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Graduate Course Outlines, Summer 2012-Spring 2013

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Summer 2012

  • MATH 4999-1: Problem Lab in Real Analysis-1—practice for PhD Qualifying Exam in Analysis.
  • Instructor: Prof. Cygan.
  • Prerequisite: Math 7311.
  • Text: Online Test Bank.
  • MATH 7999-1: Problem Lab in Topology-1—practice for PhD Qualifying Exam in Topology.
  • Instructor: Prof. Adkins.
  • Prerequisite: Math 7510.
  • Text: Online Test Bank.
  • MATH 7999-2: Problem Lab in Algebra-1—practice for PhD Qualifying Exam in Algebra.
  • Instructor: Prof. Adkins.
  • Prerequisite: Math 7210.
  • Text: Online Test Bank.

Fall 2012

  • MATH 4997-1: Vertically Integrated Research:
  • Instructor: Profs. Yakimov and Muller
In recent years representation theory of algebras has played a key role in many developments in cluster algebras, mathematical physics, and noncommutative geometry. The goal of this seminar-type course is to learn the fundamental constructions in this field and then study applications in the above mentioned areas. The course will naturally be split into two parts, "building fundamentals" and "investigating applications". Students will have the opportunity to give lectures in both parts of the course. The first part will cover the general representation theoretic and homological background, in particular quivers with relations and the Auslander-Reiten quivers of artin algebras. The second part of the course will cover applications to categorification of cluster algebras, Jacobian algebras, and noncommutative projective geometry.
  • MATH 4997-2: Vertically Integrated Research: Reflection Groups.
  • Instructor: Profs. Sage and Achar.
  • Prerequisites: Familiarity with basic group theory, such as from Math 4200
Reflection groups are a remarkable class of groups that benefit from a richly interwoven tapestry of combinatorial, geometric, and group-theoretic properties. They are connected to mathematical discoveries ranging from the very ancient (e.g., the classification of platonic solids) to the very modern (e.g, the Kazhdan-Lusztig conjectures on Lie algebra representations). In this seminar, we will study selected topics related to reflection groups. No prior familiarity with reflection groups is required.
  • MATH 4997-3: Vertically Integrated Research: Theory of distributions.
  • Instructor: Profs. Rubin and Olafsson.
  • Prerequisites: 2057 Multidimensional Calculus (2057, 4035 or equivalent).
  • Text: V.S. Vladimirov, Methods of the Theory of Generalized Functions, Taylor & Francis, 2002. Some other texts will be also used.

This course is intended to provide opportunities for undergraduate and graduate students to work in a research community, learn and create new mathematics. Possible formats include group reading and exposition, research projects, written and oral presentations. The subject of the course is introduction to the theory of distributions (or the generalized functions). This theory, created by S.L. Sobolev and L. Schwartz, is fundamental in the background of every educated mathematician. It enables one to differentiate non-differentiable functions, evaluate divergent integrals, solve differential equations of mathematical physics, and do many other useful things in analysis and applications.

  • MATH 4997-4: Vertically Integrated Research: Algorithms and computations in knot theory.
  • Instructor: Profs. Dasbach and Stoltzfus, Drs. Kearney and Tsvietkova.
  • Prerequisites: Math 2057 (Calculus of Several Variables)

Description: We will explore computational methods in knot theory, particularly of hyperbolic knots, using open source tools: SnapPea, Snap and Bar Natan's Mathematica package KnotTheory.

We particularly welcome undergraduate participants for this VIR course as we will be developing geometric concepts of three-dimensional hyperbolic geometry related to the visualization and design of solid geometric objects which can now be printed on 3D printers.

  • MATH 4997-5: Vertically Integrated Research: Resonance in wave scattering.
  • Instructor: Prof. Shipman.
  • Prerequisites: : For undergraduate students: 1-dimensional differential and integral calculus. For graduate students: A complex variables course and graduate real analysis (Math 7311).
  • Text: Notes by Maciej Zworski: Lectures on Scattering Resonances.

The phenomenon of resonance is familiar in popular and scientific tradition and commonly lies behind acoustic, electromagnetic, and mechanical processes and devices. We witness it in events such as the collapse of the Tacoma Narrows bridge, the shattering of a glass by acoustic resonance, anomalous absorption by the noble gases at specific energies, and super-sensitive frequency-dependence of light reflection from periodic surfaces. The topic of this course will be a mathematical theory that applies to a great variety of problems of resonance in wave scattering by objects in quantum and classical wave mechanics.

The primary literature reference will be the lecture notes on scattering resonances by Maciej Zworski. The classes will be run like a seminar, in which students and faculty will take turns presenting portions of the notes, related examples, and supporting material.

The mathematical theory of resonance involves sophisticated methods of complex variables and operator theory of differential equations. Even so, there is a rich array of simple and interesting models whose analysis is accessible to undergraduate students. The role of graduate students and faculty will be to learn and present the notes and supporting material. Undergraduate students will gain exposure to advanced mathematical techniques but will not be expected to grasp all the mathematics in the notes or lectures. Their role will be to work out and present models that illuminate specific resonant scattering phenomena.

  • MATH 7001: Communicating Mathematics I
  • 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
  • Instructor: Prof. Madden.
  • Prerequisites: Basic knowledge of groups and rings (the concepts necessary to state and prove the isomorphism theorems) and vector spaces (dimension, linear maps, null spaces).
  • Text: A. W. Knapp, Basic Algebra, Birkhauser 2006.
This course will cover:
  • Chapter I. Preliminaries about the Integers, Polynomials, and Matrices;
  • Chapter II. Vector Spaces over Q, R and C;
  • Chapter III. Inner-Product Spaces;
  • Chapter IV. Groups and Group Actions;
  • Chapter V. Theory of a Single Linear Transformation;
  • Chapter VIII. Commutative Rings and their Modules (sections 1-6) ;
  • topics from Chapter VI. Multilinear Algebra and Chapter VII. Advanced Group Theory (free groups, group representations, extensions of groups), as time allows.
As a supplementary text (emphasizing a more category-theoretic perspective) I will occasionally refer to P. Aluffi, Algebra: Chapter 0 , AMS Grad. Studies in Math. Vol. 104. (2009). The contents of the intended course are in the following parts of Aluffi's book: Chapter I. Set Theory and Categories, Chapter II: Groups, first encounter, Chapter III: Rings and Modules, Chapter IV. Groups, second encounter (group actions, Sylow theorems, composition series, symmetric group, exact sequences & extensions), Chapter V: Irreducibility and factorization in integral domains (UFD, Gauss's lemma, PID), Chapter VI: Linear Algebra (including finitely generated modules over a PID and applications to canonical forms).
  • MATH 7280: Introduction to Sheaf Theory.
  • Instructor: Prof. Achar.
  • Prerequisites: Math 7211 and 7512
  • Text: no required text; notes will be made available as appropriate
Sheaves provide a formalism for organizing algebraic objects (e.g., vector spaces, modules, abelian groups, etc.) locally on a topological space. Sheaf theory has its origins in algebraic topology, but now has applications throughout mathematics, and in particular plays an indispensable role in algebraic geometry. Topics for this course will include: basic definitions; Grothendieck's "six operations" on sheaves; adjointness theorems; complexes of sheaves and resolutions; and Poincare-Verdier duality. If time permits, we may also cover more advanced topics such as intersection cohomology and perverse sheaves.
  • MATH 7290-1: Algebraic Number Theory.
  • Instructor: Prof. Morales.
  • Prerequisites: Graduate algebra sequence: Math 7210-11 or equivalent. Undergraduate-level number theory.
  • Text: Algebraic Number Theory by A. Fröhlich and M. J. Taylor, Cambridge University Press 1999.
  • This is a first course in algebraic number theory. We will cover classical material such as

    • Dedekind domains
    • Absolute values and completions
    • Splitting of primes in extensions and ramification
    • Class groups and Units
    • Quadratic and cyclotomic number fields
    • Some Diophantine equations
    • Zeta functions and L-functions.
  • MATH 7290-2: Lie Algebras.
  • Instructor: Prof. Yakimov.
  • Text: J. E. Humphreys, Introduction to Lie Algebras and Representation Theory, Grad. Texts Math. 9.
Many areas of mathematics use in various ways techniques from Lie theory. The goal of this class is to introduce students to the major topics in the area of Lie algebras, most notably complex semisimple Lie algebras and their finite dimensional representations. The class will be organized in such way so it is suitable both for students who need Lie algebras for applications and for students who will specialize in the area. We will start with general facts for solvable, nilpotent, and semisimple Lie algebras. Then we will cover the Cartan-Killing classification of semisimple Lie algebras, and describe their root systems and Weyl groups. We will continue the description of the finite dimensional representations of semisimple Lie algebras. Time permitting, we will cover topics from Chevalley groups.
  • MATH 7311: Real Analysis I.
  • Instructor: Prof. Olafsson.
  • Prerequisites: Math 4032 or 4035 or the equivalent.
  • Text: L. Richardson, Measure and Integration—A Concise Introduction to Real Analysis, John Wiley & Sons, 2009, 978-0-470-25954-2

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 7325: Numerical Analysis & Applications--Finite Element Method.
  • Instructor: Prof. Sung.
  • Prerequisites: MATH 7311 or Consent of Instructor
  • Text: : The Mathematical Theory of Finite Element Methods (Third Edition) by Susanne Brenner and Ridgway Scott
This course provides an introduction to the theory of finite element methods. The following topics from chapters 1--5, 9, 10 and 12 of the text will be covered: Hilbert spaces, Sobolev spaces, variational/weak formulation of elliptic boundary value problems, finite element methods, interpolation error estimates, a priori and a posteriori error estimates, nonconforming finite element methods and mixed finite element methods.
  • MATH 7350: Complex Analysis.
  • Instructor: Prof. Antipov.
  • Prerequisites: Math 7311
  • Text: Theory of One Complex Variable by R E Green and S G Krantz, AMS.
This is a self-contained first year graduate course in complex analysis. The topics to be covered include complex line integrals, the theory of the Cauchy integral, meromorphic functions and the theory of residues, holomorphic functions, conformal mappings, harmonic functions, analytic continuation, the RIemann-Hilbert problem, Riemann surfaces and elliptic functions.
  • MATH 7360: Probability Theory.
  • Instructor: Prof. Sengupta.
  • Prerequisites: Math 7311
  • Text: None. Lecture notes will be provided.

This course is an introduction to the basic machinery of probability theory and leads to the study of stochastic processes. After reviewing measure theory and integration, we will study the fundamental mathematical structures and concepts used to model and understand random phenomena. These include random variables, distributions, sigma-algebras (which encode information), independence, and conditional expectations. Topics will include the laws of large numbers, central limit theorems, notions of convergence, infinitely divisible laws, and an initiation to stochastic processes.

  • MATH 7380: Applied Stochastic Analysis.
  • Instructor: Prof. Sundar.
  • Prerequisites: Math 7360 or its equivalent, or consent of instructor
  • Text: Arbitrage Theory in Continuous Time by Tomas Bjork
The aim of the course is to provide a mathematically precise account of arbitrage theory for financial derivatives. A self-contained treatment of stochastic differential equations and the Itô calculus will be presented, and will include the Feynman-Kac formula, and the Kolmogorov equations. Risk neutral valuation formulas and martingale measures will be introduced through Feynman-Kac representations. The Black-Scholes model, option pricing, and interest rate theory will be discussed in detail.
  • MATH 7384: Material Science -- Topics in rational mechanics: Elasticity and Plasticity
  • Instructor: Prof. Bourdin.
  • Prerequisites: Basic concepts of PDE and functional analysis.
  • Text: The course borrows ideas from the following references:
    • Bower, A. (2009). Applied Mechanics of Solids. CRC Press.
    • Ciarlet, P. G. (1983). Lectures on three-dimensional elasticity. Published for the Tata Institute of Fundamental Research, Bombay. Notes by S. Kesavan.
    • Ciarlet, P. G. (1988). Mathematical elasticity. Vol. I. North-Holland Publishing Co., Amsterdam. Three-dimensional elasticity.
    • Gurtin, M. E. (1981). An introduction to continuum mechanics. Academic Press Inc. , New York.
    • Gurtin, M. E. (1981). Topics in finite elasticity, volume 35 of CBMS-NSF Regional Conference Series in Applied Mathematics.SIAM.

In his Principia, Newton made a clear distinction between "practical" and "rational" mechanics, "[...] the science, expressed in exact propositions and demonstrations, of the motions that result from any forces whatever and of the forces that are required for any motions whatever" (The Principia : Mathematical Principles of Natural Philosophy, translation by Cohen, Whitman & Budenz)

In this course, we will apply this approach to some of the most fundamental material behavior in continuum solid mechanics: Elasticity and Plasticity.

We will begin by a simple construction of the theory of non-linear elasticity. We will show how under proper hypothesis, it can be reduced to the simpler and more familiar linearized theory of elasticity, which we will study in detail. In particular, we will highlight how this construction leads to a variational model.

Then we will focus on plasticity, introduce some of the most classical models and present recent developments.

  • MATH 7386: Theory of Partial Differential Equations.
  • Instructor: Prof. Shipman.
  • Prerequisites: Math 7311 (Real Analysis I) or the equivalent.
  • Text: Primary Text: Introduction to Partial Differential Equations by G. Folland;
    Some material will also come from: Partial Differential Equations by Lawrence C. Evans.

The course begins with an introduction to the theory and problems associated with Laplace's equation, the heat equation, and the wave equation. Then continues with a treatment of nonlinear first-order PDE including the Hamilton Jacobi equation and an introduction to conservation laws. The course finishes with an introduction to Sobolev spaces and the existence of weak solutions to second order elliptic equations.

  • MATH 7490: Matroid Theory
  • 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
  • Instructor: Prof. Cohen.
  • Prerequisite: MATH 4031 and 4200 or equivalent.
  • 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. A good supplementary reference is Chapter 1 of Algebraic Topology by Allen Hatcher, available online .

  • MATH 7520: Algebraic Topology.
  • Instructor: Prof. Dani.
  • Prerequisites: Math 7510 and 7512
  • Text: Algebraic Topology by Allen Hatcher (required)
Basic concepts of homology, cohomology, and homotopy theory.
  • MATH 7590: Braid Groups.
  • Instructor: Prof. Kearney.
  • Prerequisites: Math 7520 (Concurrent enrollment is acceptable)
  • Text: : Braid Groups, by Christian Kassel and Vladimir Turaev
We will discuss braids and braid groups, and their relationship with knots and links. We will also explore the theory of representations of braid groups.

Spring 2013

  • MATH 4997-5: Vertically Integrated Research: Reflection Groups.
  • Instructor: Profs. Sage and Achar.
  • Prerequisites: Familiarity with basic group theory, such as from Math 4200
Reflection groups are a remarkable class of groups that benefit from a richly interwoven tapestry of combinatorial, geometric, and group-theoretic properties. They are connected to mathematical discoveries ranging from the very ancient (e.g., the classification of platonic solids) to the very modern (e.g, the Kazhdan-Lusztig conjectures on Lie algebra representations). In this seminar, we will study selected topics related to reflection groups. No prior familiarity with reflection groups is required.
  • MATH 4997-6: Vertically Integrated Research: A Linear Algebra Based Approach to Quantum Computing and Quantum Information Theory
  • Instructor: Prof. Lawson
  • Prerequisites: A reasonably good background in linear algebra (e.g.Math 2085) should be considered minimal background for the course. Some very basic knowledge of quantum theory and elementary probability theory would also be helpful.
  • Text: Quantum Computing, Nakahara and Ohmi, 2008, CRC Press
  • Reference: Quantum Computing and Quantum Information, Nielsen and Chuang, 2000, Cambridge

This introductory course on the recently emerging topic of quantum computing and information theory will introduce students to major recent developments such as quantum encoding and cryptography, teleportation, error correction, and quantum computing. Basic concepts of quantum theory such as quantum states, qubits, entanglement, measurement, quantum gates etc. will be incorporated into the course. The mathematical content will center on a linear algebra approach to the subject through basic matrix theory (unitary and Hermitian matrices, positive and completely positive operators, Gram-Schmidt decomposition, etc.) together with some elementary probabilistic content.

  • MATH 4997-7: Vertically Integrated Research: Algorithms and computations in knot theory.
  • Instructor: Profs. Dasbach and Stoltzfus, Drs. Kearney and Tsvietkova.
  • Prerequisites: Math 2057 (Calculus of Several Variables)

Description: We will explore computational methods in knot theory, particularly of hyperbolic knots, using open source tools: SnapPea, Snap and Bar Natan's Mathematica package KnotTheory.

We particularly welcome undergraduate participants for this VIR course as we will be developing geometric concepts of three-dimensional hyperbolic geometry related to the visualization and design of solid geometric objects which can now be printed on 3D printers.

  • MATH 4997-8: Vertically Integrated Research: Representations of Algebras: Quivers
  • Instructor: Profs. Yakimov and Muller
Quivers are simple diagrams which can encode linear algebraic data; specifically, vector spaces and linear maps between them. In this informal seminar aimed at advanced undergraduates, we will study quivers, and how to build moduli spaces which parametrize all possible choices of data. This seminar will expose students to a casual research atmosphere and teach them concepts on the cutting edge of modern mathematics. Out-of-class obligations will be minimal.
  • MATH 7002: Communicating Mathematics II
  • 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 calculus, how to write mathematics for publication, how to give a mathematical talk, 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 7211: Algebra II.
  • Instructor: Prof. Hoffman.
  • Prerequisites: Math 7210
  • Text: A. W. Knapp, Basic Algebra, Birkhauser 2006.

This will be a continuation of Professor Madden's course in Fall 2012. The text is A. Knapp's book: Basic Algebra. We will start with chapter VIII (Commutative rings and Modules) and chapter IX (Fields and Galois Theory). If time is left, we will do some topics from Knapp's book Advanced Algebra. This will depend on the interests of the students, but it could include Semisimple rings and modules, Representations of finite groups, Introduction to homological algebra.

  • MATH 7280-1: Introduction to Schemes
  • Instructor: Prof. Muller.
  • Prerequisites: Math 7211 or permission of instructor. A previous course in algebraic geometry is not required.
  • Text: The Geometry of Schemes by Eisenbud and Harris, Springer GTM 197

Algebraic geometry began as the study of varieties - sets of solutions to systems of polynomial equations. The fundamental tool is the dictionary between the geometry of varieties and the algebra of their coordinate rings. Translating a problem from one perspective to the other often yields new techniques and intuition. Unfortunately, not every ring is the coordinate ring of a variety.

In this class, we will study schemes - a generalization of varieties which works with any commutative ring. The emphasis will be on the geometry and utility of schemes, as both can be lost in the technical details. Following the initial material, subsequent topics will be tailored to the skill and interests of the class. Homework will be assigned, with the possibility of student presentations in class.

  • MATH 7280-2: D-Modules
  • Instructor: Prof. Sage.
  • Prerequisites: Math 7211 and 7512.
  • Texts:
    • A Primer of Algebraic D-modules by S. C. Coutinho
    • D-modules, Perverse Sheaves, and Representation Theory by R. Hotta, K Takeuchi, and T. Tanisaki

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,...,xn 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: Modular Forms and Elliptic Curves.
  • Instructor: Prof. Mahlburg.
  • Prerequisites: Math 7210 and Math 4036 (or equivalent)
  • Text: Introduction to Elliptic Curves and Modular Forms (Graduate Texts in Mathematics) by Neal I. Koblitz

This course is an introduction to elliptic curves and modular forms, which underlie many of the notable results in modern number theory, including Fermat's Last Theorem and Catalan's Conjecture. Topics will include: elliptic curves, elliptic functions, elliptic curves over finite fields, L-functions, modular forms, theta functions, Eisenstein series, Hecke operators, Shimura correspondence, arithmetic applications, integer partitions

  • MATH 7320: Ordinary Differential Equations.
  • Instructor: Prof. Wolenski.
  • Prerequisites: The basics of Real Analysis (or even just Advanced Calculus) is the only prerequisite, and in particular, no a priori background in ODEs will be assumed. The course should be easily accessible to those who do not aspire to be analysts, but expect at some point to teach an undergraduate course in ODEs.
  • Text: Differential Equations: Qualitative Theory, by L. Barreira and C. Valls; published by AMS

The course will cover the qualitative theory of Ordinary Differential Equations. This includes the usual existence and uniqueness theorems, linear systems, stability theory, hyperbolic systems (the Grobman-Hartman Theorem), and if time permits, an introduction to Control Theory.

  • MATH 7330: Functional Analysis.
  • Instructor: Prof. Estrada.
  • Prerequisites: Math 7311 or equivalent.
  • Text: Topological Vector Spaces, Distributions, and Kernels, by Francois Treves, Dover 2006, ISBN 9780486453521

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 Math 7380-2: Tempered distributions and integral transforms of generalized functions.
  • Instructor: Prof. Rubin
  • Prerequisites: Multidimensional Calculus (2057, 4035 or equivalent).
  • Text: G. I. Eskin, Boundary value problems for elliptic pseudodifferential equations, Amer. Math. Soc., Providence, R.I., 1981. Some other texts will be also used.

Tempered distributions are an important class of generalized functions which are well-adapted for integral transformations of functions and distributions in several variables arising in mathematics, physics, statistics, and many other areas. The main topics of this course-seminar are the theory of tempered distributions, convolution operators, and the Fourier transform. We also plan to consider applications to several important classes of operators in Analysis. This course is a nice addition to 4997-3 in Fall 2013, however, it does not assume any familiarity with generalized functions.

  • MATH 7384: Topics in Material Science: Extreme Geometries and Optimal Materials
  • Instructor: Prof. Lipton.
  • Prerequisites: Any one of Math 3355, 3903, 4031, or 4038, or their equivalent.
  • Text: Lecture Notes. Further reading and reference:
    • Theory of Composites, Graeme Milton
    • Asymptotic Analysis for Periodic Structures, A. Bensoussan, J.-L. Lions, and G. Papanicolaou
    • Homogenization of Differential Operators and Integral Functionals, V.V. Jikov, S.M. Kozlov, and O.A. Oleinik

The study of heterogeneous media has a distinguished history involving the fundamental contributions of J.C. Maxwell and A. Einstein. Over the last thirty years there has been an explosion of activity in applied science and mathematics delivering new methods for the design of heterogeneous media with novel properties. The course provides a self contained and hands on introduction to the theory as well as a guide to the current research literature useful for understanding the mathematics and physics of complex heterogeneous media. The course begins with an introduction of variational tools and asymptotic techniques necessary for characterizing macroscopic behavior of multi-scale heterogeneous media. Next we explore methods for constructing solutions of field equations inside extreme microstructures such as the the space filling coated spheres construction of Hashin and Shtrikman and the confocal ellipsoid construction of Milton and Tartar. The third part of the course shows how to apply these tools to recover fundamental theorems that characterize extreme field behavior inside complex materials in terms of the statistics of the random medium.

  • MATH 7390-2: Stochastic Analysis.
  • Instructor: Prof. Sengupta.
  • Prerequisites: Math 7311
  • Text: None

We will study stochastic processes, with special emphasis on Brownian motion. After an examination of the nature of Brownian motion paths we will study the beautiful relationship between stochastic processes and partial differential equations. Our exploration will include the Feynman-Kac formula, which stands at the juncture of probability theory, differential equations, and functional analysis. We will also study stochastic integrals and Ito's formula for stochastic differentials. At a more abstract level the course will include an introduction to abstract Wiener spaces and analysis on such spaces. The mathematics developed in this course finds a wide range of applications, ranging from finance to quantum physics.

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 7400.

  • MATH 7490: Graph Minors.
  • 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.

This course will introduce the homology theory of topological spaces. To each space there is assigned a sequence of abelian groups, its homology groups. We will discuss methods for calculating these groups, and use them to prove results such as the Brouwer fixed-point theorem (in all dimensions), the Ham Sandwich Theorem, and generalizations of the Jordan curve theorem.

  • MATH 7550: Differential Geometry.
  • Instructor: Prof. Vela-Vick.
  • Prerequisites: Math 4032 (or equivalent) and Math 7510
  • Text: John M. Lee, Introduction to Smooth Manifolds, Springer, GTM 218

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: Knot Theory.
  • Instructor: Prof. Dasbach.
  • Prerequisites:
  • Text:

The course is an introduction to advanced knot theory. Topics that are covered are: Basic properties of knots, links, braids and 3-manifolds, state and Seifert surfaces for links, quantum invariants, knot (co-) homology theories.

  • MATH 7590-2: Topological Quantum Field Theory.
  • Instructor: Prof. Gilmer.
  • Prerequisites: Math 7510, 7512, some exposure to knots or links
  • Text: None

A TQFT is a functor from a cobordism category to a category of vector spaces (or more generally a category of modules). A cobordism category is (roughly speaking) a category whose objects are manifolds of a certain dimension ( say d) and whose morphisms are manifolds of dimension d+1 which mediate between two manifolds of dimension d. We will discuss TQFTs from an axiomatic point of view. We will discus various examples of TQFTs. We will discuss a method that constructs a TQFT starting with invariants of closed (d+1)-dimensional manifolds. We will focus mainly on the case d=2. Along the way, we will discuss invariants of 3-dimesnional manifolds from a skein theory point of view. Skeins are linearizations of the set of links in a given 3- manifold.

  • MATH 7710: Advanced Numerical Linear Algebra.
  • Instructor: Prof. Wan.
  • Prerequisites:
  • Text:
In this course, we will focus on how to perform matrix computations efficiently and accurately. Topics will include Gaussian elimination, singular value decomposition, eigenvalue solvers and iterative methods for linear systems. Both theoretical analysis and numerical experiments will be discussed. Programming language is required and Matlab is recommended.