# Graduate Course Outlines, Summer 2015-Spring 2016

## Summer 2015

• 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 2015

• MATH 4997-2: Vertically Integrated Research: Combinatorial Models in Representation Theory and Topology
• 1:30-250 T Th
• Instructor: Profs. Achar and Sage
• Prerequisites: Familiarity with basic group theory and linear algebra, such as from Math 4200 and Math 4153.
• Text: . Fulton, Young Tableaux, London Mathematical Society Student Texts No. 35, London Mathematical Society, 1996.
• This semester will be about uses of easy combinatorial models to solve longstanding open questions in representation theory and topology. (It continues last semester's theme, but last semester is not a prerequisite.) The combinatorial models include honeycombs (certain planar graphs made up of hexagons) and the closely related hives, as well as puzzles (pictures in the plane made of equilateral triangles and rhombi). These models can be used to study the representation theory of GL(n,C) and the topology of Grassmannians (the space of k-planes in Cn). One application of these methods is the solution of the Hermitian eigenvalue problem: If you know the eigenvalues of two Hermitian matrices A and B, what can you say about the possible eigenvalues of A+B?
• MATH 4997-3: Vertically Integrated Research: Big Data and Topology
• 3:30-4:20 M W F
• Instructor: Profs. Dasbach and Stoltzfus
• Prerequisites: Multivariate Calculus and Linear Algebra
• Text: Computational Topology by Herbert Edelsbrunner and John L. Harer
• In many applications data (of numerical information) can be represented by a point set in an n-dimensional space. Computational topology focuses on the computational aspects of topology, and applies topological tools to analysis of those data sets. Topological tools of simplicial homology, shape and discrete morse theory will be presented. Algorithms and software for the analysis of the topology, connectivity and shape of point sets will be explored. Projects will be either a software exploration implementation on a point data set of interest to you or a presentation of recent papers in the area.
• 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 M W F
• Instructor: Prof. Mahlburg.
• Prerequisites: MATH 4200 and 4201, or equivalents
• 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. It will cover the basic notions of group, ring, module, and field theory. Topics will include symmetric groups, the Sylow theorems, group actions, solvable groups, Euclidean domains, principal ideal domains, unique factorization domains, polynomial rings, modules over PIDs, vector spaces, field extensions, and finite fields.
• MATH 7230: Modular Forms.
• 1:30-2:20 M W F
• Instructor: Prof. Candelori.
• Prerequisites:
• Text:
• MATH 7260: Homological Algebra.
• 10:30-1150 T Th
• Instructor: Prof. Achar.
• Prerequisites: Math 7211 and 7512
• Text: Weibel, An introduction to homological algebra
• The systematic use of chain complexes and long exact sequences originated in the setting of algebraic topology, but it has now found applications throughout many areas of mathematics. In this course, we will develop these tools in a modern, algebraic way (with a focus on the language of derived categories), and we will discuss various applications in algebra and topology, depending on the interests of the class. Possible topics for applications include group cohomology, Lie algebra cohomology, or sheaf cohomology.
• MATH 7311: Real Analysis I.
• 10:30-11:50 T Th
• Instructor: Prof. Shipman.
• Prerequisites:
• Text:
• MATH 7350: Complex Analysis.
• 9:00-10:20 T Th
• Instructor: Prof. Olafsson.
• Prerequisites: Math 7311 or equivalent
• Text: Complex Analysis by Elias Stein and Rami Shakarchi, Princeton Lectures in Analysis II
• Theory of holomorphic functions of one complex variable; path integrals, Cauchy’s Theorem, power series, singularities, meromorphic functions, mapping properties, normal families, the Gamma functions and other topics.
• MATH 7360: Probability Theory.
• 9:30-10:20 M W F
• Instructor: Prof.Kuo.
• Prerequisites: Math 7311 (Real Analysis I) or equivalent
• Text: John W. Lamperti: Probability, A Survey of the Mathematical Theory, 2nd Edition, John Wiley & Sons, Inc., 1996
• A brief review of elementary probability theory will be given in the first week. Then we will cover all chapters of the textbook:
Chapter 1: Foundation
Chapter 2: Laws of Large Numbers and Random Series
Chapter 3: Limiting Distributions and the Central Limit Problem
Chapter 4: The Brownian Motion Process
Some particular topics: Kolmogorov’s extension theorem, types of convergence, laws of large numbers, convergence of random series, characteristic functions, Bochner theorem, Levy continuity theorem, Levy equivalence theorem, central limit theorem, stable and infinitely divisible laws, Brownian motion
• MATH 7365: Applied Stochastic Analysis.
• 10:30-11:20 M W F
• Instructor: Prof. Sengupta.
• Prerequisites: Math 7360 or equivalent.
• Text: None.
• We will cover some special topics with a view to applications. Possible topics are: (i) stochastic differential equations; (ii) the study of the first time when a stochastic process hits a given boundary; (iii) the Feynman-Kac formula, solving a partial differential equation in terms of stochastic integrals; (iv) a sampler from random matrices; (v) large scale behavior of correlated random variables. Applications in finance and in the study of physical phenomena will be discussed.
• MATH 7384: Topics in Material Science: Optical Metamaterials and Plasmonic Crystals
• 12:30-1:20 M W F
• Instructor: Prof. Lipton.
• Prerequisites: Any one of Math 3355, 3903, 4031, or 4038, or their equivalent.
• Text:
• In this course we provide an introduction to theory behind the design of metal-dielectric crystals 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 metamaterial crystals. 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 nobel metals. 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 necessary for designing structured media supporting backward waves and behavior associated with an effective negative index of refraction. We conclude by introducing multiscale techniques necessary for computing the effective dielectric properties of metamaterials and higher order corrections.
• MATH 7386: Theory of Partial Differential Equations.
• 1:30-2:20 M W F
• Instructor: Prof. Antipov.
• Prerequisites: Any one of Math 4340, 4031, 4038, or their equivalent.
• Texts: (1) Partial Differential Equations by Lawrence C. Evans;
(2) Partial Differential Equations of Applied Mathematics by Erich Zauderer;
(3) Lecture notes.
• Topics to be covered include:
Sobolev Spaces. Elliptic Partial Differential Equations of Second Order. Weak solutions Lax-Milgram Theorem. Regularity. Maximum principles. Eigenvalues and eigenfunctions. Applications. Second order parabolic and hyperbolic equations. Existence of weak solutions, regularity and maximum principles. Introduction into semigroup theory. Nonlinear PDEs. The Calculus of variations technique. Hamilton-Jacobi equations.
• MATH 7390-1: Iterative Methods.
• 1:30-2:50 T Th
• Instructor: Prof. Sung.
• Prerequisites: Math 7710
• Text: Iterative Methods for Sparse Linear Systems (Second Edition) by Yousef Saad
• Iterative methods are fundamental to large scale scientific computations, where the demands on memory and flops render most direct method impractical. The emphasis of this course is on the motivation, derivation and analysis of some of the most important iterative algorithms. The following topics will be covered.   Chapter 4 Basic Iterative Methods Chapter 5 Projection Methods Chapters 6 and 7 Krylov Subspace Methods Chapters 9 and 10 Preconditioning
• MATH 7390-2: Vertically Integrated Research: Analysis on Manifolds
• 1:30-2:50 T Th
• Instructor: Prof. Rubin
• Prerequisites: Math 2057, 2085 or equivalent
• Texts: 1. I. Agricola and Th. Friedrich, Global Analysis: Differential Forms in Analysis, Geometry, and Physics. American Mathematical Soc., 2002.
2. Other sources.
• This is an introductory course-seminar in Calculus on manifolds. The main topics: differential forms, vector analysis and integration on manifolds, and some other topics.
• MATH 7490: Computational Combinatorics
• 2:30-3:20 M W F
• Instructor: Prof. van Zwam.
• Prerequisites: Consent of department. A course in graph theory is strongly recommended.
• Text: None
• We study the various ways in which combinatorics and computation interact. We will study efficient algorithms for combinatorial problems such as matching in graphs, as well as heuristics for equivalence-free generation of various combinatorial objects, and the use of advanced computational tools to show (non)existence of combinatorial objects, including integer programming and positive semidefinite optimization, and algebraic tools such as Groebner basis computation. In addition to traditional homework problems, the course will include a few computational projects. Some of the lectures will be devoted to an introduction to SageMath, but other computer algebra software can be used by the students if they desire.
• MATH 7510: Topology I
• 11:30-12:20 M W F
• Instructor: Prof. Dasbach.
• Prerequisites:
• Text:
• MATH 7520: Algebraic Topology.
• 12:00-1:20 T Th
• Instructor: Prof. Cohen.
• Prerequisites: MATH 7510 and 7512, or equivalent.
• Text: Algebraic Topology by A. Hatcher
• This course continues the study of algebraic topology begun in MATH 7510 and MATH 7512. The basic idea of this subject is to associate algebraic objects to a topological space (e.g., the fundamental group, the homology groups) in such a way that topologically equivalent spaces get assigned equivalent objects (e.g., isomorphic groups). Such algebraic objects are invariants of the space, and provide a means for distinguishing between topological spaces: two spaces with inequivalent invariants cannot be topologically equivalent.

The focus of this course will be on cohomology theory, dual to the homology theory developed previously. One reason to pursue cohomology theory is that the cohomology of a space may be given a natural ring structure. This additional algebraic structure provides another topological invariant. In developing this structure, we will study several products relating homology and cohomology. These considerations will be used to study the topology of manifolds, yielding a number of duality theorems also relating homology and cohomology, with a variety of applications.

In addition to its importance within topology, cohomology theory also provides connections between topology and other subjects, including algebra and geometry. Depending on the interests of the audience, we may pursue some of these connections, such as cohomology of groups or the De Rham theorem, or other elements of algebraic topology, such as homotopy theory.

• MATH 7590: Chern-Simons Theory.
• 9:00-10:20 T Th
• Instructor: Prof. Baldridge.
• Prerequisites: Some differential forms and Lie groups knowledge is helpful
• Text: Gauge Fields, Knots and Gravity by John Baez & Javier P. Muniain
• How is knot theory related to theoretical physics? This course is a gentle introduction to the mathematical tools needed to understand the relationship between knot theory and Chern-Simons theory. First we will use differential forms to formulate Maxwell’s equations of electromagnetism. Then we introduce connections on vector bundles whose underlying structure group is from a list of certain Lie groups, and we use the connections and curvature to generalize Maxwell theory to the Yang-Mills equations. This in turn motivates Chern-Simons theory and its relationship to knot theory.

## Spring 2016

• MATH 4997-1: Vertically Integrated Research: Combinatorial Models in Representation Theory and Topology
• 10:30-11:50 T Th
• Instructor: Profs. Achar and Sage
• Prerequisites: Familiarity with basic group theory and linear algebra, such as from Math 4200 and Math 4153.
• Text: . Fulton, Young Tableaux, London Mathematical Society Student Texts No. 35, London Mathematical Society, 1996.
• This semester will be about uses of easy combinatorial models to solve longstanding open questions in representation theory and topology. (It continues the theme of the last two semesters, but those semesters are not prerequisites for this course.) The combinatorial models include honeycombs (certain planar graphs made up of hexagons) and the closely related hives, as well as puzzles (pictures in the plane made of equilateral triangles and rhombi). These models can be used to study the representation theory of GL(n,C) and the topology of Grassmannians (the space of k-planes in Cn). One application of these methods is the solution of the Hermitian eigenvalue problem: If you know the eigenvalues of two Hermitian matrices A and B, what can you say about the possible eigenvalues of A+B?
• 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. Ng.
• Prerequisites: Math 7210 or equivalent.
• Text: Abstract Algebra, 3rd ed; Dummit and Foote, Wiley 2003.
• This is the second semester of the first year graduate algebra sequence. Topics will include field theory, Galois theory, basics of commutative algebra and algebras over a field, Wedderburn’s theorem, Maschke’s theorem, tensor products and Hom for modules, possibly some introduction to homological algebra or linear representations of finite groups if time permitted.
• MATH 7290: Differential Galois Theory
• 12:00-1:20 T Th
• Instructor: Prof. Sage.
• Prerequisites: Math 7211 or permission of the instructor
• Text: Lectures on differential Galois Theory by Andy Magid, AMS University Lecture Series, Volume 7 (1994) ISBN-10: 0-8218-7004-1 ISBN-13: 978-0-8218-7004-4
• In the same way that Galois theory has its origins in studying the symmetries of the solutions of a polynomial, differential Galois theory grew out of the analogous problem for linear differential equations. More precisely, differential Galois theory studies extensions of differential fields, i.e., fields equipped with a derivation (a map obeying the Leibnitz rule). As the name suggests, there is a remarkable formal parallelism between the two theories. In particular, there is a fundamental theorem of differential Galois theory. Indeed, for appropriate differential field extensions E/F, the differential automorphisms fixing F forms a group called the differential Galois group G, and there is a one-to-one inclusion reversing correspondence between closed subgroups of G and intermediate differential field extensions. However, whereas classical Galois groups are finite (or at worst profinite), differential Galois groups are linear algebraic groups.

This class will provide an introduction to differential Galois theory. We will discuss Picard-Vessiot extensions (the differential analogue of Galois extensions) and give several different constructions of the differential Galois group. We will prove the fundamental theorem and consider when differential equations are solvable by quadratures--the differential analogue of solvability by radicals. We will further discuss the monodromy group of linear differential equations and its relationship to the differential Galois group.

• MATH 7320: Ordinary Differential Equations.
• 1:30-2:50 T Th
• Instructor: Prof. Sage.
• Prerequisites: Advanced Calculus and Linear Algebra (preferably Math 7311)
• 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 the qualitative theory of ordinary differential equations. Topics will include existence and uniqueness theory, dependence on initial conditions, linear systems, monodromy, stability, and Hamiltonian systems. Time permitting, we will give an introduction to asymptotic analysis and the Stokes phenomenon.
• MATH 7330: Functional Analysis.
• 10:30 - 11:50 T Th
• Prerequisites: Math 7311 or its equivalent.
• Text: Treves, 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.
• 12:00-1:20 T Th
• Instructor: Prof. Sundar.
• Prerequisites: Math 7360, or its equivalent
• Text: Class Notes will be distributed.
• The course will start with an introduction to Brownian motion, and martingales. This would lead to the development of stochastic integrals and stochastic differential equations. We will study a deep and fundamental connection between stochastic differential equations and a class of partial differential equations.
• MATH 7380: Topics in Elliptic Partial Differential Equations
• 12:30-1:20 M W F
• 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: Topics in Material Science:
• 1:30-2:50 T Th
• Instructor: Prof. Almog.
• Prerequisites: Math 2065 (or equivalent), Math 4036 (or equivalent)
• Text: Applied asymptotic analysis by Peter D. Miller, Graduate Studies in Mathematics, Vol. 75, AMS, (2006)
• The course revolves around the approximation of integrals and solutions of ODE involving a small parameter. A partial list of topics follows:
1. Asymptotic evaluation of integrals: Watson's lemma and Laplace's method, the steepest descent method, the method of stationary phase.
2. The WKB method
3. Singular perturbation theory
• MATH 7390: Topics in Numerical Analysis: Numerical PDE For Geometric Evolution
• 1:30-2:50 M W
• Instructor: Prof. Walker.
• Prerequisites: Theory of PDEs (MATH 4340, or MATH 7386 (e.g. weak formulations), or equivalent), numerical solution methods for PDE (MATH 4066 (e.g. finite differences), or MATH 7325 (e.g. finite elements), or equivalent).
• Text(Required): The Shapes of Things: A Practical Guide to Differential Geometry and the Shape Derivative, by S. W. Walker.
Reference Texts (not required):
1. Good all-around reference: Introduction to Shape Optimization: Theory, Approximation, and Computation, by J. Haslinger and R. A. E. Mäkinen.
2. Encyclopedia: Shapes and Geometries: Analysis, Differential Calculus, and Optimization, by Delfour and Zolesio.
• Geometric driven evolution plays a critical role in nature and industrial processes. Classic mathematical examples are mean curvature flow and Willmore flow. Related physical examples are surface tension driven droplets and bio-membranes. Many optimal shape design problems have connections to geometric evolution in the form of gradient flows.

The topics for this course are the following:

1. A review of differential geometry and surface PDEs. This is needed in order to formulate and analyze geometric evolution problems.
2. . Some numerical methods for simulating moving domain problems that are driven by geometric effects. The focus will be on finite element methods and related PDE/functional analysis issues.
3. . Discussion of selected papers in the literature on various geometric flows and their numerical simulation.
4. Basic tools of shape differential calculus for deforming domains and applications in shape optimization.
Grading in the course will be based on a semester long group project.
Professor will provide a "real" problem and guide the students on how to model it, how to formulate it mathematically and analyze it, and how to build and analyze a numerical method for it and simulate it. The necessary software framework will be provided. The final product will be a write-up describing the project, as well as plots/results with discussion, and the code needed to run it.

Prerequisites: Students should know basic elliptic PDE theory (e.g. Poisson equation, convection/diffusion equation, weak formulations, etc.) as well as some numerical methods for solving PDEs, e.g. finite difference or finite element methods, etc.; however, some discussion on numerical methods will be given in the class. Knowledge of continuum mechanics is a plus, but not required (necessary concepts will be reviewed).

• MATH 7410: Graph Theory.
• 1:30-2:20 M W F
• Instructor: Prof. Oporowski.
• Prerequisites:
• Text:
• The main theme of this course will be graph theory. We will discuss a wide range of topics, including spanning trees, eulerian trails, matching theory, connectivity, hamiltonian cycles, coloring, planarity, integer flows, surface embeddings, and graph minors. For more information see Math 7410.
• MATH 7490-1: Graph Minors.
• 10:30-11:20 M W F
• Instructor: Prof. Ding.
• Prerequisites: Math 4171 or equivalent
• Text: None (lecture notes will be distributed)
• This is an introduction to the theory of graph minors. We will discuss many problems of the following two types: determine all minor-minimal graphs that have a prescribed property; determine the structure of graphs that do not contain a specific graph as a minor. We will focus on connectivity and planarity.
• MATH 7490-2:The Tutte Polynomial for Matroids and Graphs.
• 9:00-10:20 T Th
• 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.
• 9:30-10:20 M W F
• Instructor: Prof. Gilmer.
• Prerequisites: Math 7510
• 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.
• 11:30-12:20 M W F
• Instructor: Prof. Litherland.
• Prerequisites: Math 4032 (or equivalent) 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: Mapping Class Groups.
• 10:30-11:50 T Th
• Instructor: Prof. Dani.
• Prerequisites: Math 7510 and prior completion of or concurrent enrollment in Math 7512
• Text: A primer on Mapping Class Groups by Benson Farb and Dan Margalit, Princeton University Press, 2011.
• The mapping class group of a surface is the group of its orientation preserving homeomorphisms (up to a certain equivalence relation). The study of mapping class groups is a classical subject which has exploded in the last decade.

In this course we will study the algebraic structure of mapping class groups and the detailed description of its individual elements. This includes the Nielsen – Thurston classification theorem, which gives a particularly nice representative for each element of the group. Furthermore, we will introduce spaces on which mapping class groups act, such as the complex of curves and Teichmüller space. An important theme throughout the course will be the relationships between the geometry of these spaces, the algebra of the mapping class group, and the topology of the surface.

• MATH 7590-2: Characteristic Classes.
• 9:00-10:20 T Th
• Instructor: Prof. Vela-Vick.
• Prerequisites: Math 7512 or equivalent
• Text: Characteristic Classes by Milnor and Stasheff
• Characteristic classes are incredibly powerful and useful cohomological invariants of vector bundles. In this course, we will discuss vector bundles, fiber bundles and characteristic classes with a focus on bundles over smooth manifolds. Beginning with some basic examples and constructions, we will eventually move on to the problem of classifying vector bundles. Along the way, we will construct Stiefel-Whitney, Euler, Chern and Pontryagin classes.
• MATH 7590-3: Vertically Integrated Research: Combinatorial Topology and Applications
• 3:30-4:20 M W F
• Instructor: Profs. Dasbach and Stoltzfus
• Prerequisites: MATH 2085 or equivalent
• Text:
• One of the many interesting applications of topology is the Brouwer Fixed Point Theorem. We will study proofs of the Brouwer Fixed Point Theorem, and show that it is equivalent to Sperner’s lemma and to the theorem that the Game of Hex does not end in a tie. This will lead to applications such as questions in Economics on how to divide the rent of shared apartments or the existence of a Nash equilibrium.
• MATH 7710: Advanced Numerical Linear Algebra.
• 9:00-10:20 T Th
• Instructor: Prof. Zhang.
• Prerequisites: linear algebra, advanced calculus and some programming experience (MATH 4032 or equivalent; MATH 4153 or equivalent.)
• 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).