# Graduate Courses, Summer 2022-Spring 2023

## Summer 2022

### For Detailed Course Outlines, click on course numbers.

7999-1 Problem Lab in Algebra—practice for PhD Qualifying Exam in Algebra.
• Instructor:
• Prerequisite:
• Text:
7999-2 Problem Lab in Analysis—practice for PhD Qualifying Exam in Analysis.
• Instructor:
• Prerequisite:
• Text:
7999-3 Problem Lab in Topology—practice for PhD Qualifying Exam in Topology.
• Instructor:
• Prerequisite:
• Text:
8000-n Assorted Sections of MS-Thesis Research
9000-n Assorted Sections of Doctoral Dissertation Research

## Fall 2022

### For Detailed Course Outlines, click on course numbers.

Core courses and Breadth courses are listed in bold.

4997-1 Vertically Integrated Research: An introduction to p-adic analysis. Profs Achar and Sage
• 12:00-1:20 TTh
• Instructor: Profs. Achar and Sage
• Prerequisite: Math 4031 and 4200
• Text: p-adic Analysis Compared with Real by S. Katok
• Under the usual absolute value, a rational number is small if it is close to 0 on the number line. However, there are other absolute values on the rational numbers where an integer is small if it is divisible by a large power of some fixed prime number p. In the same way that the real numbers are the completion of the rational numbers under the usual absolute value, one can use the "p-adic" absolute value to obtain a new number system called the p-adic numbers. Familiar concepts from calculus, such as differentiation, integration, and power series make sense in the p-adic numbers, but behave very differently. P-adic analysis is of fundamental importance in number theory and has increasing applications to algebraic geometry and representation theory. This seminar course is an introduction to the analytic and topological aspects of the p-adic numbers as well as their applications to number theory.
4997-2 Vertically Integrated Research: Geometry and Combinatorics of Polynomials. Profs Bibby and Cohen
• 11:30-12:20 MWF
• Instructors: Prof. Bibby and Prof. Cohen
• Prerequisite:
• Text:
• Properties (unimodality, log-concavity...) of polynomials which arise naturally in combinatorial settings (e.g., the chromatic polynomial of a graph).
4997-3 Vertically Integrated Research: Prof. Shea Vela Vick and Dr. Angela Wu
7001 Communicating Mathematics I. Prof. Oxley.
• 3:00-4:50 TTh
• Instructor: Prof. Oxley.
• Prerequisite: Consent of department. This course is required for all incoming graduate students. It is a lab course meeting for an average of two hours per week throughout the semester for one-semester-hour's credit.
• This course provides practical training in the teaching of mathematics at the pre-calculus level, how to write mathematics for publication, and treats other issues relating to mathematical exposition. Communicating Mathematics I and II are designed to provide training in all aspects of communicating mathematics. Their overall goal is to teach the students how to successfully teach, write, and talk about mathematics to a wide variety of audiences. In particular, the students will receive training in teaching both pre-calculus and calculus courses. They will also receive training in issues that relate to the presentation of research results by a professional mathematician. Classes tend to be structured to maximize discussion of the relevant issues. In particular, each student presentation is analyzed and evaluated by the class.
7210 Algebra I. Prof. Achar
• 9:00-10:20 TTh
• Instructor: Prof. Achar
• Prerequisite: Math 4200 or its equivalent
• Text: Dummit and Foote, Abstract Algebra
• This is the first semester of the first year graduate algebra sequence. It will cover the basic notions of group, ring, and module theory. Topics will include symmetric and alternating groups, Cayley's theorem, group actions and the class equation, the Sylow theorems, finitely generated abelian groups, polynomial rings, Euclidean domains, principal ideal domains, unique factorization domains, finitely generated modules over PIDs and applications to linear algebra, field extensions, and finite fields.
7230 Analytic Number Theory: Prof. Kopp
• 10:30-11:50 TTh
• Instructor: Prof. Kopp
• Prerequisite:
• Text:
7240 Algebraic Geometry, 2nd course. Prof. Hoffman
• 8:30-9:20 MWF
• Instructor: Prof. Hoffman
• Prerequisite: Familiarity with basic commutative algebra, for instance the book of Atiyah-Macdonald.
• Materials: We will follow the notes of Ravi Vakil.

Other good textbooks:

1. Hartshorne, Algebraic Geometry
2. Mumford, The Red Book of Varieties and Schemes
3. Mumford-Oda, Algebraic Geometry, II
4. Görtz-Wedhorn, Algebraic Geometry, I
• Main topics:
1. Sheaf cohomology
2. Schemes
3. Morphisms (affine, proper, flat, smooth, étale, etc. )
4. Cohomology of projective schemes.
7250 Representation Theory. Prof. Sage.
• 2:30-3:20 MWF
• Instructor: Prof. Sage
• Prerequisite: Math 7211 or permission of the instructor
• Text:
• Representation theory is the study of the ways in which a given group may act on vector spaces. Intuitively, it investigates ways in which an abstract group may be interpreted concretely as a group of matrices with matrix multiplication as the group operation. Group representations are ubiquitous in modern mathematics. Indeed, representation theory has significant applications throughout algebra, topology, analysis, and applied mathematics. It also is of fundamental importance in physics, chemistry, and material science. For example, it appears in quantum mechanics, crystallography, or any physical problem in which one studies how symmetries of a system affect the solutions.

This course is designed to give an introduction to representation theory, with an emphasis on Lie algebras and algebraic groups. The class is designed to be suitable both for students planning to specialize in representation theory and for those who need it for applications. It will start with an outline of the representation theory of finite groups over the complex numbers. We will then introduce complex algebraic groups and their Lie algebras. After discussing the basic theory of nilpotent, solvable, and semisimple Lie algebras, we will describe the classification of semisimple Lie algebras. We will continue with the universal enveloping algebra of a Lie algebra and the Poincaré-Birkhoff-Witt theorem. We will then cover highest-weight representations of Lie algebras, including Verma modules and finite-dimensional irreducible representations. We will also discuss the relationship between the representations of a semisimple algebraic group and the representations of its Lie algebra.

7311 Real Analysis I. Prof. Bulut.
• 9:30-10:20 MWF
• Instructor: Prof. Bulut.
• Prerequisite: Math 4032 or 4035 or equivalent.
• Text: Real Analysis, Modern Techniques and Their Applications, by G. B. Folland and Lecture Notes by the instructor.
• This is a standard introductory course on analysis based on measure theory and integration. We start by introducing sigma algebras and measures. We will then discuss measurable functions and integration of real and complex valued functions. As an example we discuss the Lebesgue integral on the line and n-dimensional Euclidean space. We also discuss the Lebesgue integral versus the Riemann integral. Important topics here are the convergence theorems, product measures and Fubini’s theorem and the Radon-Nikodym derivative. We give a short discussion of Banach spaces and Hilbert spaces. We then introduce Lp spaces and discuss the main properties of those spaces. Further topics include functions of bounded variations Lebesgue differentiation theorems, Lp and its dual. Other topics might be included depending on the time.

We use our own lecture notes. But those are very close to the book by Folland. There are several other very good books on analysis and measure theory:

1. R. G. Bartle: The Elements of Integration and Lebesgue Measure
2. A. Friedman: Foundations of Modern Analysis.
3. P. R. Halmos: Measure Theory (Graduate Text in Mathematics)
4. F. Jones: Lebesgue Integration on Euclidean Space.
5. L. Richardson: Measure and Integral: An Introduction to Real Analysis.
6. H. L. Royden: Real Analysis.
7350 Complex Analysis. Prof. Han.
• 10:30-11:20 MWF
• Instructor: Prof. Han.
• Prerequisite: Math 7311 or its equivalent.
• Text:
• Holomorphic and meromorphic functions of one variable including Cauchy's integral formula, theory of residues, the argument principle, and Schwarz reflection principle. Multivalued functions and applications to integration. Meromorphic functions. The Fourier transform in the complex plane. Paley-Wiener theorem. Entire function including Jensen's formula and infinite products. Conformal mapping including the Riemann mapping theorem and the Schwarz-Christoffel integral. In the case of sufficient time, further topics include the Gamma and Zeta functions and an introduction to the theory of elliptic functions.
7360 Probability Theory. Prof. Sundar
• 12:00-1:20 TTh
• Instructor: Prof. Sundar
• Prerequisite: 7311 Real Analysis I or its equivalent
• Text: Probability and Stochastics by Erhan Cinlar
• This is a self-contained introduction to modern probability theory. It starts from the concept of probability measures, and introduces random variables, and independence. After studying various modes of convergence, the Kolmogorov strong law of large numbers and results random series will be established. Weak convergence of probability measures will be discussed in detail, which would lead to the central limit theorem and its applications. A main goal of the course is to develop the concept of conditional probability and its basic properties. Stochastic processes such as Brownian motion and martingales will be introduced, and their essential features, studied.
7365 Applied Stochastic Analysis. Prof. Chen
• 12:30-1:20 MWF
• Instructor: Prof. Chen
• Prerequisite: Graduate level Probability Theory (Math 7360). Math 7360 can be taken concurrently with this course by a student, provided that he/she has a good background in measure theory and real analysis.
• Text: The course will have no fixed text book. Some references will be mentioned in the class.
• Math 7365 is a course on stochastic processes. A stochastic process can be thought as a random function of time which originates in modeling temporal dynamics of many systems. Detailed modeling of such systems requires incorporation of their inherent randomness which deterministic methods fail to capture. Examples of such systems are numerous - from biological networks to financial markets. Markov processes, in particular, form one of the most important classes of stochastic processes that are ubiquitous in probabilistic modeling. They also lead to probabilistic interpretations of a large class of PDEs. For example, Brownian motion is the underlying Markov process whose probability distribution satisfies the heat equation.

This course is a continuation/concurrence of Math 7360. Together they give a comprehensive introduction to measure theoretic probability which should be ideal for those wishing to study probability, or use it as a tool in analysis, statistics, mathematical biology, economics, finance or applied mathematics. The course also serves as a gateway to Math 7366 on stochastic analysis.

The course will cover theory of martingales and Markov processes in discrete-time. Some specific topics include Doob's decomposition theorem, Doob's inequalities, Burkholder-Davis-Gundy inequality, Kolmogorov's equations, generators, stationary measures, some elementary stability theory, ergodic theory and Poisson processes. If time allowed, we will also briefly discuss about the continuous-time case, in particularly, the Brownian motion and some stochastic algorithms like Markov Chain Monte Carlo.

7380 Seminar in Functional Analysis: Distributions. Prof. Estrada
• 1:30-2:50 TTh
• Prerequisite: Graduate level analysis. No prior knowledge of distributions is needed.
• Text: Ricardo Estrada and Ram P. Kanwal, A distributional approach to Asymptotics, Birkhäuser, Boston, 2002.
• We will start with the basic ideas on distributions and then we will consider several recent developments in the theory of distributions and in the areas of generalized functions. Topics include local distributional and Cesaro analysis of generalized functions, generalizations of the Lebesgue integral, particularly the distributional integral, thick distributions and their Fourier transforms.
7382 Introduction to Applied Math. Prof. Shipman
• 10:30-11:50 TTh
• Instructor: Prof. Shipman.
• Prerequisite:
• Text: Text TBA and notes
• Overview of the modeling and analysis of the equations of mathematical physics, such as electromagnetics, fluids, elasticity, acoustics, quantum mechanics, etc. There is a balance of breadth and rigor in developing mathematical analysis tools, such as measure theory, function spaces, Fourier analysis, operator theory, and variational principles, for understanding many differential and integral equations of physics.
7384 Topics in Material Science: Prof. Lipton.
• 9:00-10:20 TTh
• Instructor: Prof. Lipton.
• Prerequisite: Any one of Math 2065, 3355, 3903, 4031, or 4038, or their equivalent.
• Text: Will distribute course notes. Will also distribute research articles (these will be explained as part of the course.)
• In this course we provide an introduction to theory behind the design of meta-materials for the control of light. Here we develop basic ideas and intuition as well as introduce the mathematics and physics of spectral theory necessary for the rational design of meta-materials. The course provides a self contained introduction as well as a guide to the current research literature useful for understanding the mathematics and physics of wave propagation inside complex heterogeneous media. The course begins with an introduction to Bloch waves in crystals and provides an introduction to local plasmon resonance phenomena inside crystals made from nano-metallic particles. We then show how to apply these techniques to construct media with exotic properties. We provide the mathematical underpinnings for characterizing the interaction between surface plasmon spectra and Mie resonances and its effect on wave propagation. This understanding is used for design of structured media supporting backward optical and infrared waves and behavior associated with an effective negative index of refraction. Potential applications include drug delivery, optical communication, and holography.
7386 Partial Differential Equations. Prof Zhu
• 11:30-12:20 MWF
• Instructor: Prof. Zhu
• Prerequisite: Math 7311 or equivalent
• Texts: Primary Text: Partial Differential Equations by Lawrence C. Evans

• Elliptic partial differential equations by Qing Han and Fanghua Lin

• This course provides an introduction to the theory of partial differential equations. Partial differential equations are used to model fundamental physical phenomena and play an essential role in analysis and applied analysis. The most fundamental prerequisite for this course is calculus. This course prepares the necessary background knowledge for any future study related to the use of partial differential equations. Topics to be covered include:
1. Introduction of Laplace's equation, the heat equation, and the wave equation
2. Introduction of Sobolev spaces on weak derivatives, traces and Sobolev embedding
3. Elliptic equations on existence, regularity and maximum principle
4. Introduction of Calculus of Variations on Euler-Lagrange equations
5. Existence of minimizers and eigenvalue of self-adjoint elliptic operators
7390-1 Seminar in Analysis: Nonlinear Optimization Theory and Algorithms. Prof. Zhang.
• 1:30-2:20 MWF
• Instructor: Prof. Zhang.
• Prerequisite: Math 4032 or equivalent
• Text: Class Notes
• This class will cover classical nonlinear optimization theory and algorithms. Tentative topics include but not limited to Line search methods, Newton and quasi-Newton methods, Conjugate gradient methods, KKT optimality conditions, Penalty methods, Sequential quadratic programming, Trust region methods, non-smooth optimization.
7390-2 Seminar in Analysis : Feedback Control. Prof. Malisoff
• 10:30-11:20 MWF
• Instructor: Prof. Malisoff
• Prerequisite: Maths 4027 (Ordinary Differential Equations) and 4032 (Advanced Calculus) or equivalent background
• Text: (1) Sontag, E.D., Mathematical Control Theory, Deterministic Finite-Dimensional Systems, Second Edition, Texts in Applied Mathematics Vol. 6, Springer-Verlag, New York, 1998 (ISBN: 0-387-98489-5); (2) Notes from instructor.
• Control systems theory is a central and highly active research area in the applied sciences. The area focuses on methods to devise forcing functions for forced dynamical systems, in order to achieve prescribed objectives for sets of solutions of the systems. These forced dynamical systems are called control systems, and are widely used for modeling and then choosing forces to apply to engineering systems, and for choosing the manipulated parameters in disease dynamics and other biological systems. This course is devoted to feedback controls, which are forcing functions for control systems that can depend on values of the states of the systems. The focus will be on continuous time control systems that are modeled by systems of ordinary differential equations, and on discrete time control systems. This course goes beyond classical frequency domain methods, by providing more advanced methods for nonlinear systems, and explorations of case studies arising in electrical and mechanical engineering and in biological applications. However, no prerequisite background in biology or engineering will be required to understand this course.
7490 Topics in Graph Theory. Prof. Ding
• 1:30-2:50 TTh
• Instructor: Prof. Ding.
• Prerequisites: 4171 or equivalent
• Text: Algorithmic Graph Theory and Perfect Graphs (not required); lecture notes will be provided
• We will discuss graph structures involving induced subgraphs. In particular, we study the structure of interval graphs, chordal graphs, cographs, comparability graphs, and more. We will also discuss the behavior of large graphs.
7510 Topology I. Prof. Bibby
• 8:30-9:20 MWF
• Instructor: Prof. Bibby.
• Prerequisite: Advanced Calculus (Math 4031)
• Text: Topology (2nd ed.) by James R. Munkres.
• This course is a preparation course for the Core I examination in topology. It will cover general (point set) topology, the fundamental group, and covering spaces. We will also introduce simplicial complexes and manifolds, using them often as examples. A good supplementary reference is Chapter 1 of Algebraic Topology by Allen Hatcher, available online
7560 Riemannian Geometry. Prof. Dani
• 9:30-10:20 MWF
• Instructor: Prof. Dani
• Prerequisite: MATH 7550 or equivalent
• Text:
• Introduction to Riemannian geometry, the study of smooth manifolds endowed with Riemannian metrics. Topics include Riemannian metrics, connections, geodesics, curvature, Jacobi fields, completeness, spaces of constant curvature, and calculus of variations, followed by theorems that relate curvature, topology, and analysis.
7590 Complex Geometry. Prof. Zeitlin
• 12:30-2:20 MWF, Session C (second half of semester)
• Instructor: Prof. Zeitlin
• Prerequisite: Some familiarity with complex analysis (on the level of 4036), differential geometry (on the level of 7550)
• Text: D. Huybrechts "Complex Geometry: an introduction", P. Griffits, J. Harris, "Algebraic Geometry"
• This course aims to build bridges between differential geometry and algebraic geometry courses. Some topics to be covered: differential calculus on complex manifolds, divisors and line bundles, blowups, Kahler manifolds, elements of Hodge theory, vector bundles and Chern classes, Hirzebruch-Riemann-Roch theorem.
8000-n Assorted Sections of MS-Thesis Research
9000-n Assorted Sections of Doctoral Dissertation Research

## Spring 2023

### For Detailed Course Outlines, click on course numbers.

Core courses and Breadth courses are listed in bold.

4997-1 Vertically Integrated Research: Representation Theory. Profs Achar and Sage
• Instructor: Profs. Achar and Sage
• Prerequisite:
• Text:
4997-2 Vertically Integrated Research. Prof. Shea Vela Vick and Dr. Angela Wu
• 12:00-1:20 TTh
• Instructors: Prof. Vela-Vick and Dr. Wu
• Prerequisite: a first course in linear algebra, experience with topology or abstract algebra is helpful but not required
• Materials: none
• Description: This is a project-based seminar class in geometry and topology. Students work together in small groups to tackle current research problems in topics such as knot theory, differential geometry, symplectic and contact topology, etc.
4997-3 Vertically Integrated Research: Geometry and Combinatorics of Polynomials. Profs Bibby and Cohen
• Instructors: Prof. Bibby and Prof. Cohen
• Prerequisite:
• Text:
• Properties (unimodality, log-concavity...) of polynomials which arise naturally in combinatorial settings (e.g., the chromatic polynomial of a graph).
4997-4 Vertically Integrated Research: Nonsmooth Optimization. Profs Wolenski and Dr. Marazzato
7002 Communicating Mathematics II. Prof. Oxley.
• 3:00-4:50 TTh
• Instructor: Prof. Oxley.
• Prerequisite: Consent of department.
• This course is required for all incoming graduate students. It is a lab course meeting for an average of two hours per week throughout the semester for one-semester-hour's credit.
7211 Algebra II. Prof. Ng
• Instructor: Prof. Ng
• Prerequisite: Math 7210 or equivalent
• Text: Abstract Algebra, 3rd ed; Dummit and Foote, Wiley 2003
• This is the second part of our graduate algebra sequence. Topics will include field theory, Galois theory, basics of commutative algebra and algebras over a field, Wedderburn’s theorem, Maschke’s theorem, tensor products and Hom for modules, possibly some introduction to homological algebra or linear representations of finite groups if time permitted.
7230 Topics in Number Theory: Introduction to Modern Number Theory. Prof Tu
7290 Seminar in Algebra and Number Theory. Prof. Achar
• Instructor: Prof. Achar
• Prerequisite:
• Text:
7320 Ordinary Differential Equations. Prof. Tarfulea
• Instructor: Prof. Tarfulea
• Text:
• This is a standard first graduate course in differential equations. Topics include first order equations, existence and uniqueness theorems, linear equations and systems, complex linear systems, boundary value and eigenvalue problems.
7330 Functional Analysis. Prof. Walker
• Instructor: Prof. Walker
• Prerequisite: Math 7311
• Text: Banach Algebra Techniques in Operator theory GTM, 2nd edition by Ron Douglas
• In this course, we shall discuss the basics of Banach space, Banach algebra and operators on Hilbert spaces. We will then discuss the Hardy space and Toeplitz operators. We will follow the GTM book by Ron Douglas.
7366 Stochastic Analysis. Prof. Ganguly
• Instructor: Prof. Ganguly
• Prerequisite: Math 7360
• Text:
• Wiener process, stochastic integrals, stochastic differential equations.
7375 Wavelets. Prof. Ólafsson
• Instructor: Prof Ólafsson
• Prerequisite: Math 7311
• Text:
• Fourier series; Fourier transform; windowed Fourier transform or short-time Fourier transform; the continuous wavelet transform; discrete wavelet transform; multiresolution analysis; construction of wavelets.
7380 Radon Transforms and Spherical Harmonics. Prof. Rubin
7380 Elliptic PDE. Prof. Nguyen
7390-1 Seminar in Analysis. Prof Wan
7390-2 Seminar in Analysis: Optimization, Calculus of Variations, and Optimal Control. Prof Wolenski
7390-3 Seminar in Analysis: Complex Analysis II. Prof Antipov
7410 Graph Theory. Prof. Oporowski.
• Instructor: Prof. Oporowski.
• Prerequisite: MATH 2085 and MATH 4039; or equivalent
• Text:
• The main theme of this course will be graph theory. We will discuss a wide range of topics, including spanning trees, eulerian trails, matching theory, connectivity, hamiltonian cycles, coloring, planarity, integer flows, surface embeddings, and graph minors. For more information see Math 7410.
7490 Seminar in Combinatorics, Graph Theory, and Discrete Structures: Tutte Polynomial. Prof Oxley
7512 Topology II. Prof Vela-Vick.
• Instructor: Prof. Vela-Vick.
• Prerequisite: Math 7510
• Text: Algebraic Topology, by Allen Hatcher
• This course covers the basics of homology and cohomology theory. Topics discussed include singular and cellular (co)homology, Brouwer fixed point theorem, cup and cap products, universal coefficient theorems, Poincare duality, Alexander duality, Kunneth theorems, and the Lefschetz fixed point theorem.
7550 Differential Geometry. Prof Cohen
• Instructor: Prof. Cohen
• Prerequisite: Math 7210 and Math 7510
• Text: Glen E. Bredon, Topology and Geometry, Springer, GTM 139
• This course gives an introduction to the theory of manifolds. Topics to be covered include: differentiable manifolds, charts, tangent bundles, transversality, Sard's theorem, vector and tensor fields, differential forms, Frobenius's theorem, integration on manifolds, Stokes's theorem, de Rham cohomology, Lie groups and Lie group actions.
7590-1 Geometric Topology: Coxeter Groups. Prof Schreve
7590-2 Moduli Spaces of Curves. Prof Baldridge
7710 Numerical Linear Algebra. Prof. Zhang