite Element Method and its Application to Population Dynamics (PDF-2.46MB)

Niall Arthurs - An Investigation of Conservative Moving-Mesh Methods for Conservation Laws (PDF-1.1MB)

Samuel Groth - Numerical and asymptotic methods for scattering by penetrable obstacles (PDF-6.29MB)

Katherine E. Howes - Accounting for Model Error in Four-Dimensional Variational Data Assimilation (PDF-2.69MB)

Jian Zhu - Multiscale Computer Simulation Studies of Entangled Branched Polymers (PDF-1.69MB)

Tommy Liu - Stochastic Resonance for a Model with Two Pathways (PDF-11.4MB)

Matthew Paul Edgington - Mathematical modelling of bacterial chemotaxis signalling pathways (PDF-9.04MB)

Anne Reinarz - Sparse space-time boundary element methods for the heat equation (PDF-1.39MB)

Adam El-Said - Conditioning of the Weak-Constraint Variational Data Assimilation Problem for Numerical Weather Prediction (PDF-2.64MB)

Nicholas Bird - A Moving-Mesh Method for High Order Nonlinear Diffusion (PDF-1.30MB)

Charlotta Jasmine Howarth - New generation finite element methods for forward seismic modelling (PDF-5,52MB)

Aldo Rota - From the classical moment problem to the realizability problem on basic semi-algebraic sets of generalized functions (PDF-1.0MB)

Sarah Lianne Cole - Truncation Error Estimates for Mesh Refinement in Lagrangian Hydrocodes (PDF-2.84MB)

Alexander J. F. Moodey - Instability and Regularization for Data Assimilation (PDF-1.32MB)

Dale Partridge - Numerical Modelling of Glaciers: Moving Meshes and Data Assimilation (PDF-3.19MB)

Joanne A. Waller - Using Observations at Different Spatial Scales in Data Assimilation for Environmental Prediction (PDF-6.75MB)

Faez Ali AL-Maamori - Theory and Examples of Generalised Prime Systems (PDF-503KB)

Mark Parsons - Mathematical Modelling of Evolving Networks

Natalie L.H. Lowery - Classification methods for an ill-posed reconstruction with an application to fuel cell monitoring

David Gilbert - Analysis of large-scale atmospheric flows

Peter Spence - Free and Moving Boundary Problems in Ion Beam Dynamics (PDF-5MB)

Timothy S. Palmer - Modelling a single polymer entanglement (PDF-5.02MB)

Mohamad Shukor Talib - Dynamics of Entangled Polymer Chain in a Grid of Obstacles (PDF-2.49MB)

Cassandra A.J. Moran - Wave scattering by harbours and offshore structures

Ashley Twigger - Boundary element methods for high frequency scattering

David A. Smith - Spectral theory of ordinary and partial linear differential operators on finite intervals (PDF-1.05MB)

Stephen A. Haben - Conditioning and Preconditioning of the Minimisation Problem in Variational Data Assimilation (PDF-3.51MB)

Jing Cao - Molecular dynamics study of polymer melts (PDF-3.98MB)

Bonhi Bhattacharya - Mathematical Modelling of Low Density Lipoprotein Metabolism. Intracellular Cholesterol Regulation (PDF-4.06MB)

Tamsin E. Lee - Modelling time-dependent partial differential equations using a moving mesh approach based on conservation (PDF-2.17MB)

Polly J. Smith - Joint state and parameter estimation using data assimilation with application to morphodynamic modelling (PDF-3Mb)

Corinna Burkard - Three-dimensional Scattering Problems with applications to Optical Security Devices (PDF-1.85Mb)

Laura M. Stewart - Correlated observation errors in data assimilation (PDF-4.07MB)

R.D. Giddings - Mesh Movement via Optimal Transportation (PDF-29.1MbB)

G.M. Baxter - 4D-Var for high resolution, nested models with a range of scales (PDF-1.06MB)

C. Spencer - A generalization of Talbot's theorem about King Arthur and his Knights of the Round Table.

P. Jelfs - A C-property satisfying RKDG Scheme with Application to the Morphodynamic Equations (PDF-11.7MB)

L. Bennetts - Wave scattering by ice sheets of varying thickness

M. Preston - Boundary Integral Equations method for 3-D water waves

J. Percival - Displacement Assimilation for Ocean Models (PDF - 7.70MB)

D. Katz - The Application of PV-based Control Variable Transformations in Variational Data Assimilation (PDF- 1.75MB)

S. Pimentel - Estimation of the Diurnal Variability of sea surface temperatures using numerical modelling and the assimilation of satellite observations (PDF-5.9MB)

J.M. Morrell - A cell by cell anisotropic adaptive mesh Arbitrary Lagrangian Eulerian method for the numerical solution of the Euler equations (PDF-7.7MB)

L. Watkinson - Four dimensional variational data assimilation for Hamiltonian problems

M. Hunt - Unique extension of atomic functionals of JB*-Triples

D. Chilton - An alternative approach to the analysis of two-point boundary value problems for linear evolutionary PDEs and applications

T.H.A. Frame - Methods of targeting observations for the improvement of weather forecast skill

C. Hughes - On the topographical scattering and near-trapping of water waves

B.V. Wells - A moving mesh finite element method for the numerical solution of partial differential equations and systems

D.A. Bailey - A ghost fluid, finite volume continuous rezone/remap Eulerian method for time-dependent compressible Euler flows

M. Henderson - Extending the edge-colouring of graphs

K. Allen - The propagation of large scale sediment structures in closed channels

D. Cariolaro - The 1-Factorization problem and same related conjectures

A.C.P. Steptoe - Extreme functionals and Stone-Weierstrass theory of inner ideals in JB*-Triples

D.E. Brown - Preconditioners for inhomogeneous anisotropic problems with spherical geometry in ocean modelling

S.J. Fletcher - High Order Balance Conditions using Hamiltonian Dynamics for Numerical Weather Prediction

C. Johnson - Information Content of Observations in Variational Data Assimilation

M.A. Wakefield - Bounds on Quantities of Physical Interest

M. Johnson - Some problems on graphs and designs

A.C. Lemos - Numerical Methods for Singular Differential Equations Arising from Steady Flows in Channels and Ducts

R.K. Lashley - Automatic Generation of Accurate Advection Schemes on Structured Grids and their Application to Meteorological Problems

J.V. Morgan - Numerical Methods for Macroscopic Traffic Models

M.A. Wlasak - The Examination of Balanced and Unbalanced Flow using Potential Vorticity in Atmospheric Modelling

M. Martin - Data Assimilation in Ocean circulation models with systematic errors

K.W. Blake - Moving Mesh Methods for Non-Linear Parabolic Partial Differential Equations

J. Hudson - Numerical Techniques for Morphodynamic Modelling

A.S. Lawless - Development of linear models for data assimilation in numerical weather prediction .

C.J.Smith - The semi lagrangian method in atmospheric modelling

T.C. Johnson - Implicit Numerical Schemes for Transcritical Shallow Water Flow

M.J. Hoyle - Some Approximations to Water Wave Motion over Topography.

P. Samuels - An Account of Research into an Area of Analytical Fluid Mechnaics. Volume II. Some mathematical Proofs of Property u of the Weak End of Shocks.

M.J. Martin - Data Assimulation in Ocean Circulation with Systematic Errors

P. Sims - Interface Tracking using Lagrangian Eulerian Methods.

P. Macabe - The Mathematical Analysis of a Class of Singular Reaction-Diffusion Systems.

B. Sheppard - On Generalisations of the Stone-Weisstrass Theorem to Jordan Structures.

S. Leary - Least Squares Methods with Adjustable Nodes for Steady Hyperbolic PDEs.

I. Sciriha - On Some Aspects of Graph Spectra.

P.A. Burton - Convergence of flux limiter schemes for hyperbolic conservation laws with source terms.

J.F. Goodwin - Developing a practical approach to water wave scattering problems.

N.R.T. Biggs - Integral equation embedding methods in wave-diffraction methods.

L.P. Gibson - Bifurcation analysis of eigenstructure assignment control in a simple nonlinear aircraft model.

A.K. Griffith - Data assimilation for numerical weather prediction using control theory. .

J. Bryans - Denotational semantic models for real-time LOTOS.

I. MacDonald - Analysis and computation of steady open channel flow .

A. Morton - Higher order Godunov IMPES compositional modelling of oil reservoirs.

S.M. Allen - Extended edge-colourings of graphs.

M.E. Hubbard - Multidimensional upwinding and grid adaptation for conservation laws.

C.J. Chikunji - On the classification of finite rings.

S.J.G. Bell - Numerical techniques for smooth transformation and regularisation of time-varying linear descriptor systems.

D.J. Staziker - Water wave scattering by undulating bed topography .

K.J. Neylon - Non-symmetric methods in the modelling of contaminant transport in porous media. .

D.M. Littleboy - Numerical techniques for eigenstructure assignment by output feedback in aircraft applications .

M.P. Dainton - Numerical methods for the solution of systems of uncertain differential equations with application in numerical modelling of oil recovery from underground reservoirs .

M.H. Mawson - The shallow-water semi-geostrophic equations on the sphere. .

S.M. Stringer - The use of robust observers in the simulation of gas supply networks .

S.L. Wakelin - Variational principles and the finite element method for channel flows. .

E.M. Dicks - Higher order Godunov black-oil simulations for compressible flow in porous media .

C.P. Reeves - Moving finite elements and overturning solutions .

A.J. Malcolm - Data dependent triangular grid generation. .

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• Past PhD Dissertations
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Yuqi Wu Parallel Scalable Domain Decomposition Method for Simulating Blood Flows in Three-dimensional Compliant Arteries

Erin Byrne The Post-fragmentation Probability for Bacterial Aggregates

Adrianna Gillman Fast Direct Solvers for Elliptic Partial Differential Equations

Ian Grooms Asymptotic and Numerical Methods for Rapidly Rotating Buoyant Flow

Sean Nixon Development and Applications of Soliton Perturbation Theory

Kristine Snyder Tuning and Control in Human Locomotion

Kye Taylor Modeling and Analysis of the Low-dimensional Geometry of Signal and Image Patch-sets

Jinyu Li An Improved Short-DNA Elasticity Theory and a Model of the Dynamics of Biological Signaling Networks

Min Ho Park Relaxation-corrected Bootstrap Algebraic Multigrid (rBAMG)

Lei Tang Parallel Efficiency-based Adaptive Local Refinement

Patrick Young Numerical Techniques for the Solution of Partial Differential and Integral Equations on Irregular Domains with Applications to Problems in Electrowetting

James Adler Nested Iteration and First Order Systems Least Squares on Incompressible Resistive Magnetohydrodynamics

Andrew Barker Parallel Monolithic Fluid-structure Interaction Algorithms with Application to Blood Flow Simulation

Benjamin Jamroz Reduced Modeling of the Magnetorotational Instability

Christian Ketelsen Least-squares Finite Element Methods for Quantum Electrodynamics

Michael Levy A High-order Element-based Galerkin Method for the Global Shallow Water Equations

Si Liu Parallel Fully Coupled Domain Decomposition Algorithm for Some Inverse Problems

Gregory Norgard Shock regularization of conservation laws through use of spatial averaging in nonlinear terms

Nathan Aragon Flip Estimators, Cross-Entropy, and Half-Stationary Bounding Processes for Monte Carlo Simulations

Seth Claudepierre Solar Wind Driving of Magnetospheric Ultra-Low Frequency Pulsations

Terry Haut Nonlocal Formulations of Ideal Fluids and Applications

Christopher Kurcz Fast Convolutions with Helmoholtz Green's Functions and Radially Symmetric Band-Limited Kernels

Joshua Nolting Efficiency-Based Local Adaptive Refinement for FOSLS Finite Elements

Cecile Piret Analytical and Numerical Advances in Radial Basis Functions

Geoffrey Sanders Extensions to Adaptive Smooth Aggregation (alphaSA) Multigrid: Eigensolver Initialization and nonsymmetric Problems

Michael Watson  A Study of Rotationally Constrained Convection in Tall Aspect Ratio Annular Geometries

Chao Jin Parallel Domain Decomposition Methods for Stochastic Partial Differential Equations and Analysis of Nonlinear Integral Equations

Jisun Lim The Qualitative Study of a Chemical Reaction Diffusion System and Some Integral Equations

Wenjin Mao Demension Jumping and Auxiliary Variable Techniques for Markov Chain Monte Carlo Algorithms

Jonathan Pietarila-Graham Regularizations as Subgrid Models for Turbulent Flows

Thaned Rojsiraphisal A Study of the Variablility in the North Indian Ocean

Brendan Sheehan Multigrid Methods for Isotropic Neutron Transport

Jian Wang Recovering Bayesian Networks with Applications to Gene Regulatory Networks

Julia Zuev Recent Advances in Numerical Partial Differential Equations

• Cory Ahrens The Asymptotic Analysis of Communications and Wave Collapse Problems in Nonlinear Optics
• Hong Liu Rare Events, Heavy Tails, and Simulation
• Mark Hoefer Dispersive Shock Waves in Bose-Einstein Condensates and Nonlinear Nano-oscillators in Ferromagnetic Thin Films

Marcio Carvalho Applying Perfect Simulation to solve Stochastic Difference Equations that arise from certain Time Series Models

Paki Suwannajan Evaluating the Performance of Latent Semantic Indexing

Srinath Vadlamani An Algorithmic Unification of Particle-In-Cell and Continuum Methods and a Wave-Particle Description for the Electron Temperature Gradient (ETG) Instability Saturation

Neil Martinsen-Burrell Merger and Alignment of Three-dimensional Quasigeostrophic Vortices

Feng-Nan Hwang Some Parallel Linear and Nonlinear Schwarz Methods with Applications in Computational Fluid Dynamics

Eunjung Lee FOSLL* for Eddy Current Problems with Three-dimensional Edge Singularities

Scott MacLachlan Improving Robustness in Multiscale Methods

Richard McNamara Applications of Spanning Trees to Continuous-Time Markov Processes, with Emphasis on Loss Systems

Paul Mullowney Lagrangian Particle Transport/Mixing in Blinking-Roll Systems

Mark Petersen A Study of Geophysical and Astrophysical Turbulence using Reduced Equations

Oliver Roehrle Multilevel First-Order System Least Squares for Quasilinear Elliptic Partial Differential Equations

Matthew Tearle Optimal Perturbation Analysis of Stratified Shear Flows

Eric Thaler An evaluation of the operational use of numerical solutions to the quasigeostrophic diagnostic equations by weather forecasters

Chad Westphal First-Order System Least Squares for Geometrically-Nonlinear Elasticity in Nonsmooth Domains

Allison Baker Improving the performance of the linear solver restarted GMRES

Adriana Gómez-Hoyos (Dept. of Mathematics) Conservative Maps: Reversibility, Invariants and Approximation

Luke Olson Multilevel Least-Squares Finite Element Methods for Hyperbolic Partial Differential Equations

Cristina Perez Simulating the Interaction Between Intraseasonal and Interannual Variability in the Tropical Pacific with a Coupled System of Nonlinear Ordinary Differential Equations

Kristian Sandberg Forward and inverse wave propagation using bandlimited functions and a fast reconstruction algorithm for electron microscopy

Ulrike Schneider Advances and Applications in Perfect Sampling

Grady Wright Radical Basis function Interpolation: Numerical and Analytical Developments

• Eric S. Wright Modeling and Analysis of Aqueous Chemical Reactions in a Diffusive Environment

Travis Austin Advances on a Scaled Least-Squares Method for the 3-D Linear Boltzman Equation

Brian Bloechle On the Taylor Dispersion of Reactive Solutes in a Parallel-Plate Fracture-Matrix System

John Carter Stability and Existence of Traveling Wave Solutions of the Two-Dimensional Nonlinear Schrödinger Equation and its Higher-Order Generalizations

Tim Chartier Element-Based Algebraic Multigrid (AMGe) and Spectral AMGe

Andrea Codd Elasticity-Fluid Coupled Systems and Elliptic Grid Generation (EGG) based on First-Order System Least Squares (FOSLS)

Rudy Horne Collision induced timing jitter and four-wave mixing in wavelength division multiplexing soliton systems

Hugh MacMillan First-Order System Least Squares and Electrical Impedance Tomography

Bobby Phillip Asynchronous Fast Adaptive Composite Grid Methods for Elliptic Problems on Adaptively-Refined Curvilinear Grids

• Vlatcheslave Akmaev Phylogenic Approach to Molecular Structure Prediction
• Michelle Ghrist High-order finite difference methods for wave equations
• Ken Jarman Stochastic immiscible flow with moment equations
• Vanessa Robins Computational Topology at Multiple Resolutions
• David Trubatch Topics in Solitons and Inverse Scattering: I. Discretization of the vector nonlinear Schrodinger equation, and II. A new class of 'reflectionless' potentials of the nonstationary Schrodinger equation and solutions of the Kadomtsev and Petviashvili I equation
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• David Sterling Anti-integrable continuation and the destruction of chaos
• Lora Billings Dynamical Systems Methods Applied to Polynomial Factorization Families: a Study of Chaotic Attractors
• Bernard Deconinck The initial-value problem for multiphase solutions of the Kadomtsev-Petviahvili equation
• Laurie Heyer The probabilistic behavior of sequence analysis scores with application to structural alignment of RNA
• Peter Staab Three-dimensional acoustic-rotational flows in solid-fuel rocket motors
• Nicholas Coult A Multiresolution Strategy for Homogenization of Partial Differential Equations
• Robert Cramer A Multiresolution Approach to Fast Summation and Regularization of Singular Operators
• Barry Lee First-Order Systems least Squares for Electromagnetics
• Alejandro Spina Confined States in Large Aspect Ratio Thermohaline Convection
• Erik Bollt Controlling Chaos, Targeting and Transport
• James Keiser I. Wavelet Based approach to Numerical Solution of Nonlinear Partial Differential Equations, and II. Nonlinear Waves I Fully Discrete Dynamical Systems
• David Sholl Lattice Gas Models of Surface Chemistry
• Scott Herod Computer Assisted Determination of Lie Point Symmetries with Application to Fluid Dynamics
• Arthur Mizzi | Spectral Representation of the Vertical Coordinate in Three-Dimensional Atmospherical Models on Tropical B- and f- Planes
• Linda Sundbye Global Existence of Solutions for the Shallow water Equations
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Overview of the PhD Program

For specific information on the Applied Mathematics PhD program, see the navigation links to the right. 

What follows on this page is an overview of all Ph.D. programs at the School; additional information and guidance can be found on the  Graduate Policies  pages. 

General Ph.D. Requirements

• 10 semester-long graduate courses, including at least 8 disciplinary.   At least 5 of the 10 should be graduate-level SEAS "technical" courses (or FAS graduate-level technical courses taught by SEAS faculty), not including seminar/reading/project courses.  Undergraduate-level courses cannot be used.  For details on course requirements, see the school's overall PhD course requirements  and the individual program pages linked therein.
• Program Plan (i.e., the set of courses to be used towards the degree) approval by the  Committee on Higher Degrees  (CHD).
• Minimum full-time academic residency of two years .
• Serve as a Teaching Fellow (TF) in one semester of the second year.
• Oral Qualifying Examination Preparation in the major field is evaluated in an oral examination by a qualifying committee. The examination has the dual purpose of verifying the adequacy of the student's preparation for undertaking research in a chosen field and of assessing the student's ability to synthesize knowledge already acquired. For details on arranging your Qualifying Exam, see the exam policies and the individual program pages linked therein.
• Committee Meetings : PhD students' research committees meet according to the guidelines in each area's "Committee Meetings" listing.  For details see the "G3+ Committee Meetings" section of the Policies of the CHD  and the individual program pages linked therein.
• Final Oral Examination (Defense) This public examination devoted to the field of the dissertation is conducted by the student's research committee. It includes, but is not restricted to, a defense of the dissertation itself.  For details of arranging your final oral exam see the  Ph.D. Timeline  page.
• Dissertation Upon successful completion of the qualifying examination, a committee chaired by the research supervisor is constituted to oversee the dissertation research. The dissertation must, in the judgment of the research committee, meet the standards of significant and original research.

Optional additions to the Ph.D. program

Harvard PhD students may choose to pursue these additional aspects:

• a Secondary Field (which is similar to a "minor" subject area).  SEAS offers PhD Secondary Field programs in  Data Science and in  Computational Science and Engineering .   GSAS  lists  secondary fields offered by other programs.
• a Master of Science (S.M.) degree conferred  en route to the Ph.D in one of several of SEAS's subject areas.  For details see here .
• a Teaching Certificate awarded by the Derek Bok Center for Teaching and Learning .

SEAS PhD students may apply to participate in the  Health Sciences and Technology graduate program  with Harvard Medical School and MIT.  Please check with the HST program for details on eligibility (e.g., only students in their G1 year may apply) and the application process.

In Applied Mathematics

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Ph.D. Program

The degree of Doctor of Philosophy in Applied Mathematics and Computational Science is conferred in recognition of marked ability and high attainment in advanced applied and computational mathematics, including the successful completion of a significant original research project. The program typically takes four to five years to complete, although this length may vary depending on the student. Below, we describe the requirements and expectations of the program.

Written Preliminary Exam

Upon entry into the Ph.D. program, students are required to take the Written Preliminary Exam, typically scheduled the week before classes start in the Fall semester. The coverage of the exam is in Linear Algebra, Advanced Calculus, Complex Variables, and Probability at the undergraduate level. Details of the exam can be found here: Preliminary Exam Details

The student must pass the exam to continue as a Ph.D. student. The Written Exam is offered in April and August. If the student fails on the first attempt, two more attempts are granted (three attempts total).

Course Requirements

The student must take the following six core courses:

• Analysis: AMCS 6081/6091 (MATH 6080/6090)
• Numerical Analysis: AMCS 6025/6035
• Probability and Stochastic Processes: AMCS 6481/6491 (MATH 6480/6490)

These six core courses are to be completed in the first and second years of graduate studies.

Ten elective courses (a total of 14 courses) are required for graduation. These elective courses should be chosen according to the interests and/or research program of the student and must contain significant mathematical content. Whether a given course can be counted toward AMCS elective course credit will be decided in consultation with the Graduate Chair. Recent courses approved for elective credit can be discussed with your advisor and the Graduate Group Chair.

Deviations from the above may be necessary or recommended depending on the individual student; such decisions are made with the approval of the graduate chair.

Choosing an Advisor

In the first two years of graduate studies, students must choose their thesis advisor. Some students already have an advisor to whom they have committed upon entry to the program. Other students will typically start working with their prospective advisors in the latter half of the first year or the summer between the first and second year.

The purpose of the oral exam is to assess a student’s readiness to transition into full-time research and eventually write his or her dissertation. This exam will be taken by the end of the third year of graduate study.

First, an oral exam committee must be formed, consisting of three faculty members, two of whom must belong to the AMCS graduate faculty. The student must then produce a document of up to about 20 pages describing the research proposal and background material, which is then approved by the oral exam committee before the exam. In the exam, the student will give an oral presentation to the committee. A discussion with the committee follows this. In the oral exam, the committee may ask the student about the presentation as well as about necessary background material as seen fit by the committee. If the student fails this exam, the student will have one more attempt.

Dissertation and Defense

The dissertation must be a substantial original investigation in the field of applied mathematics and computational science, done under the supervision of a faculty advisor. A Ph.D. Thesis Committee consists of at least three faculty members, including the thesis advisor. When the dissertation is complete, it must be defended in a Dissertation Exam, at which the student will be expected to give a short public exposition of the results of the thesis and to satisfactorily answer questions about the thesis and related areas.

Teaching Assistant

Full-time students admitted to our Ph.D. program who are offered a financial support package for four years of study are required to be teaching assistants during the second year. Students for whom English is not their native language are required to pass a test the “Speak Test” (IELTS) demonstrating proficiency in English. More information can be found on the English Language Programs  web page.

https://www.elp.upenn.edu/institute-academic-studies/requirements

Ph.D. Program

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The range of possibilities for graduate study encompasses the areas of specialization of all of the faculty members in the field, who current number more than one hundred. The faculty members are drawn from fourteen departments in the College of Engineering, the College of Arts and Sciences, the College of Agriculture and Life Sciences and the Samuel Curtis Johnson Graduate School of Management. There is opportunity for further diversification on the choice of minor subjects.

Graduate students are admitted to the Field of Applied Mathematics from a variety of educational backgrounds that have a strong mathematics component. Generally, only students who wish to become candidates for the Doctor of Philosophy Degree are considered. About forty students are enrolled in the program, which usually requires four to five years to complete.  

A normal course load for a beginning graduate student is three courses per term. Please see field requirements for details on courses. The Director of Graduate Studies in conjunction with the student's temporary committee chair will assist first-year students in determining the appropriate courses to meet individual needs. The program allows great flexibility in the selection of courses. Most students design their own course sequences, subject to requirements, to meet their own interests. Courses are typically chosen from the math department and many applications departments. The course requirements in detail can be found under Requirements .  

Minor Subjects and Special Committee

Incoming students are assigned a temporary committee chair. Students are expected to select a permanent full committee by the end of the third semester. Students submit a "Special Committee Change and Selection Form" to the Graduate School to indicate their selection. Students may change committee members at any time by submitting a new form to the Graduate School. However, if they are post A-exam or three months within Ph.D. exam (B-exam), they must petition.

The Special Committee consists of a Chair/thesis advisor and at least one member for each of two minor subjects. One of the minor subjects must be mathematics. The other minor field can be from any area chosen by the student that is relevant to their doctoral research.  

To be admitted formally to candidacy for the Ph.D. degree, the student must pass the oral admission to candidacy examination or A exam. This must be completed before the beginning of the student's fourth year. The admission to candidacy examination is given to determine if the student is "ready to begin work on a thesis." The content and methods of examination are agreed on by the student and his/her committee before the examination. The student must be prepared to answer questions on the proposed area of research, and to pass the exam, he/she must demonstrate expertise beyond just mastery of basic mathematics covered in the standard first-year graduate courses.  

To receive an advanced degree a student must fulfill the residence requirements of the Graduate School. One unit of residence is granted for successful completion of one semester of full-time study, as judged by the chair of the Special Committee. The Ph.D. program requires a minimum of six residence units. This is not a difficult requirement to satisfy since the program generally takes four to five years to complete. A student who has done graduate work at another institution may petition to transfer residence credit but may not receive more than two such credits.  

Thesis/B Exam

The candidate must write a thesis that represents creative work and contains original results in that area. The research is carried on independently by the candidate under the supervision of the chairperson of the Special Committee. When the thesis is completed, the student presents his/her results at the thesis defense or B exam.  

Graduate Handbook

For further details on the program, see the  Graduate Handbook .

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Home > FACULTIES > Applied Mathematics > APMATHS-ETD

Applied Mathematics Theses and Dissertations

This collection contains theses and dissertations from the Department of Applied Mathematics, collected from the Scholarship@Western Electronic Thesis and Dissertation Repository

Theses/Dissertations from 2023 2023

Visual Cortical Traveling Waves: From Spontaneous Spiking Populations to Stimulus-Evoked Models of Short-Term Prediction , Gabriel B. Benigno

Spike-Time Neural Codes and their Implication for Memory , Alexandra Busch

Study of Behaviour Change and Impact on Infectious Disease Dynamics by Mathematical Models , Tianyu Cheng

Series Expansions of Lambert W and Related Functions , Jacob Imre

Data-Driven Exploration of Coarse-Grained Equations: Harnessing Machine Learning , Elham Kianiharchegani

Pythagorean Vectors and Rational Orthonormal Matrices , Aishat Olagunju

The Magnetic Field of Protostar-Disk-Outflow Systems , Mahmoud Sharkawi

A Highly Charged Topic: Intrinsically Disordered Proteins and Protein pKa Values , Carter J. Wilson

Population Dynamics and Bifurcations in Predator-Prey Systems with Allee Effect , Yanni Zeng

Theses/Dissertations from 2022 2022

A Molecular Dynamics Study Of Polymer Chains In Shear Flows and Nanocomposites , Venkat Bala

On the Spatial Modelling of Biological Invasions , Tedi Ramaj

Complete Hopf and Bogdanov-Takens Bifurcation Analysis on Two Epidemic Models , Yuzhu Ruan

A Theoretical Perspective on Parasite-Host Coevolution with Alternative Modes of Infection , George N. Shillcock

Theses/Dissertations from 2021 2021

Mathematical Modelling & Simulation of Large and Small Scale Structures in Star Formation , Gianfranco Bino

Mathematical Modelling of Ecological Systems in Patchy Environments , Ao Li

Credit Risk Measurement and Application based on BP Neural Networks , Jingshi Luo

Coevolution of Hosts and Pathogens in the Presence of Multiple Types of Hosts , Evan J. Mitchell

SymPhas: A modular API for phase-field modeling using compile-time symbolic algebra , Steven A. Silber

Population and Evolution Dynamics in Predator-prey Systems with Anti-predation Responses , Yang Wang

Theses/Dissertations from 2020 2020

The journey of a single polymer chain to a nanopore , Navid Afrasiabian

Exploration Of Stock Price Predictability In HFT With An Application In Spoofing Detection , Andrew Day

Multi-Scale Evolution of Virulence of HIV-1 , David W. Dick

Contraction Analysis of Functional Competitive Lotka-Volterra Systems: Understanding Competition Between Modified Bacteria and Plasmodium within Mosquitoes. , Nickolas Goncharenko

Phage-Bacteria Interaction and Prophage Sequences in Bacterial Genomes , Amjad Khan

The Effect of the Initial Structure on the System Relaxation Time in Langevin Dynamics , Omid Mozafar

Mathematical modelling of prophage dynamics , Tyler Pattenden

Hybrid Symbolic-Numeric Computing in Linear and Polynomial Algebra , Leili Rafiee Sevyeri

Abelian Integral Method and its Application , Xianbo Sun

Theses/Dissertations from 2019 2019

Algebraic Companions and Linearizations , Eunice Y. S. Chan

Algorithms for Mappings and Symmetries of Differential Equations , Zahra Mohammadi

Algorithms for Bohemian Matrices , Steven E. Thornton

A Survey Of Numerical Quadrature Methods For Highly Oscillatory Integrals , Jeet Trivedi

Theses/Dissertations from 2018 2018

Properties and Computation of the Inverse of the Gamma function , Folitse Komla Amenyou

Optimization Studies and Applications: in Retail Gasoline Market , Daero Kim

Models of conflict and voluntary cooperation between individuals in non-egalitarian social groups , Cody Koykka

Investigation of chaos in biological systems , Navaneeth Mohan

Bifurcation Analysis of Two Biological Systems: A Tritrophic Food Chain Model and An Oscillating Networks Model , Xiangyu Wang

Ecology and Evolution of Dispersal in Metapopulations , Jingjing Xu

Selected Topics in Quantization and Renormalization of Gauge Fields , Chenguang Zhao

Three Essays on Structural Models , Xinghua Zhou

Theses/Dissertations from 2017 2017

On Honey Bee Colony Dynamics and Disease Transmission , Matthew I. Betti

Simulation of driven elastic spheres in a Newtonian fluid , Shikhar M. Dwivedi

Feasible Computation in Symbolic and Numeric Integration , Robert H.C. Moir

Modelling Walleye Population and Its Cannibalism Effect , Quan Zhou

Theses/Dissertations from 2016 2016

Dynamics of Discs in a Nematic Liquid Crystal , Alena Antipova

Modelling the Impact of Climate Change on the Polar Bear Population in Western Hudson Bay , Nicole Bastow

A comparison of solution methods for Mandelbrot-like polynomials , Eunice Y. S. Chan

A model-based test of the efficacy of a simple rule for predicting adaptive sex allocation , Joshua D. Dunn

Universal Scaling Properties After Quantum Quenches , Damian Andres Galante

Modeling the Mass Function of Stellar Clusters Using the Modified Lognormal Power-Law Probability Distribution Function , Deepakshi Madaan

Bacteria-Phage Models with a Focus on Prophage as a Genetic Reservoir , Alina Nadeem

A Sequence of Symmetric Bézout Matrix Polynomials , Leili Rafiee Sevyeri

Study of Infectious Diseases by Mathematical Models: Predictions and Controls , SM Ashrafur Rahman

The survival probability of beneficial de novo mutations in budding viruses, with an emphasis on influenza A viral dynamics , Jennifer NS Reid

Essays in Market Structure and Liquidity , Adrian J. Walton

Computation of Real Radical Ideals by Semidefinite Programming and Iterative Methods , Fei Wang

Studying Both Direct and Indirect Effects in Predator-Prey Interaction , Xiaoying Wang

Theses/Dissertations from 2015 2015

The Effect of Diversification on the Dynamics of Mobile Genetic Elements in Prokaryotes: The Birth-Death-Diversification Model , Nicole E. Drakos

Algorithms to Compute Characteristic Classes , Martin Helmer

Studies of Contingent Capital Bonds , Jingya Li

Determination of Lie superalgebras of supersymmetries of super differential equations , Xuan Liu

Edge states and quantum Hall phases in graphene , Pavlo Piatkovskyi

Evolution of Mobile Promoters in Prokaryotic Genomes. , Mahnaz Rabbani

Extensions of the Cross-Entropy Method with Applications to Diffusion Processes and Portfolio Losses , Alexandre Scott

Theses/Dissertations from 2014 2014

A Molecular Simulation Study on Micelle Fragmentation and Wetting in Nano-Confined Channels , Mona Habibi

Study of Virus Dynamics by Mathematical Models , Xiulan Lai

Applications of Stochastic Control in Energy Real Options and Market Illiquidity , Christian Maxwell

Options Pricing and Hedging in a Regime-Switching Volatility Model , Melissa A. Mielkie

Optimal Contract Design for Co-development of Companion Diagnostics , Rodney T. Tembo

Bifurcation of Limit Cycles in Smooth and Non-smooth Dynamical Systems with Normal Form Computation , Yun Tian

Understanding Recurrent Disease: A Dynamical Systems Approach , Wenjing Zhang

Theses/Dissertations from 2013 2013

Pricing and Hedging Index Options with a Dominant Constituent Stock , Helen Cheyne

On evolution dynamics and strategies in some host-parasite models , Liman Dai

Valuation of the Peterborough Prison Social Impact Bond , Majid Hasan

Sensitivity Analysis of Minimum Variance Portfolios , Xiaohu Ji

Eigenvalue Methods for Interpolation Bases , Piers W. Lawrence

Hybrid Lattice Boltzmann - Molecular Dynamics Simulations With Both Simple and Complex Fluids , Frances E. Mackay

Ecological Constraints and the Evolution of Cooperative Breeding , David McLeod

A single cell based model for cell divisions with spontaneous topology changes , Anna Mkrtchyan

Analysis of Re-advanceable Mortgages , Almas Naseem

Modeling leafhopper populations and their role in transmitting plant diseases. , Ji Ruan

Topological properties of modular networks, with a focus on networks of functional connections in the human brain , Estefania Ruiz Vargas

Computation Sequences for Series and Polynomials , Yiming Zhang

Theses/Dissertations from 2012 2012

A Real Options Valuation of Renewable Energy Projects , Natasha Burke

Approximate methods for dynamic portfolio allocation under transaction costs , Nabeel Butt

Optimal clustering techniques for metagenomic sequencing data , Erik T. Cameron

Phase Field Crystal Approach to the Solidification of Ferromagnetic Materials , Niloufar Faghihi

Molecular Dynamics Simulations of Peptide-Mineral Interactions , Susanna Hug

Molecular Dynamics Studies of Water Flow in Carbon Nanotubes , Alexander D. Marshall

Valuation of Multiple Exercise Options , T. James Marshall

Incomplete Market Models of Carbon Emissions Markets , Walid Mnif

Topics in Field Theory , Alexander Patrushev

Pricing and Trading American Put Options under Sub-Optimal Exercise Policies , William Wei Xing

Further applications of higher-order Markov chains and developments in regime-switching models , Xiaojing Xi

Theses/Dissertations from 2011 2011

Bifurcations and Stability in Models of Infectious Diseases , Bernard S. Chan

Real Options Models in Real Estate , Jin Won Choi

Models, Techniques, and Metrics for Managing Risk in Software Engineering , Andriy Miranskyy

Thermodynamics, Hydrodynamics and Critical Phenomena in Strongly Coupled Gauge Theories , Christopher Pagnutti

Molecular Dynamics Studies of Interactions of Phospholipid Membranes with Dehydroergosterol and Penetrating Peptides , Amir Mohsen Pourmousa Abkenar

Socially Responsible Investment in a Changing World , Desheng Wu

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Ph.D. Program

Introduction.

These guidelines are intended to help familiarize graduate students with the policies governing the graduate program leading to the degrees of Doctor of Philosophy (Ph.D.) in Applied Mathematics. This material supplements the graduate school requirements found on the  Graduate Student Resources  page and the  Doctoral Degree Policies  of the graduate school. Students are expected to be familiar with these procedures and regulations.

The Doctor of Philosophy program

The Doctor of Philosophy (Ph.D.) Degree in Applied Mathematics is primarily a research degree, and is not conferred as a result of course work. The granting of the degree is based on proficiency in Applied Mathematics, and the ability to carry out an independent investigation as demonstrated by the completion of a doctoral dissertation. This dissertation must exhibit original mathematical contributions that are relevant to a significant area of application.

Course requirements for the Ph.D. program

• AMATH 561, 562, 563
• AMATH 567, 568, 569
• AMATH 584, 585, 586
• AMATH 600: two, 2-credit readings, each with a different faculty member, to be completed prior to the start of the student's second year.
• Students must take a minimum of 15 numerically graded courses. At most two of these can be at the 400 level or be cross listed with courses at the 400 level. Graduate level courses previously taken at UW (e.g., during a Master's program) count toward this requirement. Graduate level courses taken outside of UW may count toward the requirement for 15 numerically graded courses with the approval of the Graduate Program Coordinator. The entire course of study of a student and all exceptions to this list must be approved by the Graduate Program Coordinator and the student’s advisor or faculty mentors.

For students who entered the doctoral program autumn 2017 or autumn 2018, please see these degree requirements. For students who entered the doctoral program prior to autumn 2017, please see these degree requirements.  

Faculty mentoring

Upon arrival, incoming students will be assigned two faculty mentors. Until a student settles on an advisor, the faculty mentors aid the student in selecting courses, and they each guide the student through a 2-credit independent reading course on material related to the student’s research interest. The faculty mentors are not necessarily faculty in the Department of Applied Mathematics.

Faculty advisor

By the end of a student’s first summer quarter, an advisor must be determined.  T he advisor provides guidance in designing a course of study appropriate for the student’s research interests, and in formulating a dissertation topic.

A full Supervisory Committee should be formed four months prior to the student’s General Exam. The full Supervisory Committee should have a minimum of three regular members plus the Graduate School Representative , and will consist of at least two faculty members from Applied Mathematics, one of whom is to be the Chair of the Committee. If the proposed dissertation advisor is a member of the Applied Mathematics faculty, then the advisor will be the Chair. The dissertation advisor may be from another department,  or may have an  affiliate  (assistant, associate, full) professor appointment with the Applied Mathematics department  and is then also a member of the Supervisory Committee.

The Dissertation Reading Committee , formed after the General Exam,  is a subset of  at least   three members from the Supervisory Committee   who are appointed to read and approve the dissertation.  Two members of the Dissertation Reading Committee must be from the Applied Mathematics faculty. At least one of the committee members must be a member of the core  Applied Mathematics faculty. It is required that this member is present for both the general and final examination, and is included on the reading committee.

While the principal source of guidance during the process of choosing specialization areas and a research topic is the thesis advisor, it is strongly advised that the student maintain contact with all members of the Supervisory Committee. It is suggested that the student meet with the Supervisory Committee at least once a year to discuss their progress until the doctoral thesis is completed.

Examination requirements for the Ph.D. program

Students in the Ph.D. program must pass the following exams:

• The  qualifying exam
• The  general exam
• The  final exam  (defense)

Satisfactory performance and progress

At all times, students need to make satisfactory progress towards finishing their degree. Satisfactory progress in course work is based on grades. Students are expected to maintain a grade point average of 3.4/4.0 or better. Satisfactory progress on the examination requirements consists of passing the different exams in a timely manner. Departmental funding is contingent on satisfactory progress.   The Graduate School rules regarding satisfactory progress are detailed in Policy 3.7: Academic Performance and Progress .   The Department of Applied Mathematics follows these recommended guidelines of the Graduate School including an initial warning, followed by a maximum of three quarters of probation and one quarter of final probation, then ultimately being dropped from the program.    We encourage all students to explore and utilize the many available  resources  across campus.

Expected academic workload

A first-year, full-time student is expected to register for a full course load, at least three numerically graded courses, typically totaling 12-18 credits. All other students are expected to consult with their advisor and register for at least 10-18 credits per quarter.  Students who do not intend to register for a quarter must seek approved  academic leave  in order to maintain a student status.   Students who do not maintain active student status through course registration or an approved leave request need to request reinstatement to rejoin the program. Reinstatement is at the discretion of the department. Students approved for reinstatement are required to follow degree requirements active at time of reinstatement. 

Annual Progress Report

Students are required to submit an Annual Progress Report to the Graduate Program Coordinator by the second week of Spring Quarter each year. The annual progress report should contain the professional information related to the student’s progress since the previous annual report. It should contain information on courses taken, presentations given, publications, thesis progress, etc., and should be discussed with the student's advisor prior to submission. Students should regard the Annual Progress Report as an opportunity to self-evaluate their progress towards completing the PhD. The content of the Annual Progress Report is used to ensure the student is making satisfactory progress towards the PhD degree.

Financial assistance

Financial support for Doctoral studies is limited to five years after admission to the Ph.D. program in the Department of Applied Mathematics. Support for an additional period may be granted upon approval of a petition, endorsed by the student’s thesis supervisor, to the Graduate Program Coordinator.

Master of Science program

Students in the Ph.D. program obtain an M.Sc. Degree while working towards their Ph.D. degree by satisfying the  requirements for the M.Sc. degree.  

Additional Ph.D. Degree Options and Certificates

Students in the Applied Mathematics Ph.D. program are eligible to pursue additional degree options or certificates, such as the  Advanced Data Science Option  or the  Computational Molecular Biology Certificate .  Students must be admitted and matriculated to the PhD program prior to applying for these options. Option or certificate requirements are in addition to the Applied Mathematics degree requirements. Successful completion of the requirements for the option or the certificate leads to official recognition of this fact on the UW transcript.

Career resources, as well as a look at student pathways after graduation, may be found   here.

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Ph.D. Program in Mathematics

Degree requirements.

A candidate for the Ph.D. degree in mathematics must fulfill a number of different departmental requirements .

NYU Shanghai Ph.D. Track

The Ph.D. program also offers students the opportunity to pursue their study and research with Mathematics faculty based at NYU Shanghai. With this opportunity, students generally complete their coursework in New York City before moving full-time to Shanghai for their dissertation research. For more information, please visit the  NYU Shanghai Ph.D. page .

Sample course schedules (Years 1 and 2) for students with a primary interest in:

Applied Math (Math Biology, Scientific Computing, Physical Applied Math, etc.)

Additional information for students interested in studying applied math is available here .

Probability

PDE/Analysis

The Written Comprehensive Examination

The examination tests the basic knowledge required for any serious mathematical study. It consists of the three following sections: Advanced Calculus, Complex Variables, and Linear Algebra. The examination is given on three consecutive days, twice a year, in early September and early January. Each section is allotted three hours and is written at the level of a good undergraduate course. Samples of previous examinations are available in the departmental office. Cooperative preparation is encouraged, as it is for all examinations. In the fall term, the Department offers a workshop, taught by an advanced Teaching Assistant, to help students prepare for the written examinations.

Entering students with a solid preparation are encouraged to consider taking the examination in their first year of full-time study. All students must take the examinations in order to be allowed to register for coursework beyond 36 points of credit; it is recommended that students attempt to take the examinations well before this deadline. Graduate Assistants are required to take the examinations during their first year of study.

For further details, consult the page on the written comprehensive exams .

The Oral Preliminary Examination

This examination is usually (but not invariably) taken after two years of full-time study. The purpose of the examination is to determine if the candidate has acquired sufficient mathematical knowledge and maturity to commence a dissertation. The phrase "mathematical knowledge" is intended to convey rather broad acquaintance with the basic facts of mathematical life, with emphasis on a good understanding of the simplest interesting examples. In particular, highly technical or abstract material is inappropriate, as is the rote reproduction of information. What the examiners look for is something a little different and less easy to quantify. It is conveyed in part by the word "maturity." This means some idea of how mathematics hangs together; the ability to think a little on one's feet; some appreciation of what is natural and important, and what is artificial. The point is that the ability to do successful research depends on more than formal learning, and it is part of the examiners' task to assess these less tangible aspects of the candidate's preparation.

The orals are comprised of a general section and a special section, each lasting one hour, and are conducted by two different panels of three faculty members. The examination takes place three times a year: fall, mid-winter and late spring. Cooperative preparation of often helpful and is encouraged. The general section consists of five topics, one of which may be chosen freely. The other four topics are determined by field of interest, but often turn out to be standard: complex variables, real variables, ordinary differential equations, and partial differential equations. Here, the level of knowledge that is expected is equivalent to that of a one or two term course of the kind Courant normally presents. A brochure containing the most common questions on the general oral examination, edited by Courant students, is available at the Department Office.

The special section is usually devoted to a single topic at a more advanced level and extent of knowledge. The precise content is negotiated with the candidate's faculty advisor. Normally, the chosen topic will have a direct bearing on the candidate's Ph.D. dissertation.

All students must take the oral examinations in order to be allowed to register for coursework beyond 60 points of credit. It is recommended that students attempt the examinations well before this deadline.

The Dissertation Defense

The oral defense is the final examination on the student's dissertation. The defense is conducted by a panel of five faculty members (including the student's advisor) and generally lasts one to two hours. The candidate presents his/her work to a mixed audience, some expert in the student's topic, some not. Often, this presentation is followed by a question-and-answer period and mutual discussion of related material and directions for future work.

Summer Internships and Employment

The Department encourages Ph.D. students at any stage of their studies, including the very early stage, to seek summer employment opportunities at various government and industry facilities. In the past few years, Courant students have taken summer internships at the National Institute of Health, Los Alamos National Laboratory, Woods Hole Oceanographic Institution, Lawrence Livermore National Laboratory and NASA, as well as Wall Street firms. Such opportunities can greatly expand students' understanding of the mathematical sciences, offer them possible areas of interest for thesis research, and enhance their career options. The Director of Graduate Studies and members of the faculty (and in particular the students' academic advisors) can assist students in finding appropriate summer employment.

Mentoring and Grievance Policy

For detailed information, consult the page on the Mentoring and Grievance Policy .

Visiting Doctoral Students

Information about spending a term at the Courant Institute's Department of Mathematics as a visiting doctoral student is available on the Visitor Programs  page.

Thesis Defenses

Julius baldauf.

Date: Thursday, March 28, 2024 | 2:10pm | Room: 2-449 | Zoom Link

Committee: Bill Minicozzi (Thesis Advisor and Examination Committee Chair), Tristan Collins, Tristan Ozuch

The Ricci Flow on Spin Manifolds

This thesis studies the Ricci flow on manifolds admitting harmonic spinors. It is shown that Perelman's Ricci flow entropy can be expressed in terms of the energy of harmonic spinors in all dimensions, and in four dimensions, in terms of the energy of Seiberg-Witten monopoles. Consequently, Ricci flow is the gradient flow of these energies. The proof relies on a weighted version of the monopole equations, introduced here. Further, a sharp parabolic Hitchin-Thorpe inequality for simply-connected, spin 4-manifolds is proven. From this, it follows that the normalized Ricci flow on any exotic K3 surface must become singular.

Date: Wednesday, April 24, 2024 | 3:00pm | Room: 2-142

Committee: Wei Zhang, Julee Kim, Zhiwei Yun

Local newforms and spherical characters for unitary groups

We first prove a smooth transfer statement analogous to Jacquet–Rallis’s fundamental lemma and use it to compute the special value of a local spherical character that appears in the Ichino–Ikeda conjecture at a test vector. Then we provide a uniform definition of newforms for representations of both even and odd dimensional unitary groups over p-adic fields. This definition is compatible with the one given by Atobe, Oi, and Yasuda in the odd dimensional case. Using the nonvanishing of the local spherical character at the test vector, we prove the existence of the representation containing newforms in every tempered Vogan L-packet. We also show the uniqueness of such representations in Vogan L-packets and give an explicit description of them using local Langlands correspondence.

Patrik Gerber

Date: Friday, April 26, 2024 | 9:30am | Room: 2-361

Committee: Philippe Rigollet (advisor), Yury Polyanskiy, Martin Wainwright

Likelihood-Free Hypothesis Testing and Applications of the Energy Distance

The first part of this thesis studies the problem of likelihood-free hypothesis testing: given three samples X,Y and Z with sample sizes n,n and m respectively, one must decide whether the distribution of Z is closer to that of X or that of Y. We fully characterize the problem's sample complexity for multiple distribution classes and with high probability. We uncover connections to two-sample, goodness of fit and robust testing, and show the existence of a trade-off of the form mn ~ k/ε^4, where k is an appropriate notion of complexity and ε is the total variation separation between the distributions of X and Y. We demonstrate that the family of "classifier accuracy" tests are not only popular in practice but also provably near-optimal, recovering and simplifying a multitude of classical and recent results. We generalize our problem to allow Z to come from a mixture of the distributions of X and Y, and propose a kernel-based test for its solution. Finally, we verify the existence of a trade-off between m and n on experimental data from particle physics.

In the second part we study applications of the energy distance to minimax statistics. We propose a density estimation routine based on minimizing the generalized energy distance, targeting smooth densities and Gaussian mixtures. We interpret our results in terms of half-plane separability over these classes, and derive analogous results for discrete distributions. As a consequence we deduce that any two discrete distributions are well-separated by a half-plane, provided their support is embedded as a packing of a high-dimensional unit ball. We also scrutinize two recent applications of the energy distance in the two-sample testing literature.

Jae Hee Lee

Date: Monday, April 1, 2024 | 3:00pm | Room: 2-361 | Zoom Link

Committee: Prof. Paul Seidel (thesis advisor), Prof. Pavel Etingof, Prof. Denis Auroux (External, Harvard)

Equivariant quantum connections in positive characteristic

Calder Morton-Ferguson

Date: Friday, April 26, 2024 | 1:30pm | Room: 2-449 | Zoom Link

Committee: Roman Bezrukavnikov (advisor), Zhiwei Yun, Ivan Loseu

Kazhdan-Laumon categories and representations

In 1988, D. Kazhdan and G. Laumon constructed the \emph{Kazhdan-Laumon category}, an abelian category $\mathcal{A}$ associated to a reductive group $G$ over a finite field, with the aim of using it to construct discrete series representations of the finite Chevalley group $G(\mathbb{F}_q)$. The well-definedness of their construction depended on their conjecture that this category has finite cohomological dimension. This was disproven by R. Bezrukavnikov and A. Polishchuk in 2001, who found a counterexample for $G = SL_3$.

Since the early 2000s, there has been little activity in the study of Kazhdan-Laumon categories, despite them being beautiful objects with many interesting properties related to the representation theory of $G$ and the geometry of the basic affine space $G/U$. In the first part of this thesis, we conduct an in-depth study of Kazhdan-Laumon categories from a modern perspective. We first define and study an analogue of the Bernstein-Gelfand-Gelfand Category $\mathcal{O}$ for Kazhdan-Laumon categories and study its combinatorics, establishing connections to Braverman-Kazhdan's Schwartz space on the basic affine space and the semi-infinite flag variety. We then study the braid group action on $D^b(G/U)$ (the main ingredient in Kazhdan and Laumon's construction) and show that it categorifies the \emph{algebra of braids and ties}, an algebra previously studied in knot theory; we then use this to provide conceptual and geometric proofs of new results concerning this algebra.

After Bezrukavnikov and Polishchuk's counterexample to Kazhdan and Laumon's original conjecture, Polishchuk made an alternative conjecture: though this counterexample shows that the Grothendieck group $K_0(\mathcal{A})$ is not spanned by objects of finite projective dimension, he noted that a graded version of $K_0(\mathcal{A})$ can be thought of as a module over Laurent polynomials and conjectured that a certain localization of this module is generated by objects of finite projective dimension. He suggested that this conjecture could lead toward a proof that Kazhdan and Laumon's construction is well-defined, and he proved this conjecture in Types $A_1, A_2, A_3$, and $B_2$. In the final chapter of this thesis, we prove Polishchuk's conjecture for all types, and prove that Kazhdan and Laumon's construction is indeed well-defined, giving a new geometric construction of discrete series representations of $G(\mathbb{F}_q)$.

Date: Wednesday, April 3, 2024 | 3:30pm | Room: 2-449

Committee: Prof. Yufei Zhao (advisor and chair), Prof. Dor Minzer, and Prof. Philippe Rigollet

Random and exact structures in combinatorics

We aim to show various developments related to notions of randomness and structure in combinatorics and probability. One central notion, that of the pseudorandomness-structure dichotomy, has played a key role in additive combinatorics and extremal graph theory. In a broader view, randomness (and the pseudorandomness notions which resemble it along various axes) can be viewed as a type of structure in and of itself which has certain typical and global properties that may be exploited to exhibit or constrain combinatorial and probabilistic behavior.

These broader ideas often come in concert to allow the construction or extraction of exact behavior. We look at three particular directions: the singularity of discrete random matrices, thresholds for Steiner triple systems, and improved bounds for Szemerédi's theorem. These concern central questions of the areas of random matrices, combinatorial designs, and additive combinatorics.

George Stepaniants

Date: Thursday, April 25, 2024 | 2:30pm | Room: 4-149 | Zoom Link

Committee: Philippe Rigollet, Jörn Dunkel, Sasha Rakhlin

Inference from Limited Observations in Statistical, Dynamical, and Functional Problems

Observational data in physics and the life sciences comes in many varieties. Broadly, we can divide datasets into cross-sectional data which record a set of observations at a given time, dynamical data which follow how observations change in time, and functional data which observe data points over a space (and possibly time) domain. In each setting, prior knowledge of statistical, dynamical systems, and physical theory allow us to constrain the inferences and predictions we make from observational data. This domain knowledge becomes of paramount importance when the data we observe is limited: due to missing labels, small sample sizes, unobserved variables, and noise corruption.

This thesis explores several problems in physics and the life sciences, where the interplay of domain knowledge with statistical theory and machine learning allows us to make inferences from such limited data. We begin in Part I by studying the problem of feature matching or dataset alignment which arises frequently when combining untargeted (unlabeled) biological datasets with low sample sizes. Leveraging the fast numerical methods of optimal transport, we develop an algorithm that gives a state-of-the-art solution to this alignment problem with optimal statistical guarantees. In Part II we study the problem of interpolating the dynamics of points clouds (e.g., cells, particles) given only a few sparse snapshot recordings. We show how tools from spline interpolation coupled with optimal transport give efficient algorithms returning smooth dynamically plausible interpolations. Part III of our thesis studies how dynamical equations of motion can be learned from time series recordings of dynamical systems when only partial observations of these systems are captured in time. Here we develop fast routines for gradient optimization and novel tools for model comparison to learn such physically interpretable models from incomplete time series data. Finally, in Part IV we address the problem of surrogate modeling, translating expensive solvers of partial differential equations for physics simulations into fast and easily-trainable machine learning algorithms. For linear PDEs, our prior knowledge of PDE theory and the statistical theory of kernel methods allows us to learn the Green's functions of various linear PDEs, offering more efficient ways to simulate physical systems.

Date: Wednesday, April 3, 2024 | 2:00pm | Room: 2-255

Committee: Scott Sheffield (advisor), Alexei Borodin, Nike Sun

Conformal welding of random surfaces from Liouville theory

Liouville quantum gravity (LQG) is a natural model describing random surfaces, which arises as the scaling limit for random planar maps. Liouville conformal field theory (LCFT) is the underlying 2D CFT that governs LQG. Schramm-Loewner evolution (SLE) is a random planar curve, which describes the scaling limits of interfaces in many statistical physics models. As discovered by Sheffield (2010), one of the deepest results in random geometry is that SLE curves arises as the interfaces under conformal welding of LQG surfaces.

In this thesis, we present some new results on conformal welding of LQG surfaces as well as their applications towards the theory of SLE. We first define a three-parameter family of random surfaces in LQG which can be viewed as the quantum version of triangles. Then we prove the conformal welding result of a quantum triangle and a two-pointed quantum disk, and deduce integrability results for chordal SLE with three force points.

The second main result is regarding the conformal welding of a multiple number of LQG surfaces, where under several scenarios, we prove that the output surfaces can be described in terms of LCFT, and the random moduli of the surface is encoded in terms of the partition functions for the SLE curves.

The third part is about the conformal welding of the quantum disks with forested boundary, where we prove that this conformal welding gives a two-pointed quantum disk with an independent SLE$_\kappa$ for $\kappa\in(4,8)$. We further extend to the conformal welding of a multiple number of forested quantum disks, where as an application, for $\kappa\in(4,8)$, we prove the existence of the multiple SLE partition functions, which are smooth functions satisfying a system of PDEs and conformal covariance. This was open for $\kappa \in (6,8)$ and $N\ge 3$ prior to our work.

The conformal loop ensemble (CLE) is a random collection of planar loops which locally look like SLE. For $\kappa \in (4,8)$, the loops are non-simple and may touch each other and the boundary. As a second application, we derive the probability that the loop surrounding a given point in the non-simple conformal loop ensemble touches the domain boundary.

Mathematical Modeling Doctor of Philosophy (Ph.D.) Degree

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The mathematical modeling Ph.D. enables you to develop mathematical models to investigate, analyze, predict, and solve the behaviors of a range of fields from medicine, engineering, and business to physics and science.

STEM-OPT Visa Eligible

Overview for Mathematical Modeling Ph.D.

Mathematical modeling is the process of developing mathematical descriptions, or models, of real-world systems. These models can be linear or nonlinear, discrete or continuous, deterministic or stochastic, and static or dynamic, and they enable investigating, analyzing, and predicting the behavior of systems in a wide variety of fields. Through extensive study and research, graduates of the mathematical modeling Ph.D. will have the expertise not only to use the tools of mathematical modeling in various application settings, but also to contribute in creative and innovative ways to the solution of complex interdisciplinary problems and to communicate effectively with domain experts in various fields.

Plan of Study

The degree requires at least 60 credit hours of course work and research. The curriculum consists of three required core courses, three required concentration foundation courses, a course in scientific computing and high-performance computing (HPC), three elective courses focused on the student’s chosen research concentration, and a doctoral dissertation. Elective courses are available from within the School of Mathematics and Statistics as well as from other graduate programs at RIT, which can provide application-specific courses of interest for particular research projects. A minimum of 30 credits hours of course work is required. In addition to courses, at least 30 credit hours of research, including the Graduate Research Seminar, and an interdisciplinary internship outside of RIT are required.

Students develop a plan of study in consultation with an application domain advisory committee. This committee consists of the program director, one of the concentration leads, and an expert from an application domain related to the student’s research interest. The committee ensures that all students have a roadmap for completing their degree based on their background and research interests. The plan of study may be revised as needed. Learn more about our mathematical modeling doctoral students and view a selection of mathematical modeling seminars hosted by the department.

Qualifying Examinations

All students must pass two qualifying examinations to determine whether they have sufficient knowledge of modeling principles, mathematics, and computational methods to conduct doctoral research. Students must pass the examinations in order to continue in the Ph.D. program.

The first exam is based on the Numerical Analysis I (MATH-602) and Mathematical Modeling I, II (MATH-622, 722). The second exam is based on the student's concentration foundation courses and additional material deemed appropriate by the committee and consists of a short research project.

Dissertation Research Advisor and Committee

A dissertation research advisor is selected from the program faculty based on the student's research interests, faculty research interest, and discussions with the program director. Once a student has chosen a dissertation advisor, the student, in consultation with the advisor, forms a dissertation committee consisting of at least four members, including the dissertation advisor. The committee includes the dissertation advisor, one other member of the mathematical modeling program faculty, and an external chair appointed by the dean of graduate education. The external chair must be a tenured member of the RIT faculty who is not a current member of the mathematical modeling program faculty. The fourth committee member must not be a member of the RIT faculty and may be a professional affiliated with industry or with another institution; the program director must approve this committee member.

The main duties of the dissertation committee are administering both the candidacy exam and final dissertation defense. In addition, the dissertation committee assists students in planning and conducting their dissertation research and provides guidance during the writing of the dissertation.

Admission to Candidacy

When a student has developed an in-depth understanding of their dissertation research topic, the dissertation committee administers an examination to determine if the student will be admitted to candidacy for the doctoral degree. The purpose of the examination is to ensure that the student has the necessary background knowledge, command of the problem, and intellectual maturity to carry out the specific doctoral-level research project. The examination may include a review of the literature, preliminary research results, and proposed research directions for the completed dissertation. Requirements for the candidacy exam include both a written dissertation proposal and the presentation of an oral defense of the proposal. This examination must be completed at least one year before the student can graduate.

Dissertation Defense and Final Examination

The dissertation defense and final examination may be scheduled after the dissertation has been written and distributed to the dissertation committee and the committee has consented to administer the final examination. Copies of the dissertation must be distributed to all members of the dissertation committee at least four weeks prior to the final examination. The dissertation defense consists of an oral presentation of the dissertation research, which is open to the public. This public presentation must be scheduled and publicly advertised at least four weeks prior to the examination. After the presentation, questions will be fielded from the attending audience and the final examination, which consists of a private questioning of the candidate by the dissertation committee, will ensue. After the questioning, the dissertation committee immediately deliberates and thereafter notifies the candidate and the mathematical modeling graduate director of the result of the examination.

All students in the program must spend at least two consecutive semesters (summer excluded) as resident full-time students to be eligible to receive the doctoral degree.

Maximum Time Limitations

University policy requires that doctoral programs be completed within seven years of the date of the student passing the qualifying exam. All candidates must maintain continuous enrollment during the research phase of the program. Such enrollment is not limited by the maximum number of research credits that apply to the degree.

National Labs Career Fair

Hosted by RIT’s Office of Career Services and Cooperative Education, the National Labs Career Fair is an annual event that brings representatives to campus from the United States’ federally funded research and development labs. These national labs focus on scientific discovery, clean energy development, national security, technology advancements, and more. Students are invited to attend the career fair to network with lab professionals, learn about opportunities, and interview for co-ops, internships, research positions, and full-time employment.

Students are also interested in: Applied and Computational Mathematics MS

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Learn what you need to apply

The College of Science consistently receives research grant awards from organizations that include the National Science Foundation , National Institutes of Health , and NASA , which provide you with unique opportunities to conduct cutting-edge research with our faculty members.

Faculty in the School of Mathematics and Statistics conducts research on a broad variety of topics including:

• applied inverse problems and optimization
• applied statistics and data analytics
• biomedical mathematics
• discrete mathematics
• dynamical systems and fluid dynamics
• geometry, relativity, and gravitation
• mathematics of earth and environment systems
• multi-messenger and multi-wavelength astrophysics

Learn more by exploring the school’s mathematics research areas .

Michael Cromer

Basca Jadamba

Moumita Das

Carlos Lousto

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Curriculum for 2023-2024 for Mathematical Modeling Ph.D.

Current Students: See Curriculum Requirements

Mathematical Modeling, Ph.D. degree, typical course sequence

Concentrations, applied inverse problems, biomedical mathematics, discrete mathematics, dynamical systems and fluid dynamics, geometry, relativity and gravitation, admissions and financial aid.

This program is available on-campus only.

Full-time study is 9+ semester credit hours. International students requiring a visa to study at the RIT Rochester campus must study full‑time.

Application Details

To be considered for admission to the Mathematical Modeling Ph.D. program, candidates must fulfill the following requirements:

• Complete an online graduate application .
• Submit copies of official transcript(s) (in English) of all previously completed undergraduate and graduate course work, including any transfer credit earned.
• Hold a baccalaureate degree (or US equivalent) from an accredited university or college.
• A recommended minimum cumulative GPA of 3.0 (or equivalent).
• Submit a current resume or curriculum vitae.
• Submit a statement of purpose for research which will allow the Admissions Committee to learn the most about you as a prospective researcher.
• Submit two letters of recommendation .
• Entrance exam requirements: None
• Writing samples are optional.
• Submit English language test scores (TOEFL, IELTS, PTE Academic), if required. Details are below.

English Language Test Scores

International applicants whose native language is not English must submit one of the following official English language test scores. Some international applicants may be considered for an English test requirement waiver .

International students below the minimum requirement may be considered for conditional admission. Each program requires balanced sub-scores when determining an applicant’s need for additional English language courses.

How to Apply   Start or Manage Your Application

Cost and Financial Aid

An RIT graduate degree is an investment with lifelong returns. Ph.D. students typically receive full tuition and an RIT Graduate Assistantship that will consist of a research assistantship (stipend) or a teaching assistantship (salary).

Additional Information

Foundation courses.

Mathematical modeling encompasses a wide variety of scientific disciplines, and candidates from diverse backgrounds are encouraged to apply. If applicants have not taken the expected foundational course work, the program director may require the student to successfully complete foundational courses prior to matriculating into the Ph.D. program. Typical foundation course work includes calculus through multivariable and vector calculus, differential equations, linear algebra, probability and statistics, one course in computer programming, and at least one course in real analysis, numerical analysis, or upper-level discrete mathematics.

DisCoMath Seminar: Well, well, well..."well"ness in graph theory, especially well-forcedness

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Mathematics advising guide.

Mathematics encompasses the study of patterns in nature, the development of tools to understand those patterns, and the generalization of those ideas in an abstract setting. A mathematics degree teaches a student to think, to reason, to experiment, and to learn and grow. Mathematics inspires not only science, technology, and their applications, but all aspects of society.

Students learn how to ask good questions, make connections, work with others, explain their thoughts, and find evidence to back up their reasoning.

Professors provide a solid foundation in the subject, spark interest in mathematical topics, use technology in learning, use innovative pedagogical approaches, and provide students with resources to pursue research experiences. 

Graduates leave Gustavus thoroughly prepared for graduate study, secondary school teaching, a life of service, or employment in government or industry. 

Mathematics Major

This section lists the requirements of the Mathematics major. A grade of C- or higher is necessary in all 11 courses used to satisfy the requirements of the major. Additionally, you can use the Mathematics Major Form , This form will help you plan out your mathematics courses and requirements. To declare a major, use this form :

• MCS-122 Calculus II 
• MCS-150  Discrete Mathematics
• MCS-221 Linear Algebra
• MCS-222 Multivariate Calculus
• MCS-213 Intro to Algebra 
• MCS-220  Intro to Analysis
• MCS-142 Introduction to Statistics
• MCS-177 Introduction to Computer Science I
• MCS-313 and MCS-314  Algebra
• MCS-331 and MCS-332  Analysis
• MCS-353 and MCS-357 Dynamical Systems

Electives:  Two additional mathematics courses at the 200 or 300 level. Students should consult with their advisors to discuss which courses best fit their needs. 

Mathematics Minor

A grade of C- or higher is necessary in all courses used to satisfy the requirements of the minor, which are as follows:

• MCS-122 Calculus II or MCS-132 Honors Calculus II
• MCS-150 Discrete Mathematics
• MCS-213 Intro to Algebra
• MCS-220 Intro to Analysis
• MCS-303 Geometry
• MCS-313  Modern Algebra I
• MCS-314  Modern Algebra II
• MCS-321   Elementary Theory of Complex Variables
• MCS-331 Real Analysis
• MCS-344 Topics in Advanced Math
• MCS-353  Continuous Dynamical Systems
• MCS-355 Numerical Analysis
• MCS-357 Discrete Dynamical Systems
• MCS-358 Math Model Building

Sample Student Plans

All students should ideally lay out a schedule of their own showing what courses they plan to take, and when they plan to take them. The schedule may not accurately forecast the future, but it is helpful nonetheless. A printable sample plan can be found on the  Mathematics Major Form  

Student Starter Plan

The sample plans below are useful starting points in developing an individual plan. You can select the sample plan that comes closest to fitting your own situation and then tailor it as necessary. Note that certain courses are offered on an every-other year basis; for example MCS-314 (Modern Algebra II) is offered in the spring of odd years Courses offered every other year include MCS-313, MCS-314, MCS-331, MCS-344, MCS-355, MCS-357, MCS-358, MCS-385, and MCS-394. These courses are listed with an astrix in the sample plans below. Please keep these course alterations in mind when planning out your major. Check the college catalog for when the courses you are interested in will be scheduled.

Students interested in algebra should take *MCS-313 and *MCS-314 for their  Immersive Experience and MCS-213 as their Proofs  course along with two appropriate electives. 

Students interested in analysis should take *MCS-331 and *MCS-332 for their  Immersive Experience and MCS-220 as their Proofs course and two appropriate electives.  

Applied Mathematics

Students interested in applied mathematics should take *MCS-353 and *MCS-357 for their  Immersive Experience , *MCS-358 as an elective, and an additional  Elective .  

Thinking About Graduate School in Traditional Mathematics 

Students considering graduate school in mathematics should take *MCS-313,* MCS-314, *MCS-321, and *MCS-331 for their  Immersive Experience and Electives as well as an appropriate Collaborative Experience . 

Thinking About Graduate School in Applied Mathematics

Students considering graduate school in applied mathematics should take *MCS-353 and *MCS-357 for their  Immersive   Experience , *MCS-358 as an Elective , and either *MCS-313 or  *MCS 331 as their second Elective .

Studying Mathematics Abroad

 Students traveling abroad should speak with their advisors to discuss courses and study abroad programs. Study abroad programs are listed on the MCS Resources page. 

Honors Program

In order to graduate with Honors in Mathematics, a student must complete an application for admission to the Honors program, available through the department chair, showing that the student satisfies the admission requirements, and then the requirements of the program.

The requirements for admission to the Honors program are as follows:

• Completion of steps 1 - 3 of the Mathematics Major with a grade point average greater than 3.14. 
• Approval by the Mathematics Honors committee of an Honors thesis proposal. (Guidelines are available in the Mathematics Advising Guide.)

The requirements of the honors program after admission are as follows:

• Attainment of a GPA greater than 3.14 in courses used to satisfy the requirements of the major. If a student has taken more courses than the major requires, that student may designate for consideration any collection of courses satisfying the requirements of the major.
• Approval by the Mathematics Honors Committee of an Honors thesis. The thesis should conform in general outline to the previously approved proposal (or an approved substitute proposal), should include approximately 160 hours of work, and should result in an approved written document. Students completing this requirement will receive credit for the course MCS-350, whether or not they graduate with Honors. (See the Mathematics Advising Guide for the thesis guidelines.)
• Oral presentation of the thesis in a public forum, such as the departmental seminar. This presentation will not be evaluated as a criterion for thesis approval, but is required.

Honors Thesis Guidelines

Mathematics honors thesis proposals should be written in consultation with the faculty member who will be supervising the work. The proposal and thesis must each be approved by the Mathematics Honors Committee. These guidelines are intended to help students, faculty supervisors, and the committee judge what merits approval.

The thesis should include creative work, and should not reproduce well-known results; however, it need not be entirely novel. It is unreasonable for an undergraduate with limited time and library resources to do a thorough search of the literature, such as would be necessary to ensure complete novelty. Moreover, it would be rare for any topic to be simultaneously novel, easy enough to think of, and easy enough to do.

The thesis should include use of primary-source reference material. As stated above, an exhaustive search of the research literature is impractical. None the less, the resources of inter-library loan, the faculty supervisor's private holdings, etc. must be tapped if the thesis work is to go beyond standard classroom/textbook work.

The written thesis should sufficiently explain the project undertaken and results achieved that someone generally knowledgeable about mathematics, but not about the specific topic, can understand it. The quality of writing and care in citing sources should be adequate for external distribution without embarrassment.

The thesis must contain a substantial mathematical component, though it can include other disciplines as well. If a single thesis simultaneously satisfies the requirements of this program and some other discipline's honors program, it can be used for both (subject to the other program's restrictions). However, course credit will not be awarded for work which is otherwise receiving course credit.

The Mathematics Honors Committee will maintain a file of past proposals and theses, which may be valuable in further clarifying what constitutes a suitable thesis. In order to provide some guidance of the sort before the program gets under way, here are some possible topics that appear on the surface to be suitable:

• A student could study the history surrounding Fermat's last theorem, and discuss and explain past failed attempts and the recent successful attempt to prove this theorem.
• A student could research the topic of knot theory and discuss the implications of this theory to the study of DNA and other biological materials.
• A student could study the use of wavelets in signal analysis, and the general usefulness of orthonormal families of functions in signal analysis. 

Senior Oral Exam

 As described above, every math major must either take an additional upper level math course from a specified list or alternatively submit to oral examination during the Spring semester of their final year.

A student who chooses to take the oral examination selects, in consultation with a faculty member, a topic to research. They then present a 20-minute talk on that topic to an examining committee of three faculty members. At the conclusion of the talk, the faculty question the student about the talk, and also about fundamental topics from the student's full four years' of courses. The goal is not to require recollection of details, but rather to make sure that the student is leaving with the essentials intact.

The examination committee confers privately immediately after the examination and delivers the results to the student at the conclusion of their deliberations. The outcome is either that the student is deemed to have satisfied the requirement or alternatively that the student is requested to retry the examination at a later date. In the latter case, specific suggestions for areas of improvement are provided by the faculty committee.

More information about the oral examination procedures and schedule are provided routinely to those fourth-year majors who will likely choose to take the examination.

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PhD Dissertations

In 1909 the department awarded its first PhD to  Grace M. Bareis , whose dissertation was directed by Professor Harry W. Kuhn. The department began awarding PhD degrees on a regular basis around 1930, when a formal doctoral program was established as a result of the appointment of Tibor Radó as a professor at our department. To date, the department has awarded over 800 PhD degrees. An average of approximately 15 dissertations per year have been added in recent times. Find below a list of PhD theses completed in our program since 1952. (Additionally, search Ohio State at  Math Genealogy , which also includes some theses from other OSU departments.)