# 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: p-adic analysis. Profs Achar and Sage
• 12:00-1:20 TTh
• Instructor: Profs. Achar and Sage
• Prerequisite: Math 4031 and Math 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
• TTh 3:00-4:50
• Instructor: Prof. Oxley.
• Prerequisite: Consent of department. This course is required for all first-year 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: Notes and various sources, mainly Renardy/Rogers An Introduction to Partial Differential Equations
• 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: An introduction to perfectoid spaces. Profs Achar and Sage
• 12:00-1:20 TTh
• Instructor: Profs. Achar and Sage
• Prerequisite: Math 4153 and 4200
• Text: p-adic Analysis Compared with Real by S. Katok
• A "perfect field" of characteristic p > 0 is a field in which the operation of raising to the p-th power is surjective. Finite fields are perfect, but many other commonly occurring fields, such as the field of rational functions in one variable, are not perfect. In the latter case, it turns out that passing to the "perfection" (i.e., adding in all p-th roots) gives a field that behaves in some ways like a field of characteristic 0. This fact is the starting point of the theory of "perfectoid spaces," a powerful new tool invented by Peter Scholze about 10 years ago, and for which he won the Fields Medal in 2018. The theory of perfectoid spaces links characteristic p and characteristic 0, and makes it possible to use ideas from complex analysis and geometry to study questions in number theory and algebraic geometry. This course will cover the starting points of this theory.
4997-2 Vertically Integrated Research. Prof. Shea Vela Vick and Dr. Angela Wu
• MWF 2:30-3:20
• 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
• MWF 1:30-2:20
• 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: Machine Learning. Dr. Marazzato
• MWF 12:30-1:20
• Instructor: Dr. Marazzato
• Prerequisite: Calculus and linear algebra
• Text:
• Machine learning has become a major source for companies in the last few years. This course is designed to provide you with the basics of machine learning. To prepare you for your future jobs, it is also taught as a flipped class. The idea being that you will learn how to teach yourselves about new topics, an invaluable skill when you are no longer a student. The semester is split in two parts. In the first, you will learn about deep learning and in the second you will apply it on real data by working on project in small teams. This course is also a C-I (Communication Intensive) course. You will be trained on communicating mathematical results both orally and in writing and to right rigorous proofs. The former is useful for every mathematician working in a team and the latter necessary for every good mathematician.
7002 Communicating Mathematics II. Prof. Oxley
• 3:00-4:50 TTh
• Instructor: Prof. Oxley.
• Prerequisite: Consent of department. This course is required for all first-year 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.
7211 Algebra II. Prof. Ng
• MWF 2:30-3:20
• 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
• MWF 11:30-12:20
• Instructor: Prof. Tu
• Prerequisites:
1. Math 7210, 7211 (Algebra), Math 4036 (complex analysis) or equivalent are very helpful.
2. Know how to find resources and how to use computer algebra systems (Maple, Mathematica, Magma, or SageMath).

• Materials:

A Classical Introduction to Modern Number Theory (Springer GTM 84), by Kenneth Ireland and Michael Rosen. (We will mainly follow chapters 6, 8-14, 16-19 of the textbook. )

Some useful references:

1. Gauss and Jacobi Sums, by Bruce C. Berndt, Ronald J. Evans, and Kenneth S. Williams
2. Number Fields, by Daniel A. Marcus
3. Rational Points on Elliptic Curves, by Joseph H. Silverman, and John Tate.

• The goal of this course is to give a gentle introduction to some basic topics in Number Theory, that include characters, character sums, zeta functions of varieties, Dirichlet L-functions, Diophantine equations, and elliptic curves.

Character sums and zeta functions are closely related to the L-functions and Galois representations attached to a given algebraic variety. We will look at some examples arising from Diophantine equations and also certain special elliptic curves.

7290 Seminar in Algebra and Number Theory: Infinity categories. Prof. Achar
• TTh 10:30-11:50
• Instructor: Prof. Achar
• Prerequisite: Algebra II and Topology II, including basic category theory. Some familiarity with homological algebra will be helpful but not required.
• Materials: Lecture notes to be distributed in class
• In ordinary category theory, a "commutative diagram" is essentially a kind of directed graph. In infinity category theory, a "commutative diagram" is a higher-dimensional object: roughly, a kind of simplicial complex. It turns out that infinity categories arise naturally even when one starts from questions about ordinary categories. Moreover, the theory of infinity categories elegantly overcomes some shortcomings of traditional category theory, especially in the context of homological algebra. This course will have two halves. The first half will be about the foundations of infinity category theory (in the version developed by Joyal, Lurie, and others). The second half will be about "stable infinity categories," which are the infinity-categorical counterparts of abelian or triangulated categories, and which provide the "correct" modern context for doing homological algebra.
7320 Ordinary Differential Equations. Prof. Tarfulea
• MWF 8:30-9:20
• Instructor: Prof. Tarfulea
• Text: An Introduction to Dynamical Systems, Continuous and Discrete, Second Edition by R. Clark Robinson.
• This is a standard first graduate course in differential equations. We begin with the existence/uniqueness theories as well as the methods for solving linear systems of ODEs. The bulk of the course will be in the treatment of nonlinear systems, which is at the core of the study of dynamical systems. We explore phase portraits, orbits, stability/instability near orbits, bifurcations, and attractors. The methods used will be suitably general, but the course will treat lots of individual systems (arising from models in physics/biology) where we can prove more specific qualitative properties.
7330 Functional Analysis. Prof. Walker
• TTh 12:00-1:20
• Instructor: Prof. Walker
• Prerequisite: Math 7311
• Text: Functional Analysis, by Peter Lax, and other supplemental texts.
• Basics of Banach spaces, Hahn-Banach theorem, reflexive spaces, convex sets, weak convergence, etc. Operators on Hilbert spaces, contraction mappings, fixed point theorems. Spectral theory (eigenvalues). Connections with partial differential equations will be made.
7366 Stochastic Analysis. Prof. Ganguly
• MWF 11:30-12:20
• Instructor: Prof. Ganguly
• Prerequisite: Probability theory at the level of Math 7360
• Text: Stochastic differential equations by Bernt Oksendal
• The course will cover elements of stochastic integrals and stochastic differential equations. A common choice of the integrator is a Brownian motion. A Brownian motion is a continuous random function of time whose paths are very rough. This in particular means that integrals with respect to a Brownian motion cannot be defined in the usual Riemann-Stieltjes way, and a new approach needs to be introduced to address them. Understanding this approach and the necessary materials behind it is the broad goal of this course. Along the way we will learn about Ito isometry, martingale property of stochastic integrals, Ito's lemma, etc. The last part of the course will focus on stochastic differential equations which are now extensively used in modeling time-evolution of a variety of random systems. If time permits we will also discuss some research-problems in these areas.
7375 Wavelets. Prof. Ólafsson
• TTh 9:00-10:20
• Instructor: Prof. Ólafsson
• Prerequisite: Math 7311
• Books: M. A. Pinsky: Introduction to Fourier Analysis and Wavelets. Graduate Studies in Mathematics, Vol. 102, R. Fabec and G. Olafsson: Non-Commutative Harmonic Analysis, Drexville Publishing, and own notes.
• This is a basic course in Fourier analysis and related topics. The topics that will be covered includes: Fourier series, the Fourier transform, applications to differential equations, Hermite Polynomials and functions; Heisenberg Uncertainty Principle; Windowed Fourier transform and basic wavelet theory. If there is time then we will discuss connections to other fields including group actions and representation theory.
7390-1 Seminar in Analysis: Numerical methods for partial differential equations. Prof. Wan
• TTh 1:30-2:50
• Instructor: Prof. Wan
• Text:
• This course covers both classical approaches and deep learning techniques for the approximation of partial differential equations. The classical approaches include finite difference method, finite element method as well as spectral method. The deep learning techniques are receiving more and more attention and being actively developed. This is an introductory course. The main topic is to understand the error of these methods and their applicability to some challenging PDE-related problems such as uncertainty quantification, viscoelastic flows, etc.
7390-2 Seminar in Analysis: Calculus of Variations and Optimal Control. Prof. Wolenski
• MWF 9:30-10:20
• Instructor: Prof. Wolenski
• Text: No formal text is required
• This course is an introduction to the Calculus of Variations and Optimal Control from a modern viewpoint that will emphasize the role of convexity. It is aimed graduate and advanced undergraduate students interested in the theoretical foundations of any area of applied mathematics. We will begin with a review of continuous optimization in Euclidean space; the role of convexity will be emphasized and basic tools of nonsmooth analysis will be introduced. The main topic of the course is dynamic optimization. The first half will cover the classical material of the calculus of variations, including topics such as the Euler-Lagrange equation, Weierstrass maximality condition, Erdmann corner conditions, and Jacobi conjugate points. Plenty of examples will be covered. There is a natural transition into optimal control, which will be the focus of the rest of the course from a neo-classical point of of view.
7410 Graph Theory. Prof. Oporowski
• MWF 10:30-11:20
• 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
• MWF 8:30-9:20
• Instructor: Prof. Oxley
• Prerequisite: Math 7410 and 7490 (Matroid Theory) or permission of the department.
• Text: Matroid Applications edited by Neil White (Chapter 6: The Tutte Polynomial and its Applications by Thomas Brylawski and James Oxley)
• The theory of numerical invariants for matroids is one of many aspects of matroid theory having its origins within graph theory. Most of the fundamental ideas in matroid invariant theory were developed from graphs by Veblen, Birkhoff, Whitney, and Tutte when considering colorings and flows in graphs. This course will introduce the Tutte polynomial for matroids and will consider its applications in graph theory, coding theory, percolation theory, electrical network theory, and statistical mechanics.
7512 Topology II. Prof. Vela-Vick
• TTh 9:00-10:20
• 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
• MWF 12:30-1:20
• 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
• MWF 9:30-10:20
• Instructor: Prof. Schreve
• Prerequisite: Topology 7510 and Algebra 7210
• Text: A good reference is Davis' Geometry and Topology of Coxeter groups, though it's not required, and I will be using other notes and references.
• An abstract Coxeter group has a simple presentation; all generators are involutions and pairs of generators satisfy some dihedral relation, i.e. together they generate a dihedral group. Classical examples are groups generated by finitely many reflections of the sphere, Euclidean or hyperbolic space. They are important in many areas of mathematics, including topology, geometry, and representation theory.

The beginning of this course will focus on the combinatorial structure of Coxeter groups, and then we will move to studying their geometry and topology. The main topic comes from the 1988 Ph.D thesis of Moussong, who showed that any infinite Coxeter group acts by isometries on a contractible cell complex admitting a singular, nonpositively curved metric; a coarse geometric analogue of Euclidean or hyperbolic space. Throughout the course, there will also be a focus on using Coxeter groups to construct exotic examples of manifolds and other complexes.

Possible topics for the end of the semester are buildings, various flavors of cohomological invariants, applications to combinatorics, and Artin groups.

7590-2 Moduli Spaces of Curves. Prof. Baldridge
• TTh 12:00-1:20
• Instructor: Prof. Baldridge
• Prerequisite: Math 7550 or equivalent
• Text: Graphs on Surfaces and Their Applications and related research articles
• This course is an introduction to the moduli space of stable genus g complex curves (2-dimensional Riemann surfaces) with n marked points. This moduli space is important in many areas in mathematics and physics. The introduction to the moduli space in this course is particularly gentle: we will concentrate on building a “combinatorial model’’ of the moduli space of curves using metric ribbon graphs. First, we will discuss the basic facts about Riemann surfaces that any mathematician should know. We will then describe a bijective correspondence between metric ribbon graphs and compact Riemann surfaces with meromorphic Strebel differentials. This in turn will give us a graph theory description of the moduli space of Riemann surfaces. We will then explore this moduli space using Kontsevich’s notion of a cohomological field theory via edge-contraction operations on ribbon graphs and how this relates to 2D TQFTs. Students who are interested in graph theory, representation theory, mathematical physics, topology or geometry should strongly consider this course. It will also serve as a good introduction to a possible Gromov-Witten theory course next year at LSU.
7710 Numerical Linear Algebra. Prof. Zhang
• TTh 10:30-11:50
• Instructor: Prof. Zhang
• Prerequisite: Linear Algebra, Advanced Calculus, (some) Programming Experience
• Text: David Watkins, Fundamentals of Matrix Computations (Third Edition) ISBN 978-0-470-52833-4
• This is an introductory course in numerical linear algebra at the graduate level. We will cover selected topics from the eight chapters of the text. Depending on the time, our course topics will include Mathematical Tools: norms, projectors, Gram-Schmidt process, orthogonal matrices, spectral theorem, singular value decomposition and Gerschgorin's circles Error Analysis: floating point arithmetic, round-off errors, IEEE floating point standard, backward stability and conditioning General Systems: LU decomposition, partial pivoting, Cholesky decomposition, least squares problem and QR decomposition Sparse Systems: the methods of Jacobi, Richardson, Gauss-Seidel, successive over-relaxation, steepest descent and conjugate gradient Eigenvalue Problems: power methods, Rayleigh quotient iteration, deflation and QR algorithm