Summer 2011
- 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 4999-2: Problem Lab in
Topology-1—practice for PhD Qualifying Exam in Topology.
- Instructor: Prof. Gilmer.
- Prerequisite: Math 7510.
- Text: Online Test Bank.
- MATH 4999-3: Problem Lab in
Algebra-1—practice for PhD Qualifying Exam in Algebra.
- Instructor: Prof. Perlis.
- Prerequisite: Math 7210.
- Text: Online Test Bank.
Fall 2011
- MATH 4997-1: Vertically Integrated
Research: Electromagnetic Waves in Heterogeneous Structures.
- Instructor: Dr. Welters, Prof. Lipton and Prof. Shipman
- Prerequisites:
- Audience: Graduate and undergraduate students; those interested in
partial differential equations, spectral theory, and applications.
This course centers upon the partial differential equations
of electrodynamics in periodic media. The spectral theory for fields in
periodic structures is called Floquet theory. It is associated with the
Floquet transform, which is the Fourier transform of the integer lattice
subgroup of Euclidean space. Even the one-dimensional case is
mathematically interesting, being naturally connected to
differential-algebraic equations and the spectral theory thereof. An
intriguing application is to "slow light", whose analysis requires the
analytic perturbation theory of non-selfadjoint operators and in particular
the perturbation of matrices with nontrivial Jordan blocks.
The course will be in the style of an interactive seminar, in which
undergraduate and graduate students and faculty will present related topics
or problems. Participants will learn about directions and open problems in
current mathematical research.
- MATH 4997-2: Vertically Integrated
Research: The fundamental lemma and
affine Springer fibers.
- Instructor: Profs Sage and Achar.
- Prerequisites: Math 4200 and 2085, or permission of the instructor.
We will study topics related to a major and very recent advance, the
proof of the so-called "Fundamental Lemma" in the Langlands program,
for which Ngo Bau Chao received the Fields Medal in 2010.
Specifically, we will study the approach via "affine Springer fibers"
developed in the work of Goresky-Kottwitz-MacPherson. Affine Springer
fibers are geometric objects that can often be described very
concretely–for instance, in one class of important examples, an
affine Springer fiber is just a collection of 2-spheres touching at
points. These spaces can be studied in a very combinatorial way that
lends itself to hands-on computations. No prior knowledge is presumed
beyond basic familiarity with abstract and linear algebra.
- MATH 4997-3: Vertically Integrated
Research: Cluster Algebras.
- Instructor: Prof. Yakimov and Dr.
Muller.
- Prerequisites: Permission of Instructor.
Cluster Algebras are a new and exciting intersection between a wide array
of mathematical fields. They were defined by Sergey Fomin and Andrei
Zelevinsky in 2001 in relation to problems in combinatorics and Lie groups.
Only a few years later they started playing a key role in a number of
developments in representation theory, topology, combinatorics and algebraic
geometry.
The beauty of the subject is that the fundamentals require almost no
prerequisites. We will begin with an introduction presented by students
familiar with the topic, and there will be a heavy emphasis on examples and
simplicity throughout. Thus new undergraduate students who register will be
able to understand and lecture on a number of topics. Graduate students
specializing in representation theory, topology, combinatorics and algebraic
geometry will see relations and applications of cluster algebras.
This semester of this course will focus on the algebraic side of cluster
algebras and developing topics such as quantum cluster algebras. Concrete,
computable examples will be emphasized throughout. The class will also
transition from exposition of known material to research-oriented learning
and exploration. Students will not be required to present or do homework,
but participation is strongly encouraged to help follow the material.
- MATH 4997-4: Vertically Integrated
Research: Links on a Torus and Dimer Invariants
- Instructor: Profs. Dasbach and Stoltzfus.
- Prerequisites:
The knot theory VIR course will study the properties of links with diagrams
which project to the torus fiber of the Hopf
link. Specifically, we will study the dimer models for the zig-zag links
introduced by Stienstra, their Kasteleyn matrices
and work to develop related link invariants for these links.
- MATH 4997-5: Vertically Integrated
Research: All you ever need to know about SL(2,R).
- Instructor: Dr. Harris and Profs. He and Olafsson.
- Prerequisites: For graduate students: 7311 or equivalent. For undergraduate
students 1550 and 1552 or equivalent.
Representations of semisimple Lie groups are play a central role in several
parts of mathematics and physics, including number theory, geometry, and
quantum physics. In this course we will discuss several aspects of this
broad theory but concentrate on the simples example SL(2,R). We discuss the
classification of representations, harmonic analysis on the upper half
plane, orbital integrals and other topics. We will partially follow the book
by V. S. Varadarajan: An Introduction to Harmonic Analysis on Semisimple Lie
groups.
- 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: An undergraduate course in abstract algebra or equivalent.
- Text: Nathan Jacobson, Basic Algebra I, Second Edition (Paperback). Dover
Publications, 2009.
- Recommended references:
Michael Artin, Algebra, 2nd Edition. Addison Wesley 2010.
Thomas W. Hungerford, Algebra. Graduate Texts in Mathematics, vol. 73.
Springer 1980.
Anthony W. Knapp, Basic Algebra (Cornerstones). Birkhäuser Boston 2006.
Serge Lang, Algebra 3rd edition. Springer 2002.
This course provides the algebraic foundations for graduate study,
covering the basic notions of group, ring, and module theory. Topics
include: symmetric and alternating groups, the isomorphism theorems,
group actions, the Sylow theorems. Polynomial rings, Euclidean
domains, principal ideal domains, unique factorization domains.
Modules over PIDs and applications to abelian groups and linear
algebra.
- MATH 7280: Commutative Algebra and
Algebraic Geometry.
- Instructor: Prof. Hoffman.
- Prerequisites: Math 7210 and 7510; understanding of basic algebra and topology.
- Text: Ideals, Varieties and Algorithms, by Cox, Little, O'Shea.
- Recommended: Commutative Algebra with a view towards Algebraic Geometry, by D. Eisenbud.
Introduction to Commutative Algebra and Algebraic Geometry with an
emphasis on computation. Basics about prime ideals, Nullstellensatz,
affine and projective varieties. A little of the language of sheaves
and schemes. Also we will introduce the use of software packages
especially Macaulay2 and Singular, in the context of the Sage environment.
- MATH 7290-2: Noncommutative Ring
Theory.
- Instructor: Prof. Yakimov.
- Prerequisites: Graduate algebra.
- Text:
- Recommended texts:
- K. R. Goodearl and R. B., Jr. Warfield, An introduction to noncommutative
Noetherian rings. 2nd edition, London Math. Soc. Student Texts, 61.
Cambridge Univ. Press, Cambridge, 2004.
- G. R. Krause and T. H. Lenagan, Growth of algebras and Gelfand-Kirillov
dimension. Revised ed. Grad Studies in Math, 22. Amer. Math. Soc.,
Providence, RI, 2000.
- A. Joseph, Quantum groups and their primitive ideals. Ergebnisse der Mathematik und ihrer Grenzgebiete (3) 29. Springer-Verlag, Berlin, 1995.
- K. A. Brown and K. R. Goodearl, Lectures on algebraic quantum groups.Adv. Courses in Math. CRM Barcelona. Birkhäuser Verlag, Basel, 2002.
The course will introduce basic techniques and constructions in
noncommutative
ring theory, and illustrate how they work on concrete examples from the
theory
of Quantum Groups. It will be suitable both for students who specialize in
representation or ring theory and for students who need to use ring
theoretic techniques in other areas. A partial list of topics includes: ring
spectra, skew polynomial extensions, module theory, localization,
Gelfand-Kirillov dimension. These techniques and notions will be illustrated
for two large families of rings called quantum function algebras and quantum
nilpotent algebras which received a great deal of attention in the last 20
years.
- MATH 7311: Real Analysis I.
- Instructor: Prof. Richardson.
- 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. Visit the class website for further information.
- MATH 7350: Complex Analysis.
- 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 7360: Probability Theory.
- Instructor: Prof. Sundar.
- Prerequisites: Math 7311 (Real Analysis I). Students are not
expected to have any prior knowledge of probability theory.
- Text: Probability and Measure, Patrick Billingsley.
The course is a self-contained introduction to modern probability theory.
It starts right from the concept of probability measures, random variables,
and independence. Well-known limit theorems for sums of independent random
variables such as the Kolmogorov's law of large numbers, and the
three-series theorem will be discussed in detail. Sums of independent
random variables form a prototype of an important class of stochastic
processes known as martingales, and therefore play a major role. Next, weak
convergence of probability measures will be introduced and studied in
detail which would lead us to the proof of the central limit theorem. A
main part of the course is to understand conditional probability and build
the basic theory of martingales. Brownian motion is an important example of
a continuous-time martingale, and its basic features will be briefly
discussed.
- MATH 7380-2: Applied Stochastic
Analysis.
- Instructor: Prof. Kuo.
- Prerequisites: Math 7311
- Textbooks:
- Kuo, H.-H.: Introduction to Stochastic Integration.
Universitext, Springer, 2006.
- Kuo, H.-H.: Gaussian Measures in Banach Spaces.
Lecture Notes in Math., Vol. 463, Springer, 1975.
(Reprinted by BookSurge Publishing, 2006)
- Kuo, H.-H.: White Noise Distribution Theory.
CRC Press, 1996.
In this course we will cover the following topics:
- Stochastic differential equations.
- Applications to mathematical finance.
- Gaussian processes.
- Abstract Wiener space.
- White noise theory.
- General theory of stochastic integration.
- MATH 7380-3: Topics in Computational
Mathematics—Shape Optimization.
- 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): Introduction to Shape Optimization: Theory,
Approximation, and Computation, by J. Haslinger and R. A. E. Mäkinen.
- Reference Texts (not required):
Shapes and Geometries: Analysis,
Differential Calculus, and Optimization, by Delfour and Zolesio.
Perspectives in Flow Control and Optimization, by Gunzburger.
Variational Methods in Shape Optimization Problems, by Bucur and
Buttazzo.
Numerical Methods in Sensitivity Analysis and Shape
Optimization, by Laporte and LeTallec.
PDE-constrained optimization is a field of growing importance, especially
with recent advances in scientific computing. Examples are drag
minimization, structural/mechanical design, fluid flow control, process
control, finance, and inverse problems. In this course, we will study a
particular class of PDE-constrained optimization called shape
optimization. Topics will include:
- Some classic examples of optimal shape problems.
- Theory of optimal control problems with PDE constraints; calculus of
variations, adjoint equations.
- Basic tools of shape differential calculus for deforming domains.
- Methods for handling free boundaries/moving domains:
front-tracking/ALE methods, level set methods.
- Solution methods for solving shape optimization problems with
constraints, such as variational/finite element methods for saddle point
systems.
- (time permitting): Introduction to time-dependent active shape control
problems, or topology optimization.
The tools in this course can be generalized to a wide range of problems
(almost anything that uses continuum models). Grading will consist of a
small number of homework assignments, assigned literature reading, and a
significant final project instead of exams.
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.; this is necessary for the project. Knowledge of
continuum mechanics and optimization is a plus, but not required
(necessary concepts will be reviewed).
- MATH 7380-4: Finite Difference
Methods.
- Instructor: Prof. Wan.
- Prerequisites: Math 4065 or equivalent
- Text: Time dependent problems and difference method by Gustafsson, B. ,
Kreiss, Heinz-Otto, Oliger, Joseph.
This course will focus on finite difference methods for time dependent
partial difference equations (PDEs).
Specific topics include: Fourier series and trigonometric interpolation,
well-posedness of PDEs, finite difference schemes
for parabolic and hyperbolic equations, stability and convergence theory for
difference methods.
- MATH 7384: Riemann surfaces and modeling
- Instructor: Prof. Antipov.
- Prerequisites: Permission of the Instructor
- Text: Lecture notes
Riemann surfaces are widely used in mathematical physics (finite-gap
solutions of nonlinear equations, random matrices and orthogonal
polynomials), fluid mechanics, elasticity, and electromagnetics. This course
will give an introduction to the theory of algebraic functions (abelian
integrals and differentials, theta function, Cauchy-type kernels on
hyperelliptic and $n$-sheeted surfaces), the Riemann-Hilbert problem on a
compact Riemann surface, and Jacobi inversion problem. Then the course will
concentrate on applications of the theory of Riemann surfaces. We will
discuss
periodic solutions to the Korteweg - de Vries equation, a generalized
hodograph method for multiply connected domains and supercavitating flow,
and Wiener-Hopf matrix factorization in fracture mechanics and diffraction
of electromagnetic waves.
- MATH 7386: Theory of Partial Differential
Equations.
- Instructor: Prof. Lipton.
- Prerequisites: Math 7311 (Real Analysis I) or the equivalent.
- Text: 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 7390: Topics in Lie Groups and
Their Applications.
- Instructor: Prof. He.
- Prerequisites: a Lie group or Lie algebra course
- Text:
- Recommended reference:
- Harmonic Analysis in Phase Space, (Annals of Mathematical studies 122),
by G. Folland.
- The Weil representation, Maslov index and theta series, (Progress in
Mathematics 6), by G. Lion and M. Vergne.
- Representations and Invariants of the Classical Groups, by R. Goodman
and N. Wallach.
In this course, we shall discuss several important
applications of Lie groups in number theory, quantum mechanics and
invariant theory:
- Representation of the Heisenberg group
- Weil Representation and Theta series
- Theta lift and Automorphic forms
- The oscillator representation and lowest energy representations with
symplectic symmetries
- Classical Invariant theory and Howe's Duality
If we have time, we will discuss representations with nonzero cohomology.
- MATH 7490: Linear and Integer
Programming.
- Instructor: Prof. Ding.
- Prerequisites: Linear Algebra (Math 2085 or equivalent)
- Text: Theory of Linear and Integer Programming, by Schrijver. This text is recommended since it is the best book in the field and since
most materials of this course will come from this book. Since this book is
very comprehensive, we will only be able to cover about 40% of it. Other
materials will come from another good book: Integer Programming, by
Wolsey.
The purpose of this course is to introduce the theory
of linear and integer programming. The focus will be the mathematical
theory rather than that actual numerical computation. Main topics of this
course will include: linear inequalities, the structure of polyhedra,
linear programming and its algorithms, well solved integer programs, other
algorithms for integer programming (cutting plane, branch-and-bound, etc).
- MATH 7510: Topology I
- Instructor: Prof. Litherland.
- 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. Dasbach.
- Prerequisites: Math 7510 and 7512
- Text:
This course continues the study of algebraic topology begun in MATH 7510 and
MATH 7512.
While MATH 7510 developed the theory of fundamental groups and MATH 7512
developed homology theories for topological spaces the focus of this course
will be cohomology theory which is dual to homology theory. However, one of
the advantages of developing cohomology theory of spaces is that they are
naturally equipped with a ring structure.
The course will be mainly based on Hatcher's book Algebraic topology, and
additional material that we will hand out.
- MATH 7590: Differential Topology.
- Instructor: Prof. Gilmer.
- Prerequisites: 7550
- Text: Differential Topology (Graduate Texts in Mathematics, Vol 33) by
Morris W. Hirsch, published by Springer
We will study differential manifolds with an emphasis on topological results
and tools used in topology. One of the first results that we will prove is
that an abstractly defined smooth manifold of dimension n embeds in
Euclidean space of dimension 2n+1. We will study transversality, vector
bundles, intersection numbers etc.
Spring 2012
- MATH 4997-1:
Vertically Integrated Research: All you ever need to know about SL(2,R).
- Instructor: Dr. Harris and Profs. He and Olafsson
- Prerequisites:
- Text: Chapter V in Unitary Representations and Harmonic Analysis - An Introduction, M. Sugiura and An Introduction to Harmonic Analysis on Semisimple Lie groups by V. S. Varadarajan
Representations of semisimple Lie groups play a central role in several parts of mathematics and physics, including number theory, geometry, and quantum physics. In this course we will discuss several aspects of this broad theory but concentrate on the simples example SL(2,
R). In this course our main goal is to prove the Plancherel theorem for SL(2,
R). We will cover the theory of invariant eigendistributions as well as some results on orbital integrals.
- MATH 4997-2:
Vertically Integrated Research: The fundamental lemma and affine Springer fibers.
- Instructor: Dr. Bremer, Prof. Achar and Prof. Sage
- Prerequisites: 4200 and 2085, or permission of the instructor.
- Text: None
We will study topics related to a major and very recent advance, the
proof of the so-called
Fundamental Lemma in the Langlands program,
for which Ngo Bau Chao received the Fields Medal in 2010.
Specifically, we will study the approach via
affine Springer fibers
developed in the work of Goresky-Kottwitz-MacPherson. Affine Springer
fibers are geometric objects that can often be described very
concretely--for instance, in one class of important examples, an
affine Springer fiber is just a collection of 2-spheres touching at
points. These spaces can be studied in a very combinatorial way that
lends itself to hands-on computations. No prior knowledge is presumed
beyond basic familiarity with abstract and linear algebra. Attendance
in this seminar in Fall 2011 is not required.
- MATH 4997-6:
Vertically Integrated Research: Cluster Algebras.
- Instructor: Dr. Muller, Prof. Yakimov and Prof. Dan-Cohen
- Prerequisites:
- Text:
Cluster Algebras are a new and exciting intersection between a wide array of mathematical fields. They were defined by Sergey Fomin and Andrei Zelevinsky in 2001 in relation to problems in combinatorics and Lie groups. Only a few years later they started playing a key role in a number of developments in representation theory, topology, combinatorics and algebraic geometry.
The beauty of the subject is that the fundamentals require almost no prerequisites. We will begin with an introduction presented by students familiar with the topic, and there will be a heavy emphasis on examples and simplicity throughout. Thus new undergraduate students who register will be able to understand and lecture on a number of topics. Graduate students specializing in representation theory, topology, combinatorics and algebraic geometry will see relations and applications of cluster algebras.
This semester of this course will begin to develop the categorical and homological aspects of category theory, as we see how elementary manipulations of diagrams lead to operations on certain categories. Concrete, computable examples will be emphasized throughout. The class will also transition from exposition of known material to research-oriented learning and exploration. Students will not be required to present or do homework, but participation is strongly encouraged to help follow the material.
- MATH 4997-4:
Vertically Integrated Research: Knot invariants coming from Lie algebras.
- Instructor: Prof. Dasbach, Prof. Stoltzfus, and Dr. Kearney
- Prerequisites: Math 7510: Topology I
- Text: Research Literature
We will study knot invariants arising from Lie algebras
including the proof of the MMR conjecture stated by Melvin and Morton (and elucidated further by Rozansky) saying that the Alexander-Conway polynomial of a knot can be read from some of the coefficients of the Jones polynomials of cables of that knot (i.e., coefficients of the "colored" Jones polynomial). encapsulated in the paper of Dror Bar-Natan and Stavros Garoufalidis
In addition, we will read the paper of Saleur & Kauffman relating the Lie algebra GL(1|1) to the Alexander Polynomial and other approaches to the MMR conjecture by XS Lin and A. Vaintrob.
- MATH 4997-5:
Vertically Integrated Research: Domain Decomposition Methods.
- Instructor: Dr. Barker, Prof. Brenner and Dr. Park
- Prerequisites: Permission of Instructor
- Text: None required
Domain decomposition methods are numerical methods for partial differential equations that provide a natural approach to solving large scale problems on parallel computers. We will survey the field and discuss new developments. Implementation issues will also be addressed.
- 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.
- MATH 7211-2: Algebra II.
- Instructor: Prof. Achar.
- Prerequisites: Math 7210 Algebra I.
- Text: Basic Algebra I and Basic Algebra II (second edition) by Nathan Jacobson.
This is the second semester of the first-year graduate algebra sequence. In this course, we will further develop the topics introduced in the first semester. Specific topics include: normal and separable field extensions; Galois theory and applications; solvable groups, normal series, and the Jordan-Holder theorem; tensor products and Hom for modules; noetherian rings; the Hilbert Basis Theorem; and algebras over a field, including Wedderburn's and Maschke's Theorems.
- MATH 7280: Homological Algebra.
- Instructor: Prof. Muller.
- Prerequisites: 7211 or permission of instructor. One of
7280 ,
7290-2, or 7520 recommended
- Text: An Introduction to Homological algebra, by Charles Weibel. 9780521559874
In many areas of mathematics, such as topology and representation theory, algebraic objects called chain complexes and their homology arise when studying invariants or obstructions. This class will study these objects in general, and see why the arise so frequently when studying a mathematical object with an abelian category attached to it.
This class will start with an introduction to categories and abelian categories, and then develop the machinery of homological algebra. There will be an emphasis on examples, explicit computation and classroom interaction, and homework will be assigned. Most examples will be from algebra (categories of modules or representations), with less emphasis on the examples from topology (categories of sheaves), though the material can be tailored to the students.
- MATH 7290-1: Langlands Program.
- Instructor: Prof. Sage.
- Prerequisites: The first year algebra sequence.
- Text: The Local Langlands Conjecture for GL(2) by C. Bushnell and G. Henniart
The Langlands program is a network of conjectures connecting number
theory, representation theory, harmonic analysis, and algebraic
geometry. Indeed, it links so many different areas that it is
sometimes referred to as a "grand unified theory" of mathematics. To
give a specific example, let F be a p-adic field (a finite extension
of the field of p-adic numbers). The local Langlands conjecture for
GL(n)(now proven) is a far-reaching generalization of local class field
theory. It gives a relationship between arithmetic information
about F as encapsulated in the Weil-Deligne group of F--a group
closely related to the absolute Galois group of F--and the
representation theory of the algebraic group GL
_{n}(F). Local class
field theory is the case n=1.
The goal of this course is to give a gentle introduction to some main
themes of the classical Langlands program, with an emphasis on
concrete examples. We will concentrate on GL(n) and indeed primarily
on the cases n=1 and 2. Time permitting, we will also describe and
motivate the basic premises of the geometric Langlands program.
- MATH 7290-2: Computational Number
Theory.
- Instructor: Prof. Morales.
- Prerequisites: A year of graduate-level abstract algebra such as the sequence Math 7210-11 or equivalent.
- Text: The main reference is A Course in Computational Algebraic Number Theory by H. Cohen, Springer-Verlag 1996.
This will be an introductory course in algebraic number theory, with special attention given to the computational aspects of it. In addition to studying the classical topics of algebraic number theory such as rings of integers, factorization of ideals, units, class groups, etc., we will also discuss algorithms for explicit computations. When needed, I will complement the textbook with material from more conventional texts on algebraic number theory such as the books by Froehlich-Taylor and Ireland-Rosen.
- MATH 7290-3: Mathematical Logic and
Model Theory.
- Instructor: Prof. Delzell.
- Prerequisites: Mathematical maturity; no specific courses,
but a naive understanding of cardinal numbers,
and some familiarity with groups and fields.
- Text: Mathematical Logic and Model Theory: A Brief Introduction,
by Alexander Prestel and Charles Delzell,
Springer Monographs in Mathematics, 2011 (expected in October).
We will study the basics of first-order logic,
and then move into model theory.
Topics include: the satisfaction of a first-order formula by a structure;
Goedel's completeness theorem
(every sentence true in all structures is provable);
the compactness (or "finiteness") theorem
(a set S of sentences possesses a model
if and only if every finite subset of S does);
the Lowenheim-Skolem Theorem
(every infinite structure possesses a countable,
"elementary" substructure; in particular,
if the Zermelo-Fraenkel axioms for set theory are consistent,
then they have a countable model, even though
ZF proves the existence of uncountable sets);
saturated structures; ultraproducts;
model complete theories; and complete theories.
- MATH 7320: Ordinary Differential
Equations.
- Instructor: Prof. Malisoff.
- Prerequisites: Undergraduate courses on ordinary differential equations, linear algebra, and advanced calculus. Math 7311 or its equivalent recommended.
- Text: M. Hirsch, S. Smale, and R. Devaney, Differential Equations, Dynamical Systems and an Introduction to Chaos, Second Edition, Elsevier, New York, 2004 (ISBN 978-0-123497-03-1)
This course provides a basic introduction to the qualitative theory of ordinary differential equations at the beginning graduate level.
It will focus on topics from Chapters 1-10 and 17 of the text. This will be a rigorous, proof-oriented theory course. Possible topics may include existence and uniqueness theory, Peano's theorem, applications of Gronwall's inequality, dependence of solutions on initial conditions, linear systems, stability of equilibrium points, Lyapunov functions, and phase plane analysis. The exact coverage will depend on the progress of the class.
- MATH 7330-2: Functional Analysis.
- Instructor: Prof. Davidson.
- Prerequisites: Math 7311
- Text: Functional Analysis by George Bachman and Lawrence Narici 9780486402512
Functional analysis is the study of topological vector spaces and provides fundamental tools needed for research in pure and applied mathematics. Topics include: Normed vector space; Banach spaces and their generalizations; Baire category, Banach-Steinhaus, open mapping, closed graph, and Hahn-Banach theorems; duality in Banach spaces, weak topologies; Hilbert Spaces; and other topics such as commutative Banach algebras, spectral theory, and Gelfand theory.
- MATH 7380-1: Partial Differential
Equations.
- 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):
- Elliptic partial differential equations of second order by David Gilbarg and Neil S. Trudinger. Springer Verlag 2001. ISBN-10: 3540411607,
ISBN-13: 9783540411604.
- 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 to 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.
- Instructor: Prof. Shipman.
- Prerequisites: : A good analysis course and basic complex variables.
- Text: Various papers, notes, and texts
Problems of the guiding of electromagnetic, acoustic, and elastic waves by material structures pervade mathematical physics, and their analysis penetrates a rich swath of classical and modern mathematics. This course will concentrate on a number of specific problems for which techniques in complex and functional analysis will be developed and applied.
(1) A waveguide containing an obstacle.
The spectral theory for self-adjoint operators in Hilbert space is realized in a concrete way in this problem. The continuous spectrum is associated with extended fields--those resulting from the scattering of a guided wave by the obstacle in the guide. The eigenvalues are frequencies of trapped modes--those fields whose energy is exponentially trapped around the obstacle. Eigenvalues embedded in the continuous spectrum are unstable with respect to perturbations of the structure. This instability causes anomalous scattering behavior, which we will analyze by means of complex-analytic perturbation techniques.
(2) A bifurcated waveguide. Exact solutions for waves in split waveguides can be obtained by sophisticated methods of complex variables called Wiener-Hopf techniques. They involve Tauberian and Abelian theorems of Fourier analysis.
(3) Periodically layered waveguides. The monodromy matrix, which transfers field data across one period, is the fundamental object of analysis. Through it, one develops the spectral theory for periodic media, known as the Floquet theory, which is the Fourier transform of the subgroup of translations in the real line. The salient phenomenon is the existence of stop-bands--frequency intervals in which the coherent scattering by the periodic structure prohibits the propagation of waves. Localized defects admit trapped modes with frequency in a stop-band.
(4) Nonlinear waveguides. High resonant amplification of fields makes the study of nonlinearities necessary. We will see how even weak nonlinearity has pronounced effects, such as bistability, in the presence of resonance.
See the course webpage for a more in-depth description of the course.
- MATH 7390: Stochastic Analysis.
- Instructor: Prof. Sundar.
- Prerequisites: Math 7311, Math 7360 or consent of the instructor.
- Text: Typed Lecture Notes will be provided
The aim of the course is to learn the basic theory of stochastic differential equations. The course starts with a study of Wiener processes and martingales. Stochastic integration with respect to a Wiener process will be developed with full details.
Next, the Itô formula will be proved and its applications discussed.
Stochastic differential equations (SDEs) will be introduced,
and the martingale problems posed by SDEs will be studied. Next, the course will focus on topics such as (i) connection between SDEs and partial differential equations, (ii) large deviations principle for diffusion processes, and (iii) invariant measures and ergodicity.
Throughout the course, several examples will be given to illustrate the theory.
- MATH 7490: The Tutte Polynomial
for Matroids and Graphs.
- Instructor: Prof. Oxley.
- Prerequisites: Math 7400 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.
- Instructor: Prof. Dani.
- Prerequisites: Math 7510
- Text: Algebraic Topology by A. Hatcher
A fundamental problem in topology is that of determining whether or not two spaces are topologically equivalent. The basic idea of algebraic topology is to associate algebraic objects (groups, rings,
etc.) to a topological space in such a way that topologically equivalent spaces get assigned isomorphic objects. These algebraic objects are invariants of the space, and provide a means for distinguishing between topological spaces. The focus of this course is one such class of invariants, namely the homology groups of a topological space. Topics include simplicial, singular, and cellular homology, computations of homology groups and applications such as the Brouwer Fixed point theorem and generalizations of the Jordan curve theorem.
- MATH 7550: Differential Geometry.
- Instructor: Prof. Olafsson.
- Prerequisites: 4032 (or equivalent) and 7510
- Text: We will mainly follow Introduction to Smooth Manifolds (2003) by John M. Lee. Foundations of Differentiable Manifolds and Lie groups by Frank W. Warner is also good.
We will cover basic concepts such as: manifolds, submanifolds, tangent vactors, vector fields, vector bundles, transversality, Lie groups and other topics. For more information see
syllabus
- MATH 7590-1: Geometric
Topology—Topological Robotics.
- Instructor: Prof. Cohen.
- Prerequisites: MATH 7512 (MATH
7520 and/or MATH
7590 would be useful, but we will develop requisite tools from e.g., cohomology theory and Morse theory as necessary)
- Text: Invitation to Topological Robotics by M. Farber 978-3-03719-054-8
Topological robotics, part of what one might call applied algebraic topology, is a relatively new area studying topological problems inspired by robotics and engineering, as well as problems of robotics requiring topological tools.
Topics we will pursue in the course include configuration spaces of various types (e.g., configuration spaces of points in the plane, or on a surface, or on a graph, configuration spaces of linkages), and the motion planning problem from robotics. This last problem motivates a homotopy type invariant, the topological complexity of a space. We will investigate this invariant for spaces including some of the configuration spaces mentioned previously.
- MATH 7590-2: Theoretical & Computational Aspects
of 2 and 3-manifolds.
- Instructor: Prof. Stoltzfus.
- Prerequisites: Math 7510: Topology (and fundamental group)
- Text: Notes by Farb/Margalit and Thurston
This course will be devoted to the structure theorems on automorphisms of 2-dimensional manifolds and the 3-dimensional manifolds, their inter-relationships and computation of invariants that distinguish between two such objects. Central example will be 3-manifolds that fiber over a circle (e.g. fibered knots) and Seifert cyclic fiber spaces. To aid in the computation of invariants, various computational programs will be introduced: Twister (for surface diffeomorphisms), Dror Bar-Natan's KnotTheory Mathematica package and Snappea. The course will also introduce the use of William Stein's SAGE package as a common interface supporting coercion of output between packages, a notebook interface for saving computations and native graphical and group theoretical computations.
The course will include both the presentation of the theoretical development of the ideas and the concomitant computation aspects of the various structure theorems.
Student projects may be theoretical, computation or both.
- MATH 7710: Advanced Numerical Linear Algebra.
- Instructor: Prof. Sung.
- Prerequisites: Linear Algebra, Advanced Calculus
- Text: Fundamentals of Matrix Computations (Third Edition) by D.S. Watkins
Topics:
- Mathematical Tools: norms, projectors, Gram-Schmidt process,
orthogonal matrices, spectral theorem, singular value decomposition
and Gerschgorin's circles
- Error Analysis: floating point arithmetics, round-off errors, IEEE
standards, backward stability and conditioning
- General Systems: LU factorization, partial pivoting, Cholesky
factorization, least squares problems and QR factorization
- Sparse Systems: the methods of Jacobi, Richardson, Gauss-Seidel,
successive over-relaxation, steepest descent and conjugate gradient
- Eigenvalue Problems: power methods, Rayleigh quotient iteration,
deflation and QR algorithm