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# Graduate Courses, Summer 2021-Spring 2022

## Contact

## Summer 2021

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

*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:

## 7999-n Assorted Individual Reading Classes

## 8000-n Assorted Sections of MS-Thesis Research

## 9000-n Assorted Sections of Doctoral Dissertation Research

## Fall 2021

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

*course numbers.*

Core courses and Breadth courses are listed in **bold**.

## 4997-1 Vertically Integrated Research: Cryptography. Profs. Achar and Sage

- 12:00-1:20 TTh
- Instructor: Profs. Achar and Sage
- Prerequisite: Linear algebra and at least one upper-level math course
- Text: Shemanske,
*Modern Cryptography and Elliptic Curves* - Cryptography is the study of methods of sending confidential communications in the presence of adversaries. Historically, the emphasis of cryptography was on linguistic patterns such as letter frequency or common combinations of letters. Famous illustrations of such methods can be found in literary works like Edgar Allan Poe's
*The Gold Bug*and Sir Arthur Conan Doyle's*The Adventure of the Dancing Men*. By contrast, modern cryptography is primarily mathematical, drawing on number theory, abstract algebra, probability, among other fields. This course is an introduction to cryptography, with an emphasis on the algebra and number theory on which modern methods are based.

## 4997-2 Vertically Integrated Research: Prof. Vela Vick

- 12:00-1:20 TTh
- Instructor:
- Prerequisite:
- Text:

## 4997-3 Vertically Integrated Research: Parallel Computational Math. Drs. Patrick Diehl and Hartmut Kaiser

- 9:00-10:20 TTh
- Instructor: Drs. Patrick Diehl and Hartmut Kaiser
- Prerequisite: None, but some basic knowledge about programming is beneficial
- Text:
- This course will focus on the parallel implementation of computational mathematics problems using modern accelerated C++. The aim of this course is to learn how to quickly write useful efficient C++ programs.
The students will not learn low-level C/C++ instead they will learn how to use high-level data structures, iterators, generic strings, and streams (including interactive and file I/O) of the C++ ISO Standard library. In addition, highly-optimized linear algebra libraries are introduced since the course teaches to solve problems, instead of explaining low-level C++ and computer science algorithms, like sorting algorithms, which are provided in the C++ standard library.
The first part, provides a brief overview of the containers, strings, streams, input/output, and the numeric library of the C++ standard library. For linear algebra, we will look into Blaze which is an open-source, high-performance C++ math library for dense and sparse arithmetic.

The second part will solve computational mathematics problems based-on the the previous introduced features of the C++ standard library.

The third part will focus on the parallel features provided by the C++ standard library. Here, the implemented computational problems in the second part of the course will be parallized using the C++ standard library for parallelism and concurrency.

Since programming skills can only be improved by doing, there will be weekly programming exercises and a small project. After this course students have a basic overview of the C++ standard library to solve efficiently computational mathematics problems without using low-level C/C++.

## 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. Sage.

- 10:30-1150 TTh
- Instructor: Prof. Sage.
- 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.

## 7260 Homological Algebra. Prof. Achar

- 1:30-2:50 TTh
- Instructor: Prof. Achar
- Prerequisite:
- Text:

## 7290 Hopf Algebras & Finite Groups. Prof. Ng

- 10:30-11:50
- Instructor: Prof. Ng
- Prerequisite:
- Text:
- Hopf algebras, sometimes also known as quantum groups, are generalizations of groups and Lie algebras. Their representation categories are tensor categories which have many important applications to areas in mathematics and physics. In this course, we present an introduction to the theory of Hopf algebras and their representations. The basic theory of representations of finite groups will be introduced as motivation at the beginning of the course, and the course will be a great continuation of Math 7211. Prerequisites are some notions on finite groups and rings, and a good knowledge of linear algebra covered in Math 7210 or equivalent.

## 7311 **Real Analysis I.** Prof. Olafsson.

- 9:00-10:20 TTh
- Instructor: Prof. Olafsson.
- 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:

- R. G. Bartle:
*The Elements of Integration and Lebesgue Measure* - A. Friedman:
*Foundations of Modern Analysis.* - P. R. Halmos:
*Measure Theory (Graduate Text in Mathematics)* - F. Jones:
*Lebesgue Integration on Euclidean Space.* - L. Richardson:
*Measure and Integral: An Introduction to Real Analysis.* - H. L. Royden:
*Real Analysis.*

- R. G. Bartle:

## 7350 **Complex Analysis.** Prof. Antipov.

- 10:30-11:20 MWF
- Instructor: Prof. Antipov.
- Prerequisite: Math 7311 or its equivalent.
- Text:
*Complex Analysis*by Elias Stein and Rami Shakarchi, Princeton Lectures in Analysis II. - 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. Chen.

- 12:30-1:20 MWF
- Instructor: Prof. Chen.
- Prerequisite: 7311 Real Analysis I
- Text:
*Probability: Theory and Examples. 5th Edition*by Rick Durrett. - In this course we first introduce probability measures, random variables, distributions and various modes of convergence in a measure-theoretic framework. It then proceeds to discuss independence, convergence theorems for sums of independent random variables and strong law of large numbers. Weak convergence of probability measures and central limit theorem are presented with full details. An important part of this course is to introduce and develop conditional probability and its applications.

## 7365 Applied Stochastic Analysis. Prof. Ganguly.

- 9:00-10:20 TTh
- Instructor: Prof. Ganguly.
- 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 of 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, for example, through differential equations fail to capture. Examples of such systems are numerous and wide ranging - 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. 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 and some elementary stability theory. If time permits, we will also make some remarks about the continuous-time case and briefly discuss some stochastic algorithms like Markov Chain Monte Carlo, importance sampling, stochastic approximation methods which are instrumental in probabilistic approach to data-science. The course is also a gateway to the course on stochastic analysis (Math 7366)

## 7380 Introduction to Applied Math. Prof. Shipman

- 1:30-2:50 TTh
- Instructor: Prof. Shipman.
- Prerequisite:
- Text:
*An Introduction to Partial Differential Equations,*M Renardy and RC Rogers, Second edition - The purpose of this course is to introduce students interested in applied mathematics to the mathematical (differential) equations of physics and the basic techniques that are used. It does not take the place of courses in PDEs or real analysis or Fourier analysis. Instead, it teaches how a breadth of mathematical techniques and subjects bear, often simultaneously, upon the study of applied problems. Material on specific equations of physics, such as those of electromagnetics, fluids, acoustics, quantum mechanics, and other areas, will supplement the main textbook. Topics will come from the following list. Some will be dealt with in more detail than others, but I hope to at least introduce the significance of each.
Equations:

- Maxwell equations of E&M
- Navier-Stokes equations of fluids
- Schrödinger equations of QM
- Many more elementary and more complicated equations related to these

- Elliptic/Parabolic/Hyperbolic classification
- Boundary-value and initial-value problems
- Well-posedness of solutions
- Conservation Laws
- Maximum principles
- Linear equations
- Nonlinear equations

- Method of characteristics
- Weak formulations of PDEs in function spaces
- Distributional solutions
- Fourier analysis
- Operator theory
- Green functions
- Energy methods

## 7384 Topics in Material Science: Prof. Lipton.

- 2:30-3:20 MWF
- 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. Bulut.

- 12:00-1:20 TTh
- Instructor: Prof. Bulut.
- Prerequisite:
- Text:

## 7390 Topics in Convex and Stochastic Optimization. Prof. Zhang.

- 10:30-11:50 TTh
- Instructor: Prof. Zhang.
- Prerequisite: Math 2057 Multidimensional Calculus; Math 2085 Linear Algebra
- Text: Class Notes
- Convex and stochastic optimization have played important role in modern optimization. This course will focus on the theory and algorithm development for solving convex optimization problems and optimization problems with stochastic features. Tentative topics include Convex sets, Convex functions, Duality theory, gradient and stochastic gradient methods for solving convex and nonconvex optimization problems.

## 7490 Matroid Theory. Prof. Oxley

- 8:30-9:20 MWF
- Instructor: Prof. Oxley.
- Prerequisites: Permission of the Department
- Text: J. Oxley,
*Matroid Theory, Second edition,*Oxford, 2011. - What is the essence of the similarity between forests in a graph and linearly independent sets of columns in a matrix? Why does the greedy algorithm produce a spanning tree of minimum weight in a connected graph? Can one test in polynomial time whether a matrix is totally unimodular? Matroid theory examines and answers questions like these.
This course will provide a comprehensive introduction to the basic theory of matroids. This will include discussions of the basic definitions and examples, duality theory, and certain fundamental substructures of matroids. Particular emphasis will be given to matroids that arise from graphs and from matrices.

## 7510 **Topology I.** Prof. Dasbach.

- 9:30-10:20
- Instructor: Prof. Dasbach.
- 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

## 7520 Algebraic Topology. Prof. Vela-Vick.

- 9:00-10:20 TTh
- Instructor: Prof. Vela-Vick.
- Prerequisite:
- Text:
*Algebraic Topology*, by Allen Hatcher - This course will cover basic topics in homotopy theory, including the following: Whitehead's theorem, cellular approximation, excision theorem, Hurewicz theorem, long exact sequence of fiber bundles and Gysin sequences. We will also discuss spectral sequences and some of their applications to topology.

## 7590 Arrangements and Configuration Spaces. Prof. Bibby.

- 11:30-12:20 MWF
- Instructor: Prof. Bibby
- Prerequisite: Math 7512 Topology II
- Text: None
- A hyperplane arrangement is a set of codimension-one subspaces in a given vector space. Whether over the real numbers, complex numbers, or a finite field, an arrangement of hyperplanes and its complement have combinatorially, topologically, and algebraically rich structure.
Over the real numbers, how many connected components does the complement have? Over the complex numbers, what is the cohomology of the complement? Over a finite field, how many points does the complement have? Many questions such as these can actually be answered using the underlying combinatorics (a matroid).

This course will focus on the interplay between combinatorics and topology in the context of hyperplane arrangements and related objects, such as configuration spaces. After an initial introduction to the theme, students will present on a related topic of their choice. Topics may include the questions above; the cohomology ring or fundamental group; matroids in topology; arrangements associated to reflection groups. Most students in areas related to combinatorics, topology, or algebra are likely to find something of interest to them, but familiarity with topology (through Topology II) will be assumed.

## 7999-n Assorted Individual Reading Classes

## 8000-n Assorted Sections of MS-Thesis Research

## 9000-n Assorted Sections of Doctoral Dissertation Research

## Spring 2022

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

*course numbers.*

Core courses and Breadth courses are listed in **bold**.

## 4997-1 Vertically Integrated Research:

- Instructor:
- Prerequisite:
- Text:

## 4997-2 Vertically Integrated Research:

- Instructor:
- Prerequisite:
- Text:

## 7002 Communicating Mathematics II. Prof. Oxley.

- Instructor: Prof. Oxley.
- Prerequisite:
- Text:

## 7211 **Algebra II.** Prof. Long.

- Instructor: Prof. Long.
- Prerequisite:
- Text:

## 7220 Commutative Algebra &: Algebraic Geometry. Prof. Hoffman

- Instructor: Prof. Hoffman.
- Prerequisite:
- Text:

## 7230 Topics in Number Theory. Prof. Tu

- Instructor: Prof. Tu.
- Prerequisite:
- Text:

## 7320 **Ordinary Differential Equations.** Prof. Estrada

- Instructor: Prof. Estrada.
- Prerequisite:
- Text:

## 7330 **Functional Analysis.** Prof. He.

- Instructor:
- Prerequisite: Prof. He.
- Text:

## 7366 Stochastic Analysis. Prof. Sundar.

- Instructor: Prof. Sundar.
- Prerequisite:
- Text:

## 7380 Topics in **C**^{*} algebras. Prof. Olafsson.

- Instructor: Prof. Olafsson.
- Prerequisite:
- Text:

## 7384 Topics in Material Science: Prof. Shipman.

- Instructor: Prof. Shipman.
- Prerequisite:
- Text:

## 7390-1 Topics in Analysis. Prof. Walker.

- Instructor: Prof. Walker.
- Prerequisite:
- Text:

## 7390-2 Topics in Elliptic Partial Differential Equations. Prof. Zhu.

- Instructor: Prof. Zhu.
- Prerequisite:
- Text:

## 7410 **Graph Theory.** Prof. Oporowski.

- Instructor: Prof. Oporowski.
- Prerequisite:
- Text:

## 7490 Combinatorial Optimization. Prof. Ding

- Instructor: Prof. Ding.
- Prerequisite:
- Text:

## 7512 **Topology II.** Prof. Dani.

- Instructor: Prof. Dani.
- Prerequisite:
- Text:

## 7550 **Differential Geometry.** Prof. Baldridge.

- Instructor: Prof. Baldridge.
- Prerequisite:
- Text:

## 7590-1 Geometric Topology: Braids. Prof. Cohen.

- Instructor: Prof. Cohen.
- Prerequisite:
- Text:

## 7590-2 Geometry and Physics. Prof. Zeitlin

- Instructor: Prof. Zeitlin.
- Prerequisite: Some basic knowledge of differential geometry: calculus on manifolds, differential forms, Lie groups (MATH 7550 is more than enough).
- Text: I will distribute my own notes.
- This course is a rigorous introduction to quantum field theory aimed at mathematicians. The primary target is path integral techniques, including Feynman diagrams, Batalin-Vilkovisky (BV) formalism, localization. The goal of the course is to understand basic notions needed to read papers in Quantum Field Theory and String Theory. The material will be also useful for students in the physics department interested in High Energy Physics.

## 7710 **Numerical Linear Algebra.** Prof. Sung.

- Instructor: Prof. Sung.
- Prerequisite:
- Text: