MATH 3331 and BIOL 3306 or consent of instructor.

Topics in mathematical biology, epidemiology, population models, models of genetics and evolution, network theory, pattern formation, and neuroscience. Students may not receive credit for both MATH 4309 and BIOL 4309.

Introduction to basic concepts, results, methods, and applications of graph theory.

Additional Description: How does information propagate between friends and acquaintances on social media? How do diseases spread, and when do epidemics start? How should we design power grids to avoid failures, and systems of roads to optimize traffic flow? These questions can be addressed using network theory . Students in the course will develop a sound knowledge of the basics of graph theory, as well as some of the computational tools used to address the questions above. Course topics include basic structural features of networks, generative models of networks, centrality, random graphs, clustering, and dynamical processes on graphs.

Further development and applications of concepts from MATH 4331. Topics may vary depending on the instructor's choice. Possibilities include: Fourier series, point-set topology, measure theory, function spaces, and/or dynamical systems.

MATH 3331 and COSC 1410 or equivalent or consent of instructor.

Instructor's Prerequisite Notes: 

1. MATH 2331, In depth knowledge of Math 3331 (Differential Equations) or Math 3321 (Engineering Mathematics)

2. Ability to do computer assignments in FORTRAN, C, Matlab, Pascal, Mathematica or Maple.

Numerical Analysis (9th edition), by R.L. Burden and J.D. Faires, Brooks-Cole Publishers, ISBN:9780538733519

Linear systems of equations, matrices, determinants, vector spaces and linear transformations, eigenvalues and eigenvectors.

Additional Notes: This is a proof-based course. It will cover Chapters 1-4 and the first two sections of Chapter 5. Topics include systems of linear equations, vector spaces and linear transformations (developed axiomatically), matrices, determinants, eigenvectors and diagonalization.

Linear systems of equations, matrices, determinants, vector spaces and linear transformations, eigenvalues and eigenvectors.

Additional Description: In addition to the material listed, I will cover diagonalization, inner products and norms, orthogonality, the Gram-Schmidt process and orthogonal projection.

Syllabus

Similarity of matrices, diagonalization, Hermitian and positive definite matrices, normal matrices, and canonical forms, with applications.

Instructor's Additional notes: This is the second semester of Advanced Linear Algebra. I plan to cover Chapters 5, 6, and 7 of textbook. These chapters cover Eigenvalues, Eigenvectors, Diagonalization, Cayley-Hamilton Theorem, Inner Product spaces, Gram-Schmidt, Normal Operators (in finite dimensions), Unitary and Orthogonal operators, the Singular Value Decomposition, Bilinear and Quadratic forms, Special Relativity (optional), Jordan Canonical form.

Abstract Algebra , A First Course by Dan Saracino. Waveland Press, Inc. ISBN 0-88133-665-3
(You can use the first edition. The second edition contains additional chapters that cannot be covered in this course.)

Groups, rings and fields; algebra of polynomials, Euclidean rings and principal ideal domains. Does not apply toward the Master of Science in Mathematics or Applied Mathematics. 

Other Notes: This course is meant for  students who wish to pursue a Master of Arts in Mathematics (MAM). Please contact me  in order to find out whether this course is suitable for you and/or your degree plan. Notice that this course cannot be used for  MATH 3330, Abstract Algebra. 

Linear and nonlinear systems of ordinary differential equations; existence, uniqueness and stability of solutions; initial value problems; higher dimensional systems; Laplace transforms. Theory and applications illustrated by computer assignments and projects. Applies toward the Master of Arts in Mathematics degree; does not apply toward the Master of Science in Mathematics or the Master of Science in Applied Mathematics degrees.

A survey of the concepts of limit, continuity, differentiation and integration for functions of one variable and functions of several variables; selected applications.

Simple and multiple linear regression, linear models, inferences from the normal error model, regression diagnostics and robust regression, computing assignments with appropriate software. Applies toward Master of Arts in Mathematics degree; does not apply toward the Master of Science in Mathematics or the Master of Science in Applied Mathematics degrees. 

Note: This course is VEE approved for the regression component only. Approval Code: 4458-11008. For more information on VEE approved courses, click here.

Steven H. Strogatz: Nonlinear Dynamics and Chaos (with Applications to Physics, Biology, Chemistry, and Engineering) Second Edition, 2014.

Print ISBN: 9780813349107

Ebook ISBN: 9780813349114

We will discuss applications of nonlinear dynamics, following the book by Strogatz. Topics that will be considered include (for more details, check the book's table of contents): an introduction to Ordinary Differential Equations (ODE's), one-dimensional ODE's and their bifurcations; two-dimensional ODE's (linear case, limit cycles and the Poincare-Bendixson Theorem, the Hopf bifurcation), chaotic systems (logistic family, Lorenz equations, Henon map). For visualization we will use tools that do not require programming, with the option to additionally run/write Matlab code.

 Graduate standing. MATH 4333 or MATH 4378 

Additional Prerequisites: students should be comfortable with basic measure theory, groups rings and fields, and point-set topology

No textbook is required.

Topics from the theory of groups, rings, fields, and modules. 

Additional Description: This is primarily a course about analysis on topological groups. The aim is to explain how many of the techniques from classical and harmonic analysis can be extended to the setting of locally compact groups (i.e. groups possessing a locally compact topology which is compatible with their algebraic structure). In the first part of the course we will review basic point set topology and introduce the concept of a topological group. The examples of p-adic numbers and the Adeles will be presented in detail, and we will also spend some time discussing SL_2(R). Next we will talk about characters on topological groups, Pontryagin duality, Haar measure, the Fourier transform, and the inversion formula. We will focus on developing details in specific groups (including those mentioned above), and applications to ergodic theory and to number theory will be discussed.

Transformations, eigenvalues and eigenvectors.

Additional Notes: This is a proof-based course. It will cover Chapters 1-4 and the first two sections of Chapter 5. Topics include systems of linear equations, vector spaces and linear transformations (developed axiomatically), matrices, determinants, eigenvectors and diagonalization.

Transformations, eigenvalues and eigenvectors. An expository paper or talk on a subject related to the course content is required

Syllabus

Graduate standing. MATH 4332 or consent of instructor.

Instructor's Prerequisite Notes: MATH 6320

Primary (Required): Real Analysis for Graduate Students, Richard F. Bass

Supplementary (Recommended): Real Analysis: Modern Techniques and Their Applications, Gerald Folland (2nd edition); ISBN: 9780471317166.

Lebesque measure and integration, differentiation of real functions, functions of bounded variation, absolute continuity, the classical Lp spaces, general measure theory, and elementary topics in functional analysis. 

Instructor's Additional Notes: Math 6321 is the second course in a two-semester sequence intended to introduce the theory and techniques of modern analysis. The core of the course covers elements of functional analysis, Radon measures, elements of harmonic analysis, the Fourier transform, distribution theory, and Sobolev spaces. Additonal topics will be drawn from potential theory, ergodic theory, and the calculus of variations.

Graduate standing. MATH 4331

No required text.

Some topics are covered in:
-R.E. Showalter, Hilbert Space methods in Partial Differential equations. Dover.

-E. Zeidler, Nonlinear Functional Analysis and its Applications, Vol IIA, Springer.

Existence and uniqueness theory in partial differential equations; generalized solutions and convergence of approximate solutions to partial differential systems.

Additional Description: This course will develop the basic tools needed to prove existence uniqueness results for weak solutions of linear evolution problems. The results will be obtained using Galerkin and spectral methods in a Hilbert space setting. This will be used to study linear initial boundary value problems for parabolic equations (heat and diffusion equations). Also some properties of the solutions including regularity results and maximum principles.

If time permits, some results on linear hyperbolic equations will also be described.

Design and Analysis of Experiments, 8th Ed., Douglas C. Montgomery, ISBN: 9781118146927

- D.P. Bertsekas; Dynamic Programming and Optimal Con- trol, Vol. I, 4th Edition. Athena Scientific, 2017, ISBN-10: 1-886529-43-4

- J.R. Birge and F.V. Louveaux; Introduction to Stochastic Programming. Springer, New York, 1997, ISBN: 0-387-98217-

Constrained and unconstrained finite dimensional nonlinear programming, optimization and Euler-Lagrange equations, duality, and numerical methods. Optimization in Hilbert spaces and variational problems. Euler-Lagrange equations and theory of the second variation. Application to integral and differential equations.

Additional Description: This course consists of two parts. The first part is concer- ned with an introduction to Stochastic Linear Programming (SLP) and Dynamic Programming (DP). As far as DP is concerned, the course focuses on the theory and the appli- cation of control problems for linear and nonlinear dynamic systems both in a deterministic and in a stochastic frame- work. Applications aim at decision problems in finance. In the second part, we deal with continuous-time systems and optimal control problems in function space with em- phasis on evolution equations.

Finite difference, finite element, collocation and spectral methods for solving linear and nonlinear elliptic, parabolic, and hyperbolic equations and systems with applications to specific problems.

Graduate standing.  MATH 3334, MATH 3338 and MATH 4378.

Recommended Text: John A. Rice : Mathematical Statistics and Data Analysis, 3^rd editionBrooks / Cole, 2007. ISBN-13: 978-0-534-39942-9. 

Reference Texts: 

-P. MuCullagh and J.A. Nelder: Generealized Linear Models, 2^nd ed. 1999 Chapman Hall/CRC. ISBN: 978-0412317606

-Raymond H. Myers, Douglas C. Montgomery, G. Geoffrey Vining, Timothy J. Robinson, Generalized Linear Models: with Applications in Engineering and the Sciences, 2^nd ed. Wiley, 2010. ISBN: 978-0-470-45463-3. 

A survey of probability theory, probability models, and statistical inference. Includes basic probability theory, stochastic processes, parametric and nonparametric methods of statistics. 

Instructor's Description: This course is designed for graduate students who have been exposed to basic probability and statistics and would like to learn more advanced statistical theory and techniques in modelling data of various types, including continuous, binary, counts and others. The selected topics will include basic probability distributions, likelihood function and parameter estimation, hypothesis testing, regression models for continuous and categorical response variables, variable selection methods, model selection, large sample theory, shrinkage models, ANOVA and some recent advances.

Stochastic calculus, Brownian motion, change of measures, Martingale representation theorem, pricing financial derivatives whose underlying assets are equities, foreign exchanges, and fixed income securities, single-factor and multi-factor HJM models, and models involving jump diffusion and mean reversion.

Additional Description: The course is an introduction to continuous-time models in finance. We first cover tools for pricing contingency claims. They include stochastic calculus, Brownian motion, change of measures, and martingale representation theorem. We then apply these ideas in pricing financial derivatives whose underlying assets are equities, foreign exchanges, and fixed income securities. In addition, we will study models involving jump diffusion and mean reversion and the use of levy processes in finance.

Graduate standing. 

Additional Prerequisite: Basic probability and statistics understanding, familiarity with R programming is encouraged.

May be repeated with approval of chair.

Additional Description: Course will deal with a select variety of statistical methodologies used in medical research. Just to name a few - survival analysis, longitudinal data modeling, logistic regression, sample size calculation, analysis of DNA microarray data. It won’t be overly heavy on medical terminology, predominantly focusing on description of main ideas and applications for most ubiquitous statistical techniques and models. Course may come in handy for both the math/statistics/engineering students in order to acquaint themselves with medical applications, and for biology/chemistry students to better understand the statistical approaches used in medical research. The main software used throughout the course will be R Statistical Computing language, for which there will be a brief introduction during the first couple of lectures. All-in-all, at the end of the course, a successful student should:

• have a conceptual grasp of most popular statistical techniques used in medical research problems

• be able to use R statistical programming as a tool for conducting research and data analysis

• feel relatively comfortable when given a data file and asked to carry out particular analysis (be it comparison of medical treatments or performing logistic regression, among others

Software: Make sure to download R and RStudio (which can’t be installed without R) before the course starts. Use the link https://www.rstudio.com/products/rstudio/download/ to download it from the mirror appropriate for your platform. Let me know via email in case you encounter diculties.

Syllabus (PDF)

Graduate standing.

Additional Prerequisite: Two years of Calculus, Math 6308 Advanced Linear Algebra I,  Math 5386 Regression and Linear Models, or equivalent. 

May be repeated with approval of chair. A survey of probability distributions, multi-variate normal distributions, principal component analysis, classification, and clustering.

Additional Description: This course is designed for graduate students who have been exposed to basic probability and statistics and would like to learn more advanced statistical methods and techniques in jointly modelling data of multiple variables. The selected topics will include basic probability distributions, multi-variate normal distributions, principal component analysis, classification, and clustering.

May be repeated with approval of chair.

Additional Description: The course is about time analysis with special emphases on financial and energy data. The course covers ARIMA models, ARCH/GARCH models, nonlinear models, high frequency data analysis, parameter estimation for diffusion and related processes, multivariate time series, extreme value analysis, Copulas, Levy processes, and an introduction of Markov chain Monte Carlo Methods. We will use R for computing.

Differentiable Manifolds, tangent space, tangent bundle, vector bundle, Riemannian metric, connections, curvature, completeness geodesics, Jacobi fields, spaces of constant curvature, and comparison theorems.

Additional Description: This course is an introduction to the theory of smooth manifolds, with an emphasis on their geometry. The first third of the course will cover the basic definitions and examples of smooth manifolds, smooth maps, tangent spaces, and vector fields. Later in the semester we will use Euclidean, spherical, and hyperbolic geometry to introduce the notion of a Riemannian metric; we will study parallel transport, geodesics, the exponential map, and curvature. Other topics will include Lie theory and differential forms, including exterior differentiation and Stokes theorem.

The textbook highlights connections of this theory to dynamical systems; these may be mentioned in lectures but are not the focus of this course: this is first and foremost a course in differential geometry, which is oriented towards the associated preliminary exam.