Graduate Courses, Summer 2021 – Spring 2022

Summer 2021

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:

7999-n Assorted Individual Reading Classes
  • No additional information.

8000-n Assorted Sections of MS-Thesis Research
  • No additional information.

9000-n Assorted Sections of Doctoral Dissertation Research
  • No additional information.

Fall 2021

For Detailed Course Outlines, click on 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 parallelized 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:

    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. 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
    Concepts:
    • Elliptic/Parabolic/Hyperbolic classification
    • Boundary-value and initial-value problems
    • Well-posedness of solutions
    • Conservation Laws
    • Maximum principles
    • Linear equations
    • Nonlinear equations
    Techniques:
    • 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: Math 7311 (Real Analysis I) or equivalent, or consent of department
  • Texts: Primary Text: Partial Differential Equations by Lawrence C. Evans

    Additional optional references:

    • Elliptic partial differential equations by Qing Han and Fanghua Lin
    • Semilinear Schrödinger Equations by Thierry Cazenave.
  • This course provides an introduction to the theory of partial differential equations. Topics to be covered include:
    1. Introduction to elliptic, parabolic, and hyperbolic partial differential equations
    2. Examples: Laplace's equation, the heat equation, and the wave equation
    3. Introduction to Sobolev spaces, weak derivatives, existence and uniqueness of solutions.
    4. Elliptic equations: existence, regularity and the maximum principle.
    5. Selected additional topics (chosen according to time and class interest).
    Additional topics may include:
    a) Nonlinear dispersive PDE with an emphasis on the nonlinear Schrodinger equation.
    b) Introduction to the Calculus of Variations: Euler-Lagrange equations, existence of minimizers, eigenvalues of self-adjoint elliptic operators.

7390-1 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.

7390-2 Large Deviations. Prof. Ganguly.

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
  • No additional information.

8000-n Assorted Sections of MS-Thesis Research
  • No additional information.

9000-n Assorted Sections of Doctoral Dissertation Research
  • No additional information.

Spring 2022

For Detailed Course Outlines, click on course numbers. Core courses and Breadth courses are listed in bold.

4997-1 Vertically Integrated Research: Enumerative geometry. Prof. Achar
  • 12:00-1:20 TTh
  • Instructor: Pramod Achar
  • Prerequisite: Math 4200. Also Math 4039 and Math 4201 will be helpful, but not required
  • Text: 1st half: lecture notes to be distributed in class. 2nd half: Sheldon Katz, Enumerative geometry and string theory
  • In 1848, Steiner asked the following question: given 5 "generic" conic curves in the plane, how many conic curves are there that are tangent to all 5? Enumerative geometry is the study of questions of this kind. The first part of the semester will be spent studying the tools needed to answer Steiner's question (the answer is 3,264). In the second part of the semester, we will study the surprising connections between enumerative geometry and theoretical physics (and in particular, string theory) that were discovered in the 1990s.

4997-2Vertically Integrated Research. Prof. Vela-Vick and Dr. Wu
  • 1:30-2:50 TTh
  • Instructors: Shea Vela-Vick and Angela 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. Prof. Wolenski and Dr. Marazzato
  • 1:30-2:50 TTh
  • Instructors: Peter Wolenski and Frèdèric Marazzato
  • Prerequisite:
  • Text:

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. Long.
  • 10:30-11:20 MWF
  • Instructor: Prof. Long.
  • 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.

7220 Commutative Algebra &: Algebraic Geometry. Prof. Hoffman
  • 9:30-10:20 MWF
  • Instructor: Prof. Hoffman.
  • Prerequisite: Math 7210, 7510
  • Text: The main reference is D. Eisenbud, Commutative Algebra: with a View Toward Algebraic Geometry, Graduate Texts in Mathematics, 150, 1995. Corr. 3rd Edition. This will be supplemented by material from various Algebraic Geometry books. Also, computations with the software systems Macaulay2 and Singular will be introduced.
  • We will cover the basics of Commutative Algebra and give an introduction to the elements of Algebraic Geometry of affine and projective varieties. This will be a preparation for learning the theory of Schemes. Topics include:

    1. Affine and projective varieties
    2. Prime and Maximal Ideals
    3. Localization
    4. Integral extensions
    5. Primary decomposition
    6. Dimension; Hilbert-Samuel polynomial
    7. Homological methods; Free resolutions
    8. Kähler differentials
    9. Flat, smooth and étale morphisms

7230 Topics in Number Theory: Arithmetic of Quaternion Algebras in Number Theory. Prof. Tu
  • 12:30-1:20 MWF
  • Instructor: Prof. Tu.
  • Prerequisites:

    1. Math 7210 (Algebra), Math 4036 (complex analysis) or equivalent are very helpful. You should be comfortable with field extensions and Galois theory.

    2. Know how to find resources and how to use computer algebra systems (Maple, Mathematica, Magma, or SageMath).

  • Text: There is no required text, but we will mainly follow parts of the book “Quaternion Algebra” by John Voight (The official version is available on Springer’s website).

    Some useful references:

    1. “A course in Arithmetic” by J. P. Serre (GTM Springer)
    2. “Quaternion Orders, Quadratic Forms, and Shimura Curves” by M. Alsina and P. Bayer (CRM monograph series)
    3. “The Arithmetic of Quaternion Algebra” ("Arithmétique des algèbres de quaternions") by M.-F. Vigneras (LNM Springer)

  • The goal of this course is to give a gentle introduction to some basic properties of quaternion algebra, quaternion orders, and some related topics including quadratic forms, modular forms and Shimura curves.

    Quaternion algebras appear naturally in many problems and have applications to many different subjects. In number theory, they have direct applications in the theories of quadratic forms, Shimura curves (and modular curves), and in the theory of modular forms through the Jacquet-Langlands correspondence and theta series. All the arithmetic properties of these objects play important roles in developing number theory.

7320 Ordinary Differential Equations. Prof. Estrada
  • 3:00-4:20 TT
  • Instructor: Prof. Estrada.
  • Prerequisite: Undergraduate advanced calculus, undergraduate complex variables, and core graduate analysis.
  • Text: Walter, W., Ordinary Differential Equations, Springer Verlag, 1998
  • 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. He.
  • 9:30-10:20 MWF
  • Instructor: Prof. He.
  • 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. Sundar.
  • 2:30-3:50 WF
  • Instructor: Prof. Sundar.
  • Prerequisite: Math 7311 or its equivalent
  • Text: Stochastic differential equations (Springer Universitext) 6th Ed. by Bernt Oksendal; Reference book: An introduction to stochastic differential equations (AMS Publ.) by L. C. Evans.
  • The course starts with an introduction to Brownian motion and continuous martingales. The construction, distributional and path properties, and the semigroup of a Brownian motion will be studied. Next, our objective is to construct stochastic integrals with respect to a Brownian motion. A central role in stochastic analysis is played by the It\^o formula with far-reaching consequences. After proving the formula, the course proceeds to solvability of stochastic differential equations driven by a Brownian motion. The fundamental connection between stochastic differential equations and a class of parabolic partial differential equations will be established.

7380 Topics in C* algebras. Prof. Olafsson.
  • 9:00-10:20 TT
  • Instructor: Prof. Olafsson.
  • Prerequisite: Math 7311
  • Text: Own notes and S. Sakai, C*-Algebras and W*-algebras, Springer
  • C*-algebras are involved in several branches of mathematics and physics. This includes non-commutative geometry, non-commutative probability. the theory of Toeplitz operators, representation theory as well as quantum field theory. In this course we will in the beginning cover the basic definitions and fact. In particular we show that every abelian C*-algebra with identity is isomorphic to the algebra of continuous functions on a compact topological space. We then discuss states and basic representation theory. We will then give a basic theory of projections and classification of C*-algebras into type I (the easiest one), type II (connected to analysis on discrete groups) and type III (those that show up in quantum field theory). If there is time, then we will also give an introduction to direct integrals and disintegration of C*-algebras. We will provide all details in most cases, but sometimes just give a brief discussion and overview. Participants are welcome to propose a topic. We will use our own notes. But there are several interesting textbooks around including the standard book by Dixmier “C*-algebras”. We will mostly follow the book by S. Sakai, “C*-Algebras and W*-Algebras”, Springer, Classics in Mathematics.

7384 Topics in Material Science: Spectral Theory and Mathematics of Linear Wave Phenomena, Prof. Han.
  • 10:30-11:20 MWF
  • Instructor: Prof. Rui Han.
  • Prerequisite: Math 7311, Undergraduate complex variables
  • Text: Notes
  • The material in this course centers around wave dynamics and scattering. The main components are spectral theory of differential and integral operators, Fourier analysis, scattering theory and resonance. We will prove the abstract spectral theorem for self-adjoint operators in Hilbert space and illustrate and motivate it by some concrete models (e.g. Schrodinger equations with potential well, waveguides, quantum graphs, random potentials, etc). Towards the end of the semester, we will also discuss some modern research papers and related open problems.

7390-1 Topics in Analysis. Prof. Walker.
  • 7390-1 1:30-2:50 TT
  • Instructor: Prof. Walker.
  • Text: Course notes. Other reference texts will be mentioned throughout the course.
  • Prerequisite: Theory of PDEs (MATH 4340, or MATH 7386 (e.g. weak formulations), or MATH 7380 (Intro. to Applied Math)), numerical solution methods for PDE (MATH 4066 (e.g. finite differences), or MATH 7325 (e.g. finite elements), or equivalent)
  • Description:

    Geometric Partial Differential Equations (PDEs) are ubiquitous in many applications in mathematics, science, and engineering. The hallmark of Geometric PDEs is a crucial structural component involving geometry central to the phenomena that the PDEs model. Examples are liquid crystals, PDEs on surfaces/manifolds, and free boundary/geometric evolution problems (e.g. surface tension driven motion, mean curvature flow, Willmore flow, and shape optimization).

    The topics to be covered are the following:

    1. Mathematical modeling of liquid crystals (namely the Landau-de Gennes (LdG) model) and its associated numerical analysis. Time permitting, optimal control of LdG will also be discussed.
    2. Applications of surface PDEs, and their associated numerical analysis. Relevant tools from shape differential calculus will be discussed.
    3. Discussion of selected papers in the literature.

    This is a seminar course, so grading will be mainly based on attendance and a few small homework assignments.

7390-2 Topics in Elliptic Partial Differential Equations. Prof. Zhu.
  • 7390-2 12:00-1:20 TT
  • Instructor: Prof. Zhu.
  • Prerequisite: Math 7311 or equivalent
  • Text: Elliptic Partial Differential Equations, Second Edition, by Qing Han and Fanghua Lin. ISBN-10: 0-8218-5313-9, ISBN-13: 978-0-8218-5313-9. Reference Texts (not required): Elliptic partial differential equations of second order by David Gilbarg and Neil S. Trudinger. Springer Verlag 2001. ISBN-10: 3540411607, ISBN-13: 9783540411604
  • This course presents basic methods to obtain a priori estimates for solutions of second order elliptic partial differential equations. Topics covered include weak and viscosity solutions, Hopf and Alexandroff maximum principles, Harnack inequalities, De Giorgi-Nash-Moser regularity theory, regularity (i.e. continuity and differentiability) of solutions, basic theory in homogenization. The course can be viewed as a continuation of MATH 7386 but no prior knowledge of PDEs is necessary. It provides some necessary background knowledge for the study in applied analysis.

7410 Graph Theory. Prof. Oporowski.
  • 11:30-12:20 MWF
  • 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 Combinatorial Optimization. Prof. Ding
  • 1:30-2:50 TT
  • Instructor: Prof. Ding.
  • Prerequisite: Math 4171 or equivalent
  • Text: None. Class notes will be distributed.
  • We will begin with classical results on solving linear inequalities and then obtain the min-max duality theorem for linear programming. Under this general framework we discuss various min-max relations in combinatorics. These include results like Menger theorem, max-flow-min-cut theorem, Konig theorems, and many more.

7512 Topology II. Prof. Dani.
  • 1:30-2:20 MWF
  • Instructor: Prof. Dani.
  • 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. Baldridge.
  • 10:30-11:50 TT
  • Instructor: Prof. Baldridge.
  • 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: Braids. Prof. Cohen.
  • 12:00-1:20 TT
  • Instructor: Prof. Cohen.
  • Prerequisite: Math 7510 Topology I
  • Text: no required text
  • Braid groups are of interest in topology and related areas both as primary objects of study, and by virtue of their close connections to other objects of interest, including knots and links, homeomorphisms of surfaces, and configuration spaces. This course will provide an introduction to the theory of braid groups, and will explore a number of these connections.

7590-2 Geometry and Physics. Prof. Zeitlin
  • 2:30-3:50 MW
  • 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.
  • 9:00-10:20 TT
  • Instructor: Prof. Sung.
  • 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. They 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

7999-n Assorted Individual Reading Classes
  • No additional information.

8000-n Assorted Sections of MS-Thesis Research
  • No additional information.

9000-n Assorted Sections of Doctoral Dissertation Research
  • No additional information.