Summer 2016
- 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 2016
- MATH 4997-1: Vertically Integrated Research: Quiver Representations and Hall Algebras
- 12:00-1:20 T Th
- Instructor: Profs. Achar and Sage
- Prerequisites: Familiarity with basic group and ring theory and linear algebra, such as from Math 4200 and Math 4153
- Text:
- A quiver is just a directed graph, and a quiver representation is a rule that assigns a vector space to each vertex of a quiver and a linear transformation to each edge. The goal of this course is to study quiver representations and their applications to important problems in representation theory. We will start this semester by examining representations of several important kinds of quivers, including the Jordan quiver and the Kronecker quiver. Quiver representations over finite fields can be used to define a ring called the Hall algebra. By the end of the semester, we plan to get to deep theorems from the 1990s relating Hall algebras to quantum groups. No prior knowledge of quiver representations, Hall algebras, or quantum groups is required.
- 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
- 10:30-11:20 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, exact sequences.
- MATH 7230: Algebraic Number Theory.
- 1:30-2:20 M W F
- Instructor: Prof. Long.
- Prerequisites: Math 7210: Algebra I
- Text: A course in algebraic number theory by Robert B. Ash, a Dover book on mathematics
- This course is about various topics (norms, traces, discriminants, rings of integers, factorizations, units, et al.) of number fields, which are finite extensions of the field of rationals.
- MATH 7280: Categorification.
- 8:30-9:20 M W F
- Instructor: Prof. Yakimov.
- Prerequisites: Some knowledge of Lie algebras will be helpful but, if needed, some lectures and distributed lecture notes will cover what is needed. The emphasis will be on concrete examples instead of the most general statements for simple Lie algebras.
- Text: None
- The course will be an introduction to the basic ideas of categorification. It will be aimed both at students who would like to specialize in the area and at students from other areas who would like to have a general knowledge of the subject. The course will cover the main parts of the theory of categorification of quantum groups and their representations via Khovanov-Lauda-Rouquier algebras, and their relations to the categorification of knot invariants. The major constructions and theorems will be introduced and motivated through many concrete examples.
- MATH 7290: Lie Theory.
- 9:00-10:20 T Th
- Instructor: Prof. Achar.
- Prerequisites: Math 7211
- Text: None
- This course will cover the foundations of Lie theory from an algebraic perspective. The main objects of study are algebraic groups--which are philosophically close to Lie groups, but live in the realm of algebraic geometry rather than differential geometry--and their Lie algebras. The semester will cover three main topics: (i) structure theory; (ii) classification; and (iii) representation theory. No prior knowledge of algebraic geometry will be assumed.
- MATH 7311: Real Analysis I.
- 9:30-10:20 M W F
- Instructor: Prof. Davidson.
- Prerequisites: Math 4032 or 4035 or the equivalent.
- Text: Required: L. Richardson, Measure and Integration: A Concise Introduction to Real Analysis, John Wiley & Sons, June 2009. ISBN: 978-0-470-25954-2
Recommended Reference: H. L. Royden, Real Analysis, 3rd ed., Macmillan, ISBN 0024041513. - We will treat measure theory and integration on measure spaces. The examples of the real line and of Euclidean space will be emphasized throughout. Topics will include the Hopf extension theorem, completion of the Borel measure space, Egoroff's theorem, Lusin's theorem, Lebesgue dominated convergence, Fatou's lemma, product measures, Fubini's theorem, absolute continuity, bounded variation, Vitali's covering theorem, Lebesgue differentiation theorems, and the Radon-Nykodim theorem. Applications to Lp and its dual, and the Riesz-Markov-Saks-Kakutani theorem may be presented if there is sufficient time.
- MATH 7350: Complex Analysis.
- 1:30-2:50 T Th
- Instructor: Prof. Estrada.
- Prerequisites: Math 7311 or its equivalent.
- Text: Narasimhan, R. and Nievergelt, Y., Complex Analysis in One Variable, second edition, Birkhauser, Boston, 2001.
- A standard first year graduate course in complex analysis. Topics include holomorphic functions, covering spaces and the monodromy theorem, winding numbers, residues, Runge's theorem, Riemann mapping theorem, harmonic functions.
- MATH 7370: Lie Groups and Representation Theory.
- 3:00-4:20 T Th
- 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. A preliminary version can be found at http://www.math.lsu.edu/~olafsson/pdf_files/ln7370.pdf. Good introductory books on Lie groups include
J. Hilgert and K-H. Neeb: Structure and Geometry of Lie Groups
V. S. Varadarajan: Lie Groups, Lie Algebras, and their Representations
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 related to finite dimensional or infinite dimensional Lie groups.
We will mostly consider linear Lie groups, that makes several proofs much easier, but the statement will 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 and examples. Then we will discuss the exponential map, closed subgroups, the Lie algebra of a closed subgroup, group actions on manifolds, homogeneous spaces, and homogeneous vector bundles. Further material will depend on time and interests of the participants.
- MATH 7384: Topics in Material Science: Extreme Geometries and Optimal Functionally Graded Materials.
- 12:30-1:20 M W F
- Instructor: Prof. Lipton.
- Prerequisites: Either Math 4027, 4031, 4036, 4038 or equivalent
- Text: Course Notes and references to the literature will be provided
- Additive manufacturing techniques offer new opportunities for optimally engineered structures employing graded material properties. This course provides the underlying physical theory and mathematical techniques developed over the past 20 years for design of optimally graded thermal, electric and elastic materials. The techniques introduced in this course include elements of the calculus of variations and the methods of partial differential equations. The course concludes with an overview of modern numerical methods for the design of functionally graded composites for optimal stiffness and strength.
- MATH 7390-1: Calculus of Variations & Optimal Control.
- 1:30-2:20 M W F
- Instructor: Prof. Wolenski. and Prof. Hermosilla.
- Prerequisites: Advanced Calculus, Linear Algebra, and Ordinary Differential Equations
- Text: No formal text is required
- This course is an introduction to the Calculus of Variations and Optimal Control from a modern viewpoint that will emphasize the role of convexity. It is aimed at graduate and advanced undergraduate students interested in the theoretical foundations of any area of applied mathematics.
We will begin with a review of continuous optimization in Euclidean space; the role of convexity will be emphasized and basic tools of nonsmooth analysis will be introduced. The main topic of the course is dynamic optimization. The first half will cover the classical material of the calculus of variations, including topics such as the Euler-Lagrange equation, Weierstrass maximality condition, Erdmann corner conditions, and Jacobi conjugate points. Plenty of examples will be covered. There is a natural transition into optimal control, which will be the focus of the rest of the course from a neo-classical point of view.
- MATH 7390-2: Convex Optimization Theory and Applications
- 12:00-1:20 T Th
- Instructor: Prof. Zhang.
- Prerequisites: Math2057 Multidimensional Calculus and Math2085 Linear Algebra
- Text: : Class notes and the following two reference books:
- Convex Optimization by Stephen Boyd and Lieven Vandenberghe, ISBN 978-0-521-83378-3
- Convex Analysis and Optimization by Dimitri P. Bertsekas, ISBN 1-886529-45-0
- Depending on the time available, tentative topics include Convex sets, Convex functions, Smooth convex optimization problems, Nonsmooth convex optimization problems, Duality Theory, Applications, Interior Point Methods.
- MATH 7490: Matroid Theory
- 9:00-10:20 T Th
- 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.
- MATH 7510: Topology I
- 1:30-2:20 M W F
- Instructor: Prof. Litherland.
- Prerequisites: Advanced Calculus (Math 4031)
- Text:Topology (2nd Edition), By James Munkres
- This course is a preparation course for the Core I examination in topology. The course will quickly discuss some elementary notions in general topology before moving on to (basic) homotopy theory, the fundamental group, and covering spaces. Topological manifolds, simplicial complexes and CW complexes will also be discussed.
- MATH 7520: Algebraic Topology.
- 11:30-12:20 M W F
- Instructor: Prof. Gilmer.
- Prerequisites: Math 7510 and Math 7512 (or familiarity with the fundamental group and the homology of topological spaces)
- 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)
- This will be a continuation of Math 7512 from the Spring. We will study cohomology, Poincare duality, and hopefully some homotopy theory.
- MATH 7590: Hyperplane Arrangements.
- 10:30-11:50 T Th
- Instructor: Prof. Cohen.
- Prerequisites: The exposure to the fundamental group provided by MATH 7510 Topology I
- Text: Portions of the course will use the manuscript-in-progress Complex Arrangements: Algebra, Geometry, Topology (access will be provided)
- A hyperplane arrangement is a finite collection of (n-1)-dimensional subspaces in an n-dimensional vector space, such as lines in a plane, planes in 3-space, etc. Arrangements arise in a variety of mathematical contexts, and in applications ranging from mathematical physics to robotics.
The complement of an arrangement, what is left of space after the hyperplanes have been removed, is an object of fundamental interest in the topological study of arrangements. For instance, a collection of lines cuts the real plane into pieces, and understanding the topology of the complement amounts to counting the pieces. This depends on combinatorial aspects of the collection of lines such as parallelism, how many of the lines intersect at a given point, and so on. This example illustrates a central theme in the subject, the relationship between combinatorial and topological aspects of arrangements.
A complex hyperplane does not disconnect space, and the complement of an arrangement in a complex vector space has rich topological structure. One focus of this course will be on the (low-dimensional) topology of these spaces. Specific topics include braid groups, both as primary objects of study and as tools, configuration spaces, the Fox calculus, and Alexander-type invariants. In particular, we will develop general algorithms for computing these latter invariants, and investigate the extent to which these invariants of arrangements are combinatorially determined.
The topological aspects of the course will assume familiarity with the fundamental group, while other (combinatorial) aspects will be self-contained.
Spring 2017
- MATH 4997-1: Vertically Integrated Research: Quiver Representations and Hall Algebras
- 12:00-1:20 T Th
- Instructor: Profs. Achar and Sage
- Prerequisite: Familiarity with basic group and ring theory and linear algebra, such as from Math 4200 and Math 4153
- Text: None
- A quiver is just a directed graph, and a quiver representation is a rule that assigns a vector space to each vertex of a quiver and a linear transformation to each edge. The goal of this course is to study quiver representations and their applications to important problems in representation theory. We will start this semester by examining representations of several important kinds of quivers, including the Jordan quiver and the Kronecker quiver. Quiver representations over finite fields can be used to define a ring called the Hall algebra, which in turn has deep connections to quantum groups, combinatorics, and topics from algebraic geometry. We will explore some of these connections over the course of the semester. No prior knowledge of these topics is required. New participants (who weren't in the course in Fall 2016) are welcome!
- 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.
- 8:30-9:20 M W F
- Instructor: Prof. Hoffman.
- Prerequisites: The material taught in Math 7210: basics about groups, rings, modules and homomorphisms.
- Text: : Recommended:
- M. Artin, Algebra (2nd edition), Addison Wesley, 2010. ISBN: 978-0132413770
- D. Dummit and R. M. Foote, Abstract Algebra (3nd edition), John Wiley and Sons, ISBN: 978-0471433347
Michael Artin has an excellent online course at MIT:
https://ocw.mit.edu/courses/mathematics/18-701-algebra-i-fall-2010/index.htm
https://ocw.mit.edu/courses/mathematics/18-702-algebra-ii-spring-2011/ - There will be two main topics:
- Field and Galois Theory.
- Group Actions
There will be one in class midterm, worth 25% of the grade. The remaining 75% will be in assigned homework problems turned in, including a take-home final.
- MATH 7230: Partitions, Hypergeometric q-series and Modular Forms
- 10:30-11:50 T Th
- Instructor: Prof. Mahlburg.
- Prerequisites: Math 7210 (Algebra I). Complex analysis at the undergraduate level (Math 4036) is very helpful.
- Text: The Theory of Partitions, by George Andrews, Cambridge University Press 1998.
- An integer partition is an additive decomposition of a positive integer, such as 5 = 3 + 2 = 2 + 2 + 1. This course gives an introduction to integer partitions and their many applications, as well as the deep connections to basic hypergeometric q-series and modular forms. Topics will include simple combinatorial properties, including generating functions; Jacobi's Triple Product, the Rogers-Ramanujan identities, and other results from basic hypergeometric q-series; Ramanujan's congruences and other arithmetic properties; The Hardy-Ramanujan asymptotic expansion and other analytic properties; the role of theta functions and modular forms; applications in group theory and representation theory.
- MATH 7240: Algebraic Geometry
- 1:30-2:50 T Th
- Instructor: Prof. Sage.
- Prerequisites: The first year algebra sequence---Math 7210 and Math 7211.
- Text: Algebraic Geometry, Part 1: Schemes. With examples and exercises, by U. Gortz, T. Wedhorn,
- Algebraic geometry has its origin in the study of solutions to systems of polynomial equations. It is of fundamental importance in a wide range of areas of mathematics such as number theory, representation theory, and mathematical physics and also has surprising applications to such fields as statistics, mathematical biology, control theory, and robotics.
Modern algebraic geometry is based on the fundamental notion of a scheme. This course will give an introduction to schemes and their geometry, with particular emphasis on motivating the definitions and constructions and providing many examples. Topics covered will include algebraic varieties, sheaf theory, affine schemes, and projective schemes.
- MATH 7320: Ordinary Differential Equations.
- 10:30-11:50 T Th
- Instructor: Prof. Malisoff.
- Prerequisites: Math 2085 and 4031; or equivalent.
- Text: An Introduction to Dynamical Systems: Continuous and Discrete, Second Edition by R. Clark Robinson, AMS Pure and Applied Undergraduate Texts, Volume 19. ISBN: 978-0-8218-9135-3
- Existence and uniqueness theorems, approximation methods, linear equations, linear systems, stability theory; other topics such as boundary value problems.
- MATH 7330: Functional Analysis.
- 11:30-12:20 M W F
- Instructor: Prof. Rubin.
- Prerequisites: MATH 7311 or equivalent.
- Text: No special textbook. I will distribute my own notes.
- Functional Analysis is the language of modern mathematics. The course provides an introduction to the general theory of normed and metric spaces, linear operators and linear functionals, Banach and Hilbert spaces, spectral theory, and other important topics.
- MATH 7366: Stochastic Analysis.
- 2:30-3:20 M W F
- Instructor: Prof. Kuo.
- Prerequisites: Undergraduate probability theory
- Text: Kuo, H.-H.: Introductory Stochastic Integration.
Universitext, Springer 2006
Reference:
Kuo, H.-H.: Gaussian Measures in Banach Spaces. Lecture Notes in Math., Vol. 463, Springer-Verlag, 1975 (Reproduced by Amazon, Oct 2006.) - In this course we do not assume the previous knowledge of probability theory from Math 7360. The needed concepts will be fully explained in this course. We will cover the Ito theory of stochastic integration in full detail and sketch other theories of stochastic analysis such as abstract Wiener space, white noise analysis, and Malliavin calculus. In addition, we will study my recent theory of general stochastic integration for stochastic processes arising from the Ito part and the counterpart. I will propose several research problems for further investigation of this new stochastic integral.
- MATH 7375: Wavelets
- 9:30-10:20 M W F
- 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. If time permits, we also discuss the application of wavelets in basic function spaces.
- MATH 7384: Topics in Material Science: Spectral theory in wave dynamics: From classical theory to open problems
- 1:30-2:50 T Th
- Instructor: Prof. Shipman.
- Prerequisites:
- Text:
- Spectral theory of self-adjoint and normal operators underlies the dynamics of linear waves in classical and quantum physics. In fact, the mathematical theory is largely driven by physics. This is a huge and varied field, and I will choose the direction, flavor, and topics based on my research activity and tastes.
The material will come from my notes and from various books and articles in the literature. Enrolled students will present a special topic or problem to the class.
Here is the material I have in mind:- Spectral theorem and concrete realizations of it:
- Unitary representations and Fourier transforms for symmetry groups of a physical structure
- Connection to complex analysis and moment problems.
- Spectral theory in wave dynamics:
- Operators for wave dynamics that commute with symmetry groups, such as periodic operators with additional symmetries—the symmetries of the operator mirror the symmetries of the underlying physical structure.
- Analysis of operator resolvents and applications to the following:
- Scattering of waves by objects—modified Fourier modes caused by local perturbations of the wave operator.
- Resonance.
- Limiting amplitude principle and limiting absorption principle.
- Some relatively classical examples:
- Bound states and eigenvalues, symmetry-protected spectrally embedded eigenvalues; resonance.
- Coercivity and compactness.
- The Fermi algebraic surface for periodic media, reducibility, and its relation to embedded eigenvalues.
- Current research and open problems:
- “Negative-index” materials.
- Non-symmetry-protected embedded eigenvalues.
- Topologically protected eigenstates.
- Symmetries of Euclidean space—the wallpaper groups and the Bieberbach groups.
- Nonlinear scattering problems.
- Spectral theorem and concrete realizations of it:
- MATH 7410: Graph Theory.
- 1:30-2:20 M W F
- Instructor: Prof. Oporowski.
- Prerequisites:
- Text:
- MATH 7490: Combinatorial Optimization.
- 10:30-11:20 M W F
- Instructor: Prof. Ding.
- Prerequisites: Math 4171 or equivalent.
- Text: None
- First we cover 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.
- 9:30-10:20 M W F
- Instructor: Prof. Dasbach.
- Prerequisites: Math 7510
- Text: Algebraic Topology by Allen Hatcher
- This course will introduce the homology theory of topological spaces. To each space X and nonnegative integer k, there is assigned an abelian group the k-th homology group of X. We will learn to calculate these groups, and use them to prove topological results such as the Brouwer Fixed point theorem (in all dimensions) and generalizations of the Jordan curve theorem. Homology theory is important in many parts of modern mathematics. If time permits, we will cover the the basic ideas of cohomology as well.
- MATH 7550: Differential Geometry.
- 9:00-10:20 T Th
- Instructor: Prof. Vela-Vick.
- Prerequisites: 4032 (or equivalent) and 7510
- Text: Michael Spivak, A Comprehensive Introduction to Differential Geometry, 3rd Edition
- 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: 3-Manifolds.
- 12:00-1:20 T Th
- Instructor: Prof. Vela-Vick.
- Prerequisites: 7510 (or equivalent)
- Text: None
- This course will introduce the basic tools and techniques of 3-Manifold topology. Some topics to be covered include: normal surfaces, the prime decomposition theorem, the sphere theorem, Haken manifolds, torus decompositions, and Seifert fibered spaces. Time permitting, we may also touch on Dehn fillings and some relationships between 3 and 4-dimensional topology.
- MATH 7590-2: 4-Manifolds.
- 10:30-11:50 T Th
- Instructor: Prof. Baldridge.
- Prerequisites:
- Text: 4-Manifolds and Kirby Calculus by R. Gompf and A. Stipsicz
- In this course we study smooth closed 4-manifolds. We will describe standard examples and study how to construct exotic 4-manifolds (manifolds that are homeomorphic but not diffeomorphic to a standard example). Along the way we will work with fundamental groups, symplectic 4-manifolds, and Seiberg-Witten invariants. This course should be interesting to students who want an overview of 4-manifold constructions and want to learn how to use advance techniques in topology, geometry, and global analysis to distinguish different smooth 4-manifolds.
- MATH 7710: Advanced Numerical Linear Algebra.
- 12:30-1:30 M W F
- Instructor: Prof. Wan.
- Prerequisites:
- Text:
- In this course, we will focus on how to perform matrix computations efficiently and accurately. Depending on the time available, tentative topics include Gaussian elimination and its variants (including Cholesky decomposition, LU decomposition, etc.), QR decomposition, singular value decomposition, eigenvalue solvers and iterative methods for linear systems (including Gauss-Seidel iteration, SOR iteration, Conjugate Gradient method, etc.). Programming language is required and Matlab is recommended.