Summer 2024
For Detailed Course Outlines, click on course numbers.
7999-1
Problem Sessions in Algebra—practice for PhD Qualifying Exam in Algebra
- Time:
- Instructor:
- Prerequisite:
- Text:
7999-2
Problem Sessions in Analysis—practice for PhD Qualifying Exam in Analysis
- Time:
- Instructor:
- Prerequisite:
- Text:
7999-3
Problem Sessions in Topology—practice for PhD Qualifying Exam in Topology
- Time:
- Instructor:
- Prerequisite:
- Text:
7999-4
Problem Sessions in Applied Math—practice for PhD Qualifying Exam in Applied Math
- Time:
- 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 2024
For Detailed Course Outlines, click on course numbers. Core courses are listed in bold.
4997-1
Vertically Integrated Research: Algebraic geometry for matroids. Profs. Achar and Bălibanu
- 11:30-12:20 MWF
- Instructor: Profs. Achar and Bălibanu
- Prerequisite: Math 4200 or Math 4153 or permission of the instructors
- Text: None
- Description: Matroids are objects in combinatorics that capture the key features of the notion of "linear independence" in linear algebra, while also having connections to graph theory, Galois theory, and other subjects. In 2022, June Huh was awarded a Fields Medal in part for his work on resolving a 50-year-old problem in matroid theory called the "Dowling-Wilson conjecture," also called the "top heavy conjecture." The proof involved importing ideas from topology and algebraic geometry, such as intersection cohomology, Kazhdan-Lusztig theory, and Hodge theory. In this course, we'll study some of the ingredients of this proof, and we'll see how to place the work of Huh and his collaborators in the broader context of recent developments in representation theory and algebraic geometry.
4997-2
Vertically Integrated Research: Groups with remarkable origins. Profs. Dani and Schreve
- 1:30-2:50 TTh
- Instructors: Profs. Dani and Schreve
- Prerequisite: Math 4200 or 4023
- Text: none
- Description: Groups are ubiquitous in Mathematics. A first course in Algebra usually includes examples such as the symmetric group, matrix groups, or groups that arise as symmetries of some objects (eg. tilings of a plane). This course will introduce a variety of infinite groups that have more unusual origins, such as dynamical systems, logic and computer science, and explore their often surprising geometric and algebraic properties. Topics will include Thompson’s, Self-similar, Lamplighter, and Baumslag-Solitar groups, which are all important examples in geometric group theory.
4997-3
Vertically Integrated Research on Geometry and Topology. Profs. Vela-Vick and ...
- 12:00-1:20 TTh
- Instructors: Profs. Vela-Vick and ...
- Prerequisite: A first course in linear algebra, experience with topology or abstract algebra is helpful but not required
- Text: 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.
7001
Communicating Mathematics I. Prof. Shipman and Dr. Ledet
- TTh 3:00-4:50
- Instructor: Prof. Shipman and Dr. Ledet
- 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. Hoffman
- 8:30-9:20 MWF
- Instructor: Prof. Hoffman
- Prerequisite: Math 4200 or its equivalent
- Text: Dummit and Foote, Abstract Algebra
- Description: 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
Topics in Number Theory: Arithmetic of CM Elliptic Curves and Modular Forms. Prof. Tu
- 10:30-11:20 MWF
- Instructor: Prof. Tu
-
Prerequisite:
1. Math 7210 (Algebra), Math 4036 (complex analysis) or equivalent are very helpful. You should be comfortable with meromorphic and analytic functions, groups theory, group actions, Galois theory, and module theory.
2. Know how to find resources and references to help you to understand the material better.
3. Know how to use computer algebra systems (such as Maple, Mathematica, MATLAB, SageMath, or Magma).
- Text: We will mainly follow chapters 1 and 2 in the book Advanced Topics in the Arithmetic of Elliptic Curves by Joseph H. Silverman.
- Description: Elliptic curves with complex multiplication (CM) have many special properties. The (arithmetic) properties of CM elliptic curves play an important role in the research of number theory and arithmetic geometry, especially in the class field theory of imaginary quadratic fields, modular forms, L-functions and Galois representations. The course will begin with basic definitions and examples of CM elliptic curves, review important properties in class field theory and modular forms, and will talk about the connection with Grossencharacters (Hecke characters). In the later part of the semester, we will discuss some applications, advanced topics, and open problems, depending on students’ interest.
7240
Algebraic Geometry. Prof. X Wang
- 2:30-3:20 MWF
- Instructor: Prof. X Wang
- Prerequisite:
-
Text: There are no required text, but the following references will be helpful:
Algebraic Geometry: Notes on a Course, Michael Artin, Graduate Studies in Mathematics 222
Algebraic Geometry, Robin Hartshorne, Graduate Texts in Mathematics, Spring-Verlag, 52.
- Description: This is a beginning graduate course in the topics of Algebraic Geometry, including the introduction to classical complex algebraic varieties and its modern form using the methods of schemes and cohomology. The course goal is to enable the students to understand the basic notions and properties in algebraic varieties in an affine or projective space over an algebraically closed field and to be familiar with the methods of schemes and cohomology to study topics in the classical theory of algebraic curves and surfaces.
7311
Real Analysis (a.k.a. Analysis I). Prof. Tarfulea
- 9:30-10:20 MWF
- Instructor: Prof. Tarfulea
- Prerequisite: Undergraduate real analysis
- Text: Stein and Shakarchi : Real Analysis
- Description: 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.
7325
Numerical Analysis and Applications. Prof. Walker
- 1:30-2:50 TTh
- Instructor: Prof. Walker
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Prerequisite: Analysis (MATH 4031, 4032), Numerical Linear Algebra (MATH 7710)
Some basic functional analysis and PDE theory (MATH 4340, or MATH 7386 or equivalent) will be reviewed. Prior exposure to numerical analysis (MATH 4065 or 4066) and Python would be useful but not mandatory.
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Text: Instructor will use course notes.
Various Reference Texts: The Finite Element Method for Elliptic Problems, (Classics in Applied Mathematics, 2002) by Philippe G. Ciarlet;
Finite Elements: Theory, Fast Solvers, and Applications in Solid Mechanics, 3rd edition (2007) by D. Braess;
The Mathematical Theory of Finite Element Methods, 3rd or 4th Edition (2008) by S. Brenner, R. Scott. -
Description:
The finite element method (FEM) is one of the most successful computational tools in dealing with partial differential equations (PDE) arising in science and engineering (solid and fluid mechanics, electromagnetism, mathematical physics, etc.). The formulation of the FEM, its properties, stability, and convergence will be discussed. Implementation and programming of the FEM will also be addressed mainly via Python programming projects. We will also use open source software for this.
Additional topics will be introduced depending on the interests of the class.
7350
Complex Analysis. Prof. Antipov
- 12:30-1:20 MWF
- Instructor: Prof. Antipov
- Prerequisite: Math 7311
- Text:Lecture notes and Complex Analysis by Elias Stein and Rami Shakarchi, Princeton Lectures in Analysis II.
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Description: Theory of holomorphic functions of one complex variable; path integrals, power series, singularities, mapping properties, normal families, other topics.
More specifically: 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. Wiener-Hopf method and the Riemann-Hilbert problem. Entire functions 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
- 11:30-12:20 MWF
- Instructor: Prof. Sundar
- Prerequisite: Math 7311
- Text: Probability and Stochastics by Erhan Cinlar
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Description: Probability spaces, random variables and expectations, independence, convergence concepts, laws of large numbers, convergence of series, law of iterated logarithm, characteristic functions, central limit theorem, limiting distributions, martingales.
This is a self-contained introduction to modern probability theory. Starting from the concept of probability measures, it would introduce random variables, and independence. After a study of various modes of convergence, the Kolmogorov strong law of large numbers and results on 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. Ganguly
- 12:00-1:20 TTh
- Instructor: Prof. Ganguly
- Prerequisite: Math 7360
- Text:
- Description: Brownian motion, basic stochastic calculus, applications to finance.
7380
Seminar in Functional Analysis: Spherical Harmonics and Integral Geometry. Prof. Rubin
- 10:30-11:20 MWF
- Instructor: Prof. Rubin
- Prerequisite: Real analysis (MATH 7311); Functional Analysis (MATH 7330)
- Text: B. Rubin, Introduction to Radon transforms: With elements of fractional calculus and harmonic analysis (Encyclopedia of Mathematics and its Applications), Cambridge University Press, 2015, ISBN-10: 0521854598, ISBN-13: 978-0521854597.
- Description: This course-seminar is focused on the Fourier analysis on the unit sphere and related operators of integral geometry in the n-dimensional real Euclidean space. Operators of this kind arise in PDE, harmonic analysis, group representations, mathematical physics, geometry, and many other areas of mathematics and applications.
7382
Introduction to Applied Mathematics. Prof. Massatt
- 9:00-10:20 MWF
- Instructor: Prof. Massatt
- Prerequisite: Simultaneous enrollment in Math 7311
- Text: Lecture notes
- Description: Overview of modeling and analysis of 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 differential and integral equations of physics.
7386
Theory of Partial Differential Equations. Prof. Bulut
- 10:30-11:50 TTh
- Instructor: Prof. Bulut
- Prerequisite: Math 7330
- Text: Partial Differential Equations by L. C. Evans
- Description: Sobolev spaces. Theory of second order scalar elliptic equations: existence, uniqueness and regularity. Additional topics such as: Direct methods of the calculus of variations, parabolic equations, eigenvalue problems.
7390
Seminar in Analysis: Optimization, Calculus of Variations, and Optimal Control. Prof. Wolenski
- 1:30-2:20 MWF
- Instructor: Prof. Wolenski
- Prerequisite:
- Text: Lecture Notes Fundamentals of Optimization by R.T. Rockafellar
-
Description: This course will cover Lecture Notes written by R.T. Rockafellar.
The rough outline of the course is:
I. What is Optimization?
II. Problem Formulation
III. Unconstrained Minimization
IV. Constrained Minimization
V. Lagrange Multipliers
VI. Games and Duality
X. Exercises
7490
Seminar in Combinatorics, Graph Theory, and Discrete Structures: Probabilistic Methods in Combinatorics. Prof. Z Wang
- 12:30-1:20 MWF
- Instructor: Prof. Z Wang
- Prerequisite: Familiarity with basic background of combinatorics and probability.
-
Text: Alon and Spencer, The probabilistic method, Wiley, 4th edition, ISBN: 978-1-119-06195-3.
Other useful resources:
Luke Postle (from University of Waterloo)’s youtube recordings on Probabilistic Methods.
Yufei Zhao (from MIT)’s lecture notes on Probabilistic Methods.
-
Description: This course is a graduate-level introduction to the probabilistic method, a fundamental and powerful technique
in combinatorics and theoretical computer science. The probabilistic method is a nonconstructive method,
primarily used in combinatorics and pioneered by Paul Erdös, for proving the existence of a prescribed kind of
mathematical object. The essence of the approach is as follows: to show the existence of some combinatorial
object, we prove by showing a certain random construction works with positive probability. This method has
now been applied to other areas of mathematics such as number theory, linear algebra, and real analysis, as
well as in computer science (e.g. randomized rounding), and information theory. The course will focus on
methodology as well as combinatorial applications.
Topics: Linearity of expectations, Alteration, Second Moment Methods, Chernoff bound, Lov´asz local lemma, Correlation inequalities, Janson inequalities, Concentration of measure, Entropy method, Container method, Random graphs.
7510
Topology I. Prof. Dani
- 10:30-11:50 TTh
- Instructor: Prof. Dani
- Prerequisite: Advanced Calculus (Math 4031)
- Text: Topology (2nd ed.) by James R. Munkres.
- Description: 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. A good supplementary reference is Chapter 1 of Algebraic Topology by Allen Hatcher, available online.
7560
Riemannian Geometry. Prof. Vela-Vick
- 9:00-10:20 TTh
- Instructor: Prof. Vela-Vick
- Prerequisite: MATH 7550
- Text: Riemannian Geometry by Manfredo P. do Carmo
- Description: Riemannian metrics and connections, geodesics, completeness, Hopf-Rinow theorem, sectional curvature, Ricci curvature, scalar curvature, Jacobi fields, second fundamental form and Gauss equations, manifolds of constant curvature, first and second variation formulas, Bonnet-Myers theorem, comparison theorems, Morse index theorem, Hadamard theorem, Preissmann theorem, and further topics such as sphere theorems, critical points of distance functions.
7590
Seminar in Geometry and Algebraic Topology: Combinatorial Algebraic Topology. Prof. Bibby
- 9:30-10:20 MWF
- Instructor: Prof. Bibby
- Prerequisite: Math 7512: Topology II
- Text: Combinatorial Algebraic Topology by Kozlov
- Description: We will learn about some fundamental tools for computing algebraic invariants of a topological space with an underlying combinatorial structure (eg. a partially ordered set). Topics may include, as time and interest allows, discrete Morse theory, shellability, sheaf theory, (homotopy) colimits, and spectral sequences.
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 2025
For Detailed Course Outlines, click on course numbers.
4997
Vertically Integrated Research: Topology TBA. Profs. TBA
4997(tentative)
Vertically Integrated Research: Topics in Machine Learning. Profs. Drenska and Wolenski
7002
Communicating Mathematics II. Prof. Shipman and Dr. Ledet
- TTh 3:00-4:50
- Instructor: Prof. Shipman and Dr. Ledet
- 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
- Hour TBA
- Instructor: Prof. Ng
- Prerequisite: Math 7210 or equivalent
- Text:
- Description: Fields: algebraic, transcendental, normal, separable field extensions; Galois theory, simple and semisimple algebras, Wedderburn theorem, group representations, Maschke’s theorem, multilinear algebra.
7230
Topics in Number Theory: Hypergeometric Functions. Prof. Long
- Hour TBA
- Instructor: Prof. Long
- Prerequisites:
- Text:
- Description:
7290
Seminar in Algebra and Number Theory: Derived Algebraic Geometry. Prof. Achar
- Time TBA
- Instructor: Prof. Achar
- Prerequisite:
- Text:
- Description:
7290
Seminar in Algebra and Number Theory. Prof. He
- Time TBA
- Instructor: Prof. He
- Prerequisite:
- Text:
- Description:
7320
Ordinary Differential Equations. Prof. Wolenski
- Time TBA
- Instructor: Prof. Wolenski
- Prerequisite: Undergraduate ODEs and analysis
- Text: I will begin with my notes and then follow the text An Introduction to Dynamical Systems: Continuous and Discrete (Pure and Applied Undergraduate Texts), 2nd Edition, by R. Clark Robinson
- Description: Existence and uniqueness theorems, approximation methods, linear equations, nonlinear equations, stability theory; other topics such as boundary value problems.
7330
Functional Analysis (a.k.a. Analysis II). Prof. Estrada
- Time TBA
- Instructor: Prof. Estrada
- Prerequisite: Math 7311 or equivalent and undergraduate complex variables
- Text: Methods of modern mathematical physics, vol 1: Functional analysis” by Reed and Simon
- Description: Banach spaces and their generalizations; Baire category, Banach-Steinhaus, open mapping, closed graph, and Hahn-Banach theorems; duality in Banach spaces, weak topologies; other topics such as commutative Banach algebras, spectral theory, distributions, and Fourier transforms.
7366
Stochastic Analysis. Prof. Sundar
- Time TBA
- Instructor: Prof. Sundar
- Prerequisite: Probability theory at the level of Math 7360. Alternatively, a good knowledge of advanced undergraduate probability theory including the concept of stochastic processes and graduate level analysis is also acceptable.
- Text: Stochastic differential equations by Bernt Oksendal
- Description: 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. Nguyen
- Time TBA
- Instructor: Prof. Nguyen
- Prerequisite: Math 7311.
- Text:
- Description:
7384
Topics in Material Science. Prof. Lipton
- Time TBA
- Instructor: Prof. Lipton
- Prerequisite: Real analysis and undergraduate complex variables
- Text:
- Description:
7390
Seminar in Analysis: Topics in Dispersive Partial Differential Equations. Prof. Bulut
- Time TBA
- Instructor: Prof. Bulut
- Prerequisite: Math 7311 and 7386
- Text:
- Description:
7390
Seminar in Analysis: Prof. Wan
- Time TBA
- Instructor: Prof. Wan
- Prerequisite:
- Text:
- Description:
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7410
Graph Theory. Prof. Z Wang
- TTh 10:30-11:50
- Instructor: Prof. Z Wang
- Prerequisite: Math 2085 and Math 4039, or equivalent.
- Text:
- Description: Matchings and coverings, connectivity, planar graphs, colorings, flows, Hamilton graphs, Ramsey theory, topological graph theory, graph minors.
7512
Topology II. Prof. Cohen
- TTh 9:00-10:20
- Instructor: Prof. Cohen
- Prerequisite: Math 7510
- Text: Algebraic Topology by Hatcher
- Description: 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
- TTh 12:00-1:20
- Instructor: Prof. Baldridge
- Prerequisite: Math 7210 and 7510; or equivalent.
- Text: An Introduction to Manifolds by L. Tu.
- Description: Manifolds, vector fields, vector bundles, transversality, deRham cohomology, metrics, other topics.
7590
Seminar in Geometry and Algebraic Topology: L2 Homology. Prof. Schreve
- Time TBA
- Instructor: Prof. Schreve
- Prerequisite:
- Text:
- Description:
7590
Seminar in Geometry and Algebraic Topology: Gromov-Witten Theory. Prof. Zeitlin
- Time TBA
- Instructor: Prof. Zeitlin
- Prerequisite:
- Text:
- Description:
7710
Advanced Numerical Linear Algebra. Prof. Brenner
- Time TBA
- Instructor: Prof. Brenner
- Prerequisite: linear algebra, advanced calculus, some programming experience (but not required)
- Text: Fundamentals of Matrix Computations, 3rd edition, by D. S. Watkins.
-
Description: This course will develop and analyze linear algebra algorithms. Fundamentals will be discussed, i.e.:
Basic tools: norms, projectors, spectral theorem, singular value decomposition. Direct methods: LU factorization, Cholesky, least squares problem, QR factorization. Iterative methods: Jacobi, Richardson, Gauss-Seidel, successive over-relaxation, steepest descent, conjugate gradient. Eigenvalue problems: power methods, Rayleigh quotient iteration, deflation, QR algorithm.
As time permits, we may do some machine learning problems/examples, e.g. curve fitting, Bayesian viewpoint, classification.
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.