Graduate Course Outlines, Summer 2018-Spring 2019

Summer 2018

  • 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 2018

  • MATH 4997-1: Vertically Integrated Research: The representation theory of the Lie algebra 𝔰𝔩2(ℂ)
  • 1:30-2:50 TTh
  • Instructor: Profs. Achar and Sage
  • Prerequisites: Math 4200 and 4153
  • Text: Lectures on 𝔰𝔩2(ℂ)-modules by V. Mazorchuk
  • A representation of a Lie algebra L is a Lie algebra map from L to the space of linear maps from a vector space V to itself. The representation theory of semisimple Lie algebras is of fundamental importance in many areas of mathematics and physics, including number theory, differential equations, and quantum mechanics. For example, the finite-dimensional irreducible representations of the simplest example, 𝔰𝔩2(ℂ) (the 2x2 complex matrices with trace 0) arise naturally in studying the quantum-mechanical states of the hydrogen atom. The goal of this course is to provide an introduction to representation theory by focusing exclusively on 𝔰𝔩2(ℂ). In particular, we will give a complete description of the irreducible representations of 𝔰𝔩2(ℂ) (including the infinite-dimensional ones). No prior familiarity with Lie algebras or representation theory is required.
  • 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, many powerful invariants have been invented that allow us to better understand the topology and geometry of low-dimensional spaces. For example, they can help us distinguish different knots and links, and explore the geometry of 3-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 6303-2: Implementing Curriculum Standards for Mathematics in High School: Statistical Reasoning
  • 3:30-6:00 Tuesday. Second Floor of Prescott Hall
  • Instructor: Prof. Ferreyra.
  • Goal: be ready to teach Statistical Reasoning and AP Statistics in high school. This is also a course for undergraduate and graduate students who never took a statistics course and want to learn it for understanding and multiple applications.
  • Prerequisite: The course Statistical Reasoning for high schools only has high school Algebra I as prerequisite for high school students because it is a self-contained course. The requirements for MATH 6303 are to have sufficient pedagogical, math reasoning skills (as in the Standards of Mathematical Practice), and content knowledge to teach other high school math courses including dual enrollment college algebra. Calculus will not be used/taught in MATH 6303.
  • Statistical Reasoning is a new course written for high schools following the Louisiana Student Standards for the domain of Statistics and Probability and the Standards of Mathematical Practice. Approximately 100 detailed lessons of 50 minutes have been written and organized in 7 units. These lessons will be available to students of MATH 6303.

    The content of the high school Statistical Reasoning course is approximately the first 75% of the content of the AP Statistics course. The technology taught and utilized in the course is Microsoft Excel.

    MATH 6303 will model high school teaching. Class time will be used to go over the lessons that have been written. Students will also “teach” some of these lessons to their peers and will prepare additional materials for the high school classroom such as activities, homework and assessments. Additionally, students will provide editorial and feedback comments on some of the available lessons. Students will also create a few lessons on topics which are in the AP Statistics course but not in the Statistical Reasoning course. Some students may also choose to work on converting Microsoft Excel lessons into lessons using Google sheets.

    Students need to purchase an AP Statistics preparation book (around $15).

    Tests of the type encountered in Praxis II will count for 50% of the grade. The remaining 50% will be awarded for in-class presentations, and preparation, editing and writing of lessons and assessments. The amount of assigned work will depend on the number of credits the student has registered for.

  • 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
  • 12:00-1:20 TTh
  • Instructor: Prof. Tu.
  • Prerequisites: Math 4200 Algebra I, or equivalent
  • Text: Abstract Algebra, 3rd ed; Dummit and Foote, Wiley 2003. References:
    1. A first course in Abstract Algebra; Fraleigh
    2. Algebra: An Approach via Module Theory; Adkins and Weintraub
  • This is the first semester of the first year graduate algebra sequence, and covers the material required for the Comprehensive Exam in Algebra. It will cover the basic notions of group, ring, module, and field theory. Topics will include GROUPS: finitely generated abelian groups, group actions, and the Sylow theorems; RINGS and MODULES: Euclidean domains, principal ideal domains, unique factorization domains, polynomial rings, and modules over PIDs; FIELDS: vector spaces, applications to Jordan canonical form, field extensions, and finite fields.
  • MATH 7230: Algebraic Number Theory.
  • 1:30-2:20 MWF
  • Instructor: Prof. Mahlburg.
  • Prerequisites:
  • Text:
  • MATH 7240: Algebraic Geometry.
  • 10:30-11:20 MWF
  • Instructor: Prof. Casper.
  • Prerequisites:
  • Text:
  • MATH 7311: Real Analysis I.
  • 10:30-1150 TTh
  • Instructor: Prof. Olafsson.
  • Prerequisites: Math 4032 or 4035 or equivalent.
  • Text: Real Analysis, Modern Techniques and Their Applications, by G. B. Folland.
  • 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 then apply that to the Lp spaces. Further topics include functions of bounded variations Lebesgue differentiation theorems, Lp and its dual, and the Riesz-Markov-Saks-Kakutani theorem and Fourier series may be presented if there is sufficient time.

    We will mostly follow the book by Folland. We will also hand out notes as needed. There are several other very good books on analysis and measure theory:

    1. L. Richardson: Measure and Integral: An Introduction to Real Analysis.
    2. P. R. Halmos: Measure Theory (Graduate Text in Mathematics)
    3. A. Friedman: Foundations of Modern Analysis. Dover
  • MATH 7350: Complex Analysis.
  • 12:00-1:20 TTh
  • Instructor: Prof. Antipov.
  • Prerequisites: 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. The Fourier transform in the complex plane. Entire function including Jensen's formula and infinite products. Conformal mapping including the Riemann mapping theorem and the Schwarz-Christoffel integral. Riemann surfaces of algebraic functions (this topic is not covered by the book, and notes will be available).
  • MATH 7360-2: Probability Theory.
  • 11:300-12:20 MWF
  • Instructor: Prof. Sundar.
  • Prerequisites: Math 7311 or equivalent
  • Text: Probability and Measure by Patrick Billingsley
  • This is a self-contained introduction to modern probability theory. It starts from the concept of probability measures, and introduces random variables, distributions and independence. A study of various modes of convergence is presented. Next, the course delves into well-known limit theorems such as the Kolmogorov strong law of large numbers, three-series theorem and the law of the iterated logarithm. Weak convergence of probability measures will be discussed in detail, which leads us 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. Martingales and Brownian motion will be studied and their essential features discussed.
  • MATH 7380: Topics in Spectral Geometry and Partial Differential Equations.
  • 1:30-2:50 MWF
  • Instructor: Prof. Zhu.
  • Prerequisites: Some knowledge of real analysis Math 7311 is helpful, but it will be reviewed in the course.
  • Text: Class notes and references will be provided
  • Eigenproblems are the most basic and important equations in the study of partial differential equations (PDEs). Understanding of any useful properties for eigenfunctions and eigenvalues contributes the development of PDEs. This course will present some basic theory of PDEs and some topics in spectral geometry. The first part of the course will discuss the variational and non-variational techniques for PDEs, including Euler-Lagrange equation, fixed point theorem, and method of subsolutions and supersolutions. The second part will cover some background knowledge of Riemannian geometry and analysis of Laplace-Beltrami operator. Topics include heat kernel, isoperimetric inequality, local and global properties for eigenfunctions. No prior knowledge of PDEs is required (necessary concepts will be reviewed). Grade in the course will be based on student engagement and a final presentation. This course is suitable for everyone interested in PDEs, spectral theory, harmonic analysis and geometric analysis.
  • MATH 7384: Topics in Material Science:
  • 10:30-1150 TTh
  • Instructor: Prof. Shipman. and Prof. Junshan Lin (Auburn University)
  • Topic: Mathematics of linear wave phenomena
  • Prerequisites:
  • Text:
  • This course will be taught jointly by Profs. Stephen Shipman and Junshan Lin (Auburn University) in a live-streaming setting. Students from any SEC university may enroll. It is intended to be a two-semester course (although the first semester may be taken in isolation). The second semester will be offered in the spring of 2019.

    The material in this course builds toward open problems in mathematical physics centered around wave dynamics and scattering in electromagnetics and acoustics. The mathematical topics form a coherent body of theory and techniques. The main components are the partial differential equations (PDE) of electromagnetics and acoustics and other derived phenomena in complex media; integral equations and boundary-integral representations of solutions to PDE; Fourier analysis and the residue calculus for the study of scattering and resonance; spectral theory of differential and integral operators illustrated and motivated by examples; and asymptotic analysis. The second semester of the course will concentrate on specific problems that are motivated by modern scientific and mathematical research. A goal is that students will be able to understand open problems in this area and be equipped with the basic mathematical tools to begin trying to solve them.

  • MATH 7386: Partial Differential Equations.
  • 2:30-3:20 MWF
  • Instructor: Prof. Lipton.
  • Prerequisites: Math 2057 or equivalent
  • Text: Partial Differential Equations by Lawrence C. Evans; Chapters 2 - 10.
  • Description: Partial differential equations, both linear and nonlinear describe physical laws that govern fluid dynamics, electromagnetics, gravitation, solid mechanics, quantum mechanics, celestial mechanics, etc. Methods of explicit solution together with the theory of existence and uniqueness of solution are indispensable tools for the rational solution (computational or otherwise) of problems in economics, physics and engineering. Topics Covered:

    (1) Hyperbolic, Parabolic, and Elliptic Partial Differential Equations
    (2) Examples: The Wave Equation, The Heat Equation, The Laplace Equation
    (3) Sobolev spaces and existence and uniqueness of solution.
    (4) Spectral theory
    (5) Banach fixed point theorem
    (6) Subdifferentials and nonlinear semigroups
    (7) Hamilton Jacobi Equations

  • MATH 7390-2: Variational Inequalities.
  • 9:00-10:20 TTh
  • Instructor: Prof. Sung.
  • Prerequisites: Functional Analysis (MATH 7330)
  • Text: An Introduction to Variational Inequalities and Their Applications by David Kinderlehrer and Guido Stampacchia (SIAM 2000)
  • The theory of variational inequalities treats optimization problems with inequality constraints. In this course we study the existence, uniqueness and regularity of the solutions of elliptic and parabolic variational inequalities. Applications and numerical methods will also be discussed.
  • MATH 7390-3: Variational Analysis and Dynamic Optimization.
  • 12:30-1:20 MWF
  • Instructor: Prof. Wolenski.
  • Prerequisites: Advanced Calculus, Linear Algebra, or their equivalent; permission of the instructor.
  • Text: No formal text, but we will draw from several sources.
  • Variational Analysis is a relatively new subject that is in effect the “mathematics of optimization”. It introduces differentiable-like concepts and develops ideals that allow for a systematic treatment of functions and sets that may not be differentiable in the classical sense. The beginning third of the course will introduce normal cones of closed sets and proximal subgradients of lower semicontinuous functions within the background of a review of finite-dimensional optimization. The second third of the course will cover the classical calculus of variations, and the final third will combine the first two topics with a modern treatment of optimal control theory.
  • MATH 7510: Topology I
  • 8:30-9:20 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.
  • 12:30-1:20 MWF
  • Instructor: Prof. Litherland.
  • Prerequisites: Math 7510 and 7512
  • Text: Allen Hatcher, Algebraic Topology,CUP 2001. A free electronic version is available online.
  • This course continues the study of algebraic topology begun in MATH 7510 and MATH 7512. While MATH 7510 developed the theory of fundamental groups and MATH 7512 developed homology theories for topological spaces the focus of this course will be cohomology theory which is dual to homology theory. However, one of the advantages of developing cohomology theory of spaces is that they are naturally equipped with a ring structure.
  • MATH 7590: Heegaard Floer Homology.
  • 1:30-2:50
  • Instructor: Prof. Wong.
  • Prerequisites: 7510 Topology I recommended, and 7550 Differential Geometry recommended
  • Text: None. Suggested references will be provided during the course.
  • Heegaard Floer homology is an invariant for 3-manifolds, as well as knots and links in them, that has proved to be very powerful since its inception in 2001, capturing plenty of geometrical information. Although they were originally defined analytically, combinatorial algorithms for effective computation have been found in many cases. In this course, we will develop the invariant from both a theoretical and a computational perspective, with a view towards open problems in the subject.

Spring 2019

  • MATH 4997-1: Vertically Integrated Research: Combinatorics and geometry of flag manifolds
  • 12:00-1:20 TTh
  • Instructor: Profs. Achar and Sage
  • Prerequisites: Math 4200 and Math 4153 or permission of the instructors
  • Text:
  • A flag manifold is a generalization of projective space (the set of lines in a vector space) and of Grassmannians (the set of linear subspaces of some fixed dimension). Flag manifolds have rich connections to representation theory and combinatorics, especially through classical 19th century problems of enumerative geometry that are collectively known as "Schubert calculus." In this course, we will study both classical and modern aspects of this theory, including recent work on invariants of Schubert varieties. In particular, we will discuss how pattern avoidance in permutations can be used to describe aspects of the geometry of Schubert varieties.
  • MATH 4997-2: Vertically Integrated Research:
  • 12:00-1:20 TTh
  • Instructor: Profs. Vela-Vick and Wong
  • 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.
  • 11:30-12:20 MWF
  • Instructor: Prof. Yakimov.
  • Prerequisites:
  • Text:
  • MATH 7230: Topics in Number Theory
  • 9:30-10:20 MWF
  • Instructor: Profs. Hoffman. and Long.
  • Prerequisites: Basic Algebra and Topology; this is covered in Math 7210, 7510. Complex Analysis - familiarity with analytic functions; this is covered in Math 7350. It is helpful, but not required to have some exposure to Number Theory and Algebraic Geometry. There will be courses taught on these in Fall 2018 (Math 7230, 7240).
  • Text: No required text, but much of the material will be drawn from
    1. A First Course in Modular Forms (Graduate Texts in Mathematics, Vol. 228), Diamond & Shurman
    2. A Classical Introduction to Modern Number Theory, Ireland & Rosen, (Graduate Texts in Mathematics) (v. 84)
    3. An Introduction to the Langlands Program, Bernstein & Gelbart (eds)
  • Description: The focus is on L-functions. These are at the center of much research today. Themes: Euler products; analytic continuation and functional equations; special values; relations among different kinds of L-functions. Briefly there are two broad classes of L-functions: those that arise from automorphic forms and those that arise from geometry. There are conjectural relations between the L-functions of the first kind with those of the second kind.

    Outline:

    1. Dirichlet L-functions, analytic continuation and functional equation.
    2. The L-functions defined by modular forms.
    3. Zeta functions of varieties over finite fields; Weil Conjectures.
    4. Hasse-Weil zeta functions.
    5. Conjectural relations: Wiles's theorem, Langlands program.

    Some special topics for student projects:

    1. Tate's thesis.
    2. Adelic viewpoint: automorphic forms for GL(2).
    3. L-functions of Galois representations.
    4. Special values — Birch and Swinnerton-Dyer conjecture.
  • MATH 7250: Representation Theory
  • 3:30-4:50 MW
  • Instructor: Prof. Zeitlin.
  • Prerequisites: Math 7211, some familiarity with basic notions of differential and algebraic geometry is welcome, but not required.
  • Text: None. Notes will be distributed
  • This course is an introduction to modern representation theory. We will start with covering basic aspects of representation theory of finite groups, structure theory of semisimple Lie algebras and their representations. Then we will discuss more advanced topics, such as quantum groups, Kac-Moody algebras, vertex algebras and their applications.
  • MATH 7290: An introduction to the geometric Langlands Program
  • 1:30-2:50 TTh
  • Instructor: Prof. Sage.
  • Prerequisites: The first year graduate algebra sequence or permission of the instructor
  • Texts:
    1. Langlands correspondence for loop groups by E. Frenkel
    2. Isomonodromic deformations and Frobenius manifolds--An introduction by C. Sabbah
  • The geometric Langlands program is an outgrowth of the classical Langlands program in number theory. The latter (in the case of the rational numbers) describes a conjectural relationship between data involving the absolute Galois group of Q and the representation theory of real and p-adic groups. The geometric version, on the other hand, relates systems of differential equations with meromorphic coefficients to the representation theory of certain infinite-dimensional Lie algebras called affine Kac-Moody Lie algebras.

    The goal of this course is to give a gentle introduction to some main ideas of the geometric Langlands program, with an emphasis on concrete examples. We will also discuss the number-theoretic motivation for the geometric conjectures. We will concentrate on GL(n) and indeed primarily on GL(2).

  • MATH 7320: Ordinary Differential Equations.
  • 10:30-11:50 TTh
  • Instructor: Prof. Shipman
  • Prerequisites: Undergraduate advanced calculus, undergraduate complex variables, and core graduate analysis.
  • Text: Various selections from the literature
  • The field of ODEs is pervasive in science and mathematics and can be fascinating and beautiful, as opposed to the contrary stereotype. We will study both linear and nonlinear ODEs. These are genuinely different areas of mathematics, as the questions that are investigated are very different in nature.

    In the linear theory, one studies problems of spectral theory of differential operators, such as Schrödinger operators with different kinds of potentials on different domains. This invokes delicate properties of entire and meromorphic functions, moment problems, asymptotic analysis, and of course linear algebra. New problems in inverse spectral theory continue to be interesting--where we seek to characterize the differential operators that possess certain spectral data. I also want to include some more specialized topics, such as linear systems governed by indefinite quadratic forms--this is a part of linear algebra that seems to be new to most people.

    The overarching concepts for the nonlinear theory are flows of vector fields and dynamical systems. Upon that basis, one studies diverse phenomena such as bifurcations, separation of time scales, bursting (such as in neurobiology), hysteresis, stability, control systems, chaos, strange attractors, and the Noether theorem on conservation laws corresponding to continuous symmetries. We will introduce these notions and study some of them in depth.

    A semester course cannot do any justice to all of these topics. My goal is twofold: (1) to present the foundational rigorous theory of ODEs, and (2) to introduce a broad variety of topics in ODEs that highlight what makes the field interesting.

  • MATH 7330: Functional Analysis.
  • 3:00-4:20 MW
  • Instructor: Prof. Estrada.
  • Prerequisites: Math 7311 or its equivalent.
  • Text: Trèves, F., Topological Vector Spaces, Distributions, and Kernels; Dover, 2006.
  • A standard first course in functional analysis. Topics include Banach spaces, Hilbert spaces, Banach algebras, topological vector spaces, spectral theory of operators and the study of the topology of the spaces of distributions.
  • MATH 7366: Stochastic Analysis.
  • 10:30-11:30 MWF
  • Instructor: Prof. Kuo.
  • Prerequisites: Math 7360 or equivalent
  • Text: Class notes and the book Introductory Stochastic Integration by H.-H. Kuo, Universitext, Springer 2006
  • In this course we will first give a quick overview of the Ito theory of stochastic integration, which deals with nonanticipating stochastic processes. Then we will study a new theory of stochastic integration which I introduced in 2008. This new theory is a real extension of the Ito theory to include anticipating stochastic processes. In fact, the general stochastic processes arising from this new theory involve the Ito part (nonanticipating) 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.
    I will prepare a set of lecture notes and propose several research problems for further investigation of this new theory.
  • MATH 7380: Topics in Elliptic Partial Differential Equations
  • 1:30-2:20 MWF
  • Instructor: Prof. Nguyen.
  • Prerequisites: 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):
    1. Elliptic partial differential equations of second order by David Gilbarg and Neil S. Trudinger. Springer Verlag 2001. ISBN-10: 3540411607, ISBN-13: 9783540411604.
    2. Fully Nonlinear Elliptic Equations by Luis A. Caffarelli and Xavier Cabre. ISBN-10: 0821804375, ISBN-13: 9780821804377.
  • This course presents basic methods to obtain a priori estimates for solutions of second order elliptic partial differential equations in both divergence and non-divergence forms . Topics covered include weak and viscosity solutions, Hopf and Alexandroff maximum principles, Harnack inequalities, De Giorgi-Nash-Moser regularity theory, continuity and differentiability of solutions. The course can be viewed as a continuation of MATH 7386 but no prior knowledge of PDEs is necessary.
  • MATH 7384-1: Topics in Material Science:
  • 1:30-2:50 TTh
  • Instructor: Prof. Shipman.
  • Prerequisites:
  • Text:
  • MATH 7390: Topics Course on Convex and Stochastic Optimization
  • 10:30-11:50 TTh
  • Instructor: Prof. Zhang.
  • Prerequisites: Math2057 Multidimensional Calculus, Math2085 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 with stochastic feature. Tentative topics include Convex sets, Convex functions, Duality theory, gradient and stochastic gradient methods for solving convex and nonconvex optimization problems.
  • MATH 7410: Graph Theory.
  • 1:30-2:20 MWF
  • Instructor: Prof. Oporowski.
  • Prerequisites:
  • Text:
  • MATH 7490: Tutte Polynomial for Matroids and Graphs.
  • 8:30-9:20 MWF
  • Instructor: Prof. Oxley.
  • Prerequisites: Math 7410 and 7490 (Matroid Theory) or permission of the department.
  • Text: Matroid Applications edited by Neil White (Chapter 6: The Tutte Polynomial and its Applications by Thomas Brylawski and James Oxley)
  • The theory of numerical invariants for matroids is one of many aspects of matroid theory having its origins within graph theory. Most of the fundamental ideas in matroid invariant theory were developed from graphs by Veblen, Birkhoff, Whitney, and Tutte when considering colorings and flows in graphs. This course will introduce the Tutte polynomial for matroids and will consider its applications in graph theory, coding theory, percolation theory, electrical network theory, and statistical mechanics.
  • MATH 7512: Topology II.
  • 12:30-1:20 MWF
  • Instructor: Prof. Gilmer.
  • Text: Algebraic Topology by A. Hatcher, Cambridge Univ. Press (This is available for free download in pdf format or may be purchased as a book)
  • We will discuss the homology groups of topological spaces. To a topological space, one associates a sequence of abelian groups called their homology groups. To a continuous map, one associates a sequence of group homomorphisms in a functorial way. One application of homology is the Brouwer fixed point theorem which asserts any continuous map from an n-dimensional disk to itself has a fixed point. One also has a higher dimensional version of the Jordan curve theorem. We will learn to calculate homology groups in a variety of ways. If time permits, we will begin to discuss cohomology as well.
  • MATH 7550: Differential Geometry.
  • 9:00-10:20 TTh
  • Instructor: Prof. Baldridge.
  • Prerequisites: 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.
  • MATH 7590-1: Geometric Topology: Symplectic Geometry.
  • 1:30-2:20 TTh
  • Instructor: Prof. Vela-Vick.
  • Prerequisites: Math 7510; co-requisite: Math 7512, 7550
  • Text: Introduction to Symplectic Topology, by Dusa McDuff and Dietmar Salamon
  • An introduction to symplectic geometry and topology. Topics include symplectic manifolds, Lagrangian subspaces, the Moser theorem and Darboux theorem, Hamiltonian diffeomorphisms and flows, almost complex structures and complex vector bundles. Basic construction techniques for symplectic manifolds will also be discussed.
  • MATH 7710: Advanced Numerical Linear Algebra.
  • 9:00-10:20 TTh
  • Instructor: Prof. Walker.
  • Prerequisites: linear algebra, advanced calculus, some programming experience
  • Text: Main: Fundamentals of Matrix Computations (3rd edition), by D. S. Watkins
    Supplemental: Pattern Recognition and Machine Learning (Information Science and Statistics), by C. Bishop
    Reference: Numerical Linear Algebra, 1st Edition, by N. Trefethen
    Standard Ref: Matrix Computations, by G. Golub and C. van Loan
  • This course will develop and analyze linear algebra algorithms with an emphasis on problems from machine learning (ML). The fundamentals of ML will be worked out in tandem with numerical linear algebra theory. In particular, we will cover the following (however, some topics may be omitted depending on time).

    Machine Learning:

    Basic tools: curve fitting, probability, loss functions, inference.
    Regression: linear models, bayesian viewpoint, evidence approximation.
    Classification: discriminants, generative models, bayesian viewpoint.
    Neural Networks: basic architecture, optimization issues, back propagation, regularization, bayesian viewpoint.

    Linear Algebra:

    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.