Graduate Course Outlines, Summer 2019-Spring 2020

Summer 2019

MATH 7999-1: Problem Lab in Algebra —practice for PhD Qualifying Exam in Algebra.
Instructor:
Prerequisite: Math 7210.
Text: Online Test Bank.

MATH 7999-2: Problem Lab in Real Analysis—practice for PhD Qualifying Exam in Analysis.
Instructor:
Prerequisite: Math 7311.
Text: Online Test Bank.

MATH 7999-3: Problem Lab in Topology—practice for PhD Qualifying Exam in Topology.
Instructor:
Prerequisite: Math 7510.
Text: Online Test Bank.

Fall 2019

MATH 4997-1: Vertically Integrated Research: Buildings
12:00-1:20 TTh
Instructor: Profs. Achar and Sage
Prerequisites: Math 4200 and 4153
Text: Buildings—Theory and Applications by P. Abramenko and K. Brown
A building is a certain type of simplicial complex introduced by Jacques Tits to provide a geometric framework for studying simple algebraic groups. They have since turned out to have important applications to group theory, representation theory, geometry, and discrete mathematics. The simplest examples of buildings are (not necessarily finite) graphs with very strong regularity properties. For example, if F is a field, then the spherical building associated to SL_3(F) (the 3x3 matrices with entries in F and determinant one) is a bipartite graph with a vertex for each line and for each plane in F^3; two vertices are connected precisely when they correspond to a line contained in a plane. More generally, the building associated to SL_n(F) for n>3 is an (n-1)-dimensional simplicial complex with simplices corresponding to “flags”—chains of subspaces in F^n. In this course, we will provide an introduction to the theory of buildings, with a particular focus on concrete examples. We will also discuss various applications.

MATH 4997-2: Vertically Integrated Research: Invariants in Low-Dimensional Topology
10:30-11:50 TTh
Instructor: Profs. Vela-Vick and Wong
Prerequisites: None
Text: None. Suggested references will be provided during the course.
In recent years, research has found many interesting interactions between low-dimensional topology and contact and symplectic geometry. This has resulted in the discovery of invariants that can help us distinguish different knots and links, and explore the geometry of 3- and 4-dimensional manifolds. The study of these invariants represents a very active area of current research. In this vertically integrated research seminar, we will study some of these invariants, with a view towards computations and applications.

MATH 4997-3: Vertically Integrated Research: Parallel Computational Math
9:00-10:20 TTh
Instructor: Drs. Patrick Diehl and Hartmut Kaiser
Prerequisites:
Text:
See the course webpage Math 4997-3 for a detailed course description.

MATH 7001: Communicating Mathematics I
3:00-4:50 T Th
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.

MATH 7210: Algebra I
12:30-1:20 MWF
Instructor: Prof. Ng.
Prerequisites: Math 4200 or equivalent
Text: Abstract Algebra, 3rd ed; Dummit and Foote, Wiley 2003.
This is the first semester of the first year graduate algebra sequence, and covers the material required for the Comprehensive Exam in Algebra. Topics will include group actions and Sylow Theorems, finitely generated abelian groups; rings and modules: PIDs, UFDs, finitely generated modules over a PID, applications to Jordan canonical form.

MATH 7230: Class Field Theory.
10:30-11:50 TTh
Instructor: Prof. Mahlburg.
Prerequisites: Algebraic Number Theory is essential. A good understanding of Galois Theory at the level of Graduate Algebra II is recommended.
Text: Class Field Theory, by Nancy Childress, Springer Universitext, 2009. Freely available as an E-book through LSU Libraries.
Class Field Theory is the study of abelian extensions of number fields (which may be global or local). These extensions are described in terms of arithmetic invariants such as the ideal and ray class groups. One of the main results is Artin's Reciprocity Law, which generalizes quadratic reciprocity, and can be viewed analytically as a first case of Langlands Program.

MATH 7260: Homological Algebra.
9:00-10:20 TTh
Instructor: Prof. Achar.
Prerequisites:
Text:

MATH 7290: Modular Tensor Categories and Quantum Invariants.
10:30-11:20 MWF
Instructor: Profs. Ng and Wang
Prerequisites:
Text:
This course introduces modular tensor categories and the Reshetikhin-Turaev (RT) 3-dimensional topological quantum field theories (TQFTs). We will start with the theory of modular tensor categories and their graphical calculus. We will then give an axiomatic definition of TQFTs and discuss some examples arise from finite groups and quantum groups.

MATH 7311: Real Analysis I.
10:30-11:20 MWF
Instructor: Prof. Sundar.
Prerequisites: Math 4032 or 4035 or equivalent.
Text: Real Analysis, Modern Techniques and Their Applications, by G. B. Folland.
This is an standard course on measure theory and integration. After introducing measures, we discuss measurable functions, and develop integration of real and complex valued functions. An important example would be the Lebesgue integral on the line and n-dimensional Euclidean space. Next, we will present the connection between the Lebesgue integral and the Riemann integral, and study functions of bounded variation, absolute continuity, and Lebesgue differentiation theorems. Important topics such as Lebesgue convergence theorems, product measures, Fubini's theorem and the Radon-Nikodym derivative will be covered. We will give a short introduction of Banach and Hilbert spaces and then apply it to Lp spaces. The Riesz-Markov-Kakutani theorem will be discussed if there is sufficient time. A sampler of very good books on analysis and measure theory are:
A. Friedman: Foundations of Modern Analysis
L. Richardson: Measure and Integral: An Introduction to Real Analysis
H. L. Royden: Real Analysis
R. L. Wheeden and A. Zygmund: Measure and Integral: An Introduction to Real Analysis

MATH 7325: Finite Element Methods.
12:00-1:20 TTh
Instructor: Prof. Walker.
Prerequisites: Analysis (MATH 4031, 4032), Linear Algebra (MATH 7710)
Texts: (required): The Mathematical Theory of Finite Element Methods, 3rd Edition (2008) by S. Brenner, R. Scott; Instructor will also use course notes.
Reference Texts (not required): Finite Elements: Theory, Fast Solvers, and Applications in Solid Mechanics, 3rd edition (2007) by D. Braess; The Finite Element Method for Elliptic Problems, (Classics in Applied Mathematics, 2002) by Philippe G. Ciarlet.
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 of the FEM will also be addressed mainly via Python programming projects.
Additional topics may be introduced depending on the interests of the class.
Prerequisites: 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.

MATH 7350: Complex Analysis.
1:30-2:50 TTh
Instructor: Prof. Sage.
Prerequisites: Math 7311 or its equivalent.
Text: Narasimhan, R. and Nievergelt, Y., Complex Analysis in One Variable, second edition, Birkhauser, Boston, 2001.
This course is an introduction to complex analysis at the graduate level. Topics include holomorphic functions, covering spaces and the monodromy theorem, winding numbers, residues, Runge's theorem, the Mittag-Leffler and Weierstrass theorems on construction of functions with specified zeros or poles, the Riemann mapping theorem, and an introduction to Riemann surfaces.

MATH 7360: Probability Theory.
1030-11:50 TTh
Instructor: Prof. Ganguly.
Prerequisites: Familiarity with Measure theory (Math 7311) is desirable but not required. However a good mathematical maturity and a solid understanding of undergraduate analysis at the level of Math 4027-4032 is important.
Text:
The course is a self-contained introduction to modern probability theory. It starts from the concept of probability measures, random variables, and independence. Several important notions of convergence including almost sure convergence, convergence in probability, convergence in distribution will be introduced. Well-known limit theorems for sums of independent random variables such as the Kolmogorov's law of large numbers, Central Limit Theorem and their generalizations will be studied. Sums of independent, centered random variables form the prototype for an important class of stochastic processes known as martingales, and therefore play a major role in probability and other areas of mathematics. A part of the course will be devoted to understanding conditional probability which then forms the foundation for the theory of martingales. Brownian motion is an important example of a continuous-time martingale. If time permits, basic features of martingale theory and Brownian Motions will be briefly discussed. The course will not have any fixed text book, but some books will be cited for reference. It is important that the students take and read the class notes carefully.

MATH 7380: Spherical Harmonics and Integral Geometry.
8:30-9:20 MWF
Instructor: Prof. Rubin.
Prerequisites: Differential calculus in n-dimensional space (MATH 4035), Lebesgue integration (MATH 7311).
Text: Prof. Rubin will provide his notes following the book:
B. Rubin: Introduction to Radon transforms: With elements of fractional calculus and harmonic analysis (Encyclopedia of Mathematics and its Applications), Cambridge University Press (2015).
Syllabus: The focus of this course/seminar is introduction to the Fourier analysis on the unit sphere in the n-dimensional real Euclidean space and related problems of integral geometry. This topic plays an important role in PDE, harmonic analysis, group representations, mathematical physics, geometry, and many other areas of mathematics and applications.

MATH 7384: Topics in Material Science: Optical Metamaterials and Plasmonic Composites.
2:30-3:20 MWF
Instructor: Prof. Lipton.
Prerequisites: Any one of Math 2065, 3355, 3903, 4031, or 4038, or their equivalent.
Text:
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.

MATH 7386: Partial Differential Equations.
12:00-1:20 MW
Instructor: Prof. Bulut.
Prerequisites: Math 7311 (Real Analysis I) or equivalent, or consent of department.
Text: 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 Calculus of Variations: Euler-Lagrange equations, existence of minimizers, eigenvalues of self-adjoint elliptic operators.

MATH 7510: Topology I
9:30-10:20 MWF
Instructor: Prof. Dasbach.
Prerequisites: 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.

MATH 7520: Algebraic Topology.
11:30-12:20 MWF
Instructor: Prof. Cohen.
Prerequisites: Math 7512
Text: Algebraic Topology by Allen Hatcher (this can be obtained free online)
This is a continuation of Math 7512. We will discuss cup products in cohomology, Poincare and Lefshetz duality, and beginning homotopy theory.

MATH 7590: Trisections on smooth 4-manifolds.
1:30-2:50 TTh
Instructor: Prof. Baldridge.
Prerequisites: Differential Geometry (7550) and Topology II (7512)
Text: 4-Manifolds and Kirby Calculus by R. Gompf and A. Stipsicz, and papers by Peter Lambert-Cole on trisections
In this course, we study trisections on smooth closed 4-manifolds with the goal of understanding the minimal genus of all embedded smooth surfaces representing a given homology class (The Thom Theorem). A trisection of a 4-manifold is analogous to a Heegaard decompositions of a 3-manifold. It is a cutting-edge tool that has already shown great promise in understanding and proving some of the hardest problems in 4-manifold theory. This course will benefit students who want an overview of 4-manifold constructions and want to learn how to use advanced techniques in topology and geometry to study smooth manifolds.

Spring 2020

MATH 4997-1: Vertically Integrated Research: Buildings
12:00-1:20 TTh
Instructor: Profs. Achar and Sage
Prerequisites: Math 4200 and 4153
Text: Buildings—Theory and Applications by P. Abramenko and K. Brown
A building is a certain type of simplicial complex introduced by Jacques Tits to provide a geometric framework for studying simple algebraic groups. They have since turned out to have important applications to group theory, representation theory, geometry, and discrete mathematics. The simplest examples of buildings are (not necessarily finite) graphs with very strong regularity properties. For example, if F is a field, then the spherical building associated to SL_3(F) (the 3x3 matrices with entries in F and determinant one) is a bipartite graph with a vertex for each line and for each plane in F^3; two vertices are connected precisely when they correspond to a line contained in a plane. More generally, the building associated to SL_n(F) for n>3 is an (n-1)-dimensional simplicial complex with simplices corresponding to “flags”—chains of subspaces in F^n. In this course, we will provide an introduction to the theory of buildings, with a particular focus on concrete examples. We will also discuss various applications. . (This course continues the theme of the Fall 2019 semester, but that semester is not a prerequisite for this course, and new participants are welcome.)

MATH 4997-2: Vertically Integrated Research: Invariants in Low-Dimensional Topology
9:00-10:20 TTh
Instructor: Profs. Vela-Vick and Wong
Prerequisites: None
Text: None. Suggested references will be provided during the course.
In recent years, research has found many interesting interactions between low-dimensional topology and contact and symplectic geometry. This has resulted in the discovery of invariants that can help us distinguish different knots and links, and explore the geometry of 3- and 4-dimensional manifolds. The study of these invariants represents a very active area of current research. In this vertically integrated research seminar, we will study some of these invariants, with a view towards computations and applications.

MATH 4997-3: Vertically Integrated Research: Linear Algebra and Learning from Data
1:30-2:500 TTh
Instructor: Profs. Wolenski and Madden
Prerequisites:
Text:

MATH 7002: Communicating Mathematics II
3:00-4:50 T Th
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.

MATH 7211: Algebra II.
10:30-11:20 MWF
Instructor: Prof. Hoffman.
Prerequisites: Knowledge of Algebra at the level of Math 7210.
Text: No required text. The material we will cover can be found in
1. Dummitt and Foote, Abstract Algebra, 3rd edition
2. S. Lang, Algebra. 3rd edition, GTM 211.
3. M. Artin, Algebra, 2nd edition.
4. N. Jacobson, Algebra I, II.
5. Hungerford, Algebra, GTM 73.
There are two main topics
1. Galois Theory.
2. Representations of finite groups.
The first topic covers the theory of field extensions. The second is concerned with the Frobenius-Schur theory of finite-dimensional matrix representations of finite groups. Along the way we will treat the structure theory of finite-dimensional algebras.

MATH 7230: Introduction to Elliptic Curves and Modular Forms
9:30-12:20 MWF
Instructor: Prof. Tu.
Prerequisites: 1. Math 7210 (Algebra), Math 4036 (complex analysis) or equivalent are very helpful. You should be comfortable with meromorphic and analytic functions.
2. Know how to find resources and how to use computer algebra systems (such as Maple, Mathematica, Matlab).
Text: There is no required text, but the following references and web sources are useful.
1. Introduction to Elliptic Curves and Modular Forms (GTM) by Neal I. Koblitz
2. Elliptic Curves, Modular Forms, and their L-functions, by Álvaro Lozano-Robledo
3. A First Course in Modular Forms (GTM) by F. Diamond and J. Shurman.
4. Advanced Topics in the Arithmetic of Elliptic Curves by Joseph H. Silverman
Elliptic curves and modular forms appear naturally in connection with problems arising in many areas in math, and have applications to other broader fields such as information theory, computer science and physics. Together with theory of Galois representation, the arithmetic properties of elliptic curves and modular forms play crucial roles in the proof of Fermat's Last theorem. All of them have been extreme important in the developments of Number Theory.
The goal of this course is to give a gentle introduction to some basic properties of elliptic curves, modular forms, and their connection. In the later part of the semester, we will discuss some applications, advanced topics, and open problems, depending on audiences' interests.

MATH 7280: D-modules
1:30-2:50 TTh
Instructor: Prof. Sage.
Prerequisites: Math 7211
Text:
The theory of D-modules provides an algebraic approach to the study of linear partial differential equations. It has important applications to many fields of mathematics, including representation theory, singularity theory, and the Langlands program. In this theory, one studies solutions to systems of partial differential equations in terms of modules over rings of differential operators. For example, when the system involves n variables and has polynomial coefficients, the appropriate ring of differential operators is the Weyl algebra: the noncommutative algebra generated by the linear functions x₁,...,x_n and the corresponding partial derivatives ∂₁,...,∂_n.

In the first part of the course, we will study the Weyl algebra and its modules, i.e. D-modules on Cⁿ. We will then discuss D-modules on smooth complex algebraic varieties; here, it is necessary to introduce sheaf-theoretic methods. A main goal of the course is to discuss the Riemann-Hilbert correspondence--a vast generalization of Hilbert's 21st problem on the existence of linear differential equations with a prescribed monodromy group--in some special cases. Time permitting, we will discuss some applications to representation theory and to the geometric Langlands program.

MATH 7290: Group Schemes
11:30-12:20 MWF
Instructor: Prof. Casper.
Prerequisites: Math 7210,7211
Text: Milne Algebraic groups: the theory of group schemes of finite type over a field
This course is a modern exposition of the basic theory of affine group schemes of finite type over a field. It will cover basic definitions, examples, constructions, and theorems such as subgroups, products, quotients, isomorphism theorems, Hopf algebras, and representations. We will approximately follow the exposition of Milne's Algebraic Groups, for which the appropriate background is a little commutative algebra and some basic familiarity with algebraic geometry.

MATH 7320: Ordinary Differential Equations.
9:00-10:20 TTh
Instructor: Prof. Malisoff
Prerequisites: Advanced Calculus and Linear Algebra (preferably Math 7311 or equivalent)
Text: Ordinary Differential Equations and Dynamical System by Gerald Teschl, AMS Graduate Studies in Mathematics, Volume 140 , (2012) ISBN-10: 0-8218-8328-3 ISBN-13: 978-0-8218-8328-0
This course will cover qualitative theory of ordinary differential equations, and will include topics such as existence and uniqueness theory, dependence on initial conditions, linear systems, stability, Hamiltonian systems, and control systems.

MATH 7330: Functional Analysis.
3:30-4:20 MWF
Instructor: Prof. He.
Prerequisites: 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, operators on Hilbert spaces. We will then discuss the Hardy space and Toeplitz operators. We will follow loosely the GTM book by Ron Douglas.

MATH 7366: Stochastic Analysis.
1:30-2:20 MWF
Instructor: Prof. Kuo.
Prerequisites: Math 7360 or equivalent
Text: (1) Textbook. H.-H. Kuo: Introductory Stochastic Integration, Universitext, Springer 2006.
(2) Lecture notes for a new book under preparation.
In this course we will first give an overview of the Ito theory of stochastic integration, which deals with adapted stochastic processes. Then we will study an extension of the Ito theory to anticipating stochastic processes, which I introduced in 2008. The general stochastic processes arising from this new theory involve the Ito part (adapted) and the counterpart (instantly independent). A simple example of such stochastic process is given by the solution of a linear stochastic differential equation with anticipating initial condition. One application of this new theory is to analyze stock markets when investors have insider information. The lecture notes for this course, together with previous lecture notes, will be written up for a new book on stochastic integration.

MATH 7370: Lie Groups and Representations
10:30-11:20 MWF
Instructor: Prof. Olafsson.
Prerequisites: Math 7311. A basic knowledge of differential geometry is also helpful.
Text: We will not use any fixed textbook but mainly use our own lecture notes. The notes will be posted and upgraded regularly. Good introductory books on Lie groups include
1. J. Hilgert and K-H. Neeb: Structure and Geometry of Lie Groups
2. V. S. Varadarajan: Lie Groups, Lie Algebras, and their Representations
3. N. R. Wallach: Harmonic Analysis on Homogeneous Spaces.
This is an introductory course in Lie groups and homogeneous spaces needed for further research in harmonic analysis and representation theory We will mostly consider linear Lie groups which makes several proofs much easier, but the statement are still be valid for general Lie groups. In fact, every connected, finite dimensional Lie group is locally isomorphic to a linear Lie group.
The course starts with basic definitions of Lie groups and Lie algebras. We then discuss some examples. Then the exponential map, the Lie algebra of a closed linear groups, actions on manifolds and homogeneous spaces. Several examples will be given. The rest depends on how much time we have: finite dimensional representations, homogeneous vector bundles and the Plancherel formula for square integrable sections of homogeneous vector bundles over compact homogeneous spaces.

MATH 7375: Wavelets:
12:30-1:20 MWF
Instructor: Prof. Nguyen.
Prerequisites: Math 7311
Text: Introduction to Fourier Analysis and Wavelets by M. A. Pinsky, Graduate Studies in Mathematics, Volume 102, AMS, and instructor's own notes.
This course is a basic introduction to Fourier analysis and wavelets. We start with Fourier series of smooth and square integrable periodic functions. We then discuss the Fourier transform on the n-dimensional Euclidean space. Topics include convolution of functions, approximate identity, the Fourier transform of rapidly decreasing functions and tempered distributions. Applications include the Shannon sampling theorem, Heisenberg uncertainty principle, and the windowed/short time Fourier transform. We then introduce the basic idea behind the wavelet theory and, as an example, the Haar wavelet. The course then continues with the introduction of multiresolution analysis and the construction of various wavelets.

MATH 7384: Topics in Material Science: Inverse Problems
2:30-3:20 MWF
Instructor: Prof. Li.
Prerequisites: Advanced calculus, Multivariable calculus and Partial differential equations (all undergraduate level)
Text: None
The topic for this semester is inverse problems. We’ll introduce basic concepts and principles in inverse problems and reveal how X-ray tomography, impedance tomography, inverse transport problems and hybrid inverse problems work. The emphasis will be the physical derivations of the inversion methods, with a few classical and important theorems proved along the way. It is suitable for students in applied operator theory, applied analysis, applied PDEs and numerical analysis.

MATH 7410: Graph Theory.
1:30-2:20 MWF
Instructor: Prof. Oporowski.
Prerequisites:
Text:

MATH 7490: Combinatorial Optimization
10:30-11:50 TTh
Instructor: Prof. Ding.
Prerequisites: Math 4171 or equivalent.
Text: None
We will begin with classical min-max results like Menger theorem, max-flow-min-cut theorem, and Konig theorem. Then we establish a connection between these results and Integer Programming. Under this general framework, we discuss more min-max results concerning packing and covering of various combinatorial objects.

MATH 7512: Topology II.
10:30-11:50 TTh
Instructor: Prof. Vela-Vick.
Prerequisites: Maath 7510
Text: Algebraic Topology, by Allen Hatcher
This course covers the basics of homology and cohomology theory. Topics discussed including 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.

MATH 7550: Differential Geometry.
12:00-1:20 TTh
Instructor: Prof. Zeitlin.
Prerequisites: MATH 7210 and 7510; or equivalent.
Text: None. Instructor notes will be used.
This course is an introduction to the theory of manifolds. Topics to be covered include: differentiable manifolds, vector bundles, transversality, Sard's theorem, vector and tensor fields, differential forms, integration on manifolds, Stokes' theorem, de Rham cohomology, Lie groups and Lie group actions, elements of Riemannian geometry.

MATH 7590-1: Geometric Topology: Hyperbolic Geometry.
10:30-11:50 TTh
Instructor: Prof. Zimmer.
Prerequisites: Familiarity with basic algebraic topology (e.g. fundamental group, covering spaces) and basic real analysis (e.g. measures).
Text: None
In this course we will study geometric and dynamical problems in real hyperbolic geometry. After introducing real hyperbolic d-space, I plan on discussing the general theory of word hyperbolic groups, convex co-compact subgroups of the isometry group, Patterson-Sullivan theory, Bowen's rigidity theorem for the Hausdorff dimension of quasi-circles, the ergodicity of the geodesic flow on compact real hyperbolic manifolds, and Mostow's rigidity theorem.

MATH 7590-2: Modular categories and Reshetikhin-Turaev TQFTs
12:30-1:30 MWF
Instructor: Prof. Wang
Prerequisites: Maath 7290 (Fall 2019)
Text: Vladimir G. Turaev, Quantum invariants of knots and 3-manifolds (Third edition). De Gruyter Studies in Mathematics, 18. De Gruyter, Berlin, 2016.
This is a continuation of the course MATH 7290 on modular categories and Reshetikhin-Turaev TQFTs (RT-TQFTs) in Fall 2019. In this course, we will study the properties of the RT-TQFTs arising from SU(2) and finite groups in detail. If time permits, we will discuss the relation between the RT-TQFT and the Turaev-Viro state-sum TQFT.

MATH 7710: Advanced Numerical Linear Algebra.
11:30-12:50 MW
Instructor: Prof. Zhang.
Prerequisites: linear algebra, advanced calculus and some programming experience
Text: Fundamentals of Matrix Computations, Third Edition, D.S. Watkins, Wiley, 2010
This course will develop and analyze fundamental algorithms for the numerical solutions of problems in linear algebra. Topics include direct methods for general linear systems based on matrix factorization (LU, Cholesky and QR), iterative methods for sparse systems (Jacobi, Gauss-Seidel, SOR, steepest descent and conjugate gradient), and methods for eigenvalue problems (power methods, Rayleigh quotient iteration and QR algorithm).