Summer 2013
- MATH 7590: Random Symplicial Complexes
- Instructor: Prof. Cohen.
- Prerequisite:
- Text:
- Random graphs have been objects of interest in mathematics and applications for approximately half a century. Viewing a graph as a one-dimensional simplicial complex, it is natural to investigate random simplicial complexes of higher dimension. In this course, we will work through a number of recent papers in this subject, focusing mainly on various topological, combinatorial, and probabilistic aspects of random complexes of dimension two.
- MATH 7999-1: Problem Lab in Real Analysis-1—practice for PhD Qualifying Exam in Analysis.
- Instructor:
- Prerequisite: Math 7311.
- Text: Online Test Bank.
- MATH 7999-2: Problem Lab in Topology-1—practice for PhD Qualifying Exam in Topology.
- Instructor:
- Prerequisite: Math 7510.
- Text: Online Test Bank.
- MATH 7999-3: Problem Lab in Algebra-1—practice for PhD Qualifying Exam in Algebra.
- Instructor:
- Prerequisite: Math 7210.
- Text: Online Test Bank.
Fall 2013
- MATH 4997-6: Vertically Integrated Research: Diagrammatic Algebra and Representation Theory
- Instructor: Profs. Sage and Achar.
- Prerequisites: Familiarity with basic group theory and linear algebra, such as from Math 4200 and Math 4153
- Text:
- Ordinary "algebra" is essentially a language for keeping track of compositions of functions. (Operations like addition and multiplication are, after all, themselves functions, and a complicated expression involving many of those operations is really a composition.) "Diagrammatic algebra" is a general term that refers to various ways of working with function-like objects that can compose not just sequentially, but in a 2-dimensional way. Diagrammatic algebra has its origins in category theory and has connections to and applications in representation theory, topology, mathematical physics, and other areas. In this course, we will look at many concrete, down-to-earth examples, which will involve drawing lots of pictures. We will then concentrate on how diagrammatic algebra can be used in representation theory to study reflection groups and quantum groups.
- MATH 4997-7: Vertically Integrated Research: The Alexander Polynomial and its Relatives
- Instructor: Profs. Dasbach and Stoltzfus, Dr. Kearney.
- Prerequisites: ElementaryTopology (MATH 4039) or equivalent, or permission of instructors.
- Text:
- One of the most studied knot invariants is the Alexander polynomial. There are many different ways known to effectively compute it. Some are combinatorial in nature, some are more closely related to the topology of the knot complement. We will discuss the Alexander polynomial as well as some of its relatives, such as knot signatures, the A-polynomial and Heegaard Floer knot homology.
- 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. 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 7280: Commutative Algebra.
- Instructor: Prof. Madden.
- Text: Nathan Jacobson, Basic Algebra II: Second Edition, Dover
- Prerequisites: Math 7210
- This course will present the parts of commutative algebra that are needed as a foundation for algebraic geometry. The main text will be Chapter 7 (Commutative Ideal Theory) of volume II of Jacobson's classic survey of algebra. This includes most of the material in the standard introduction to commutative algebra by Atiyah-MacDonald. I will supplement this with examples and illustrations from combinatorial commutative algebra (e.g., toric ideals, affine semigroup rings) and real algebraic geometry. Topics from other chapters of Jacobson and/or other sources may be included to the extent they support the main goal of the course, which is to understand commutative algebra as a language for geometry. <\li>
- MATH 7290-1: Representation Theory.
- Instructor: Prof. Sage.
- Prerequisites: Math 7211 or permission of the instructor
- Text: C. Procesi, Lie Groups: An Approach through Invariants and Representations, Springer, 2007.
- Representation theory is the study of the ways in which a given group may act on vector spaces. Intuitively, it investigates ways in which an abstract group may be interpreted concretely as a group of matrices with matrix multiplication as the group operation. Group representations are ubiquitous in modern mathematics. Indeed, representation theory has significant applications throughout algebra, topology, analysis, and applied mathematics. It also is of fundamental importance in physics, chemistry, and material science.
For example, it appears in quantum mechanics, crystallography, or any physical problem in which one studies how symmetries of a system affect the solutions.
This course is designed to give an introduction to representation theory, with an emphasis on Lie algebras and algebraic groups. The class is designed to be suitable both for students planning to specialize in representation theory and for those who need it for applications. It will start with an outline of the representation theory of finite groups over the complex numbers. We will then introduce complex algebraic groups and their Lie algebras. After discussing the basic theory of nilpotent, solvable, and semisimple Lie algebras, we will describe the classification of semisimple Lie algebras. We will continue with the universal enveloping algebra of a Lie algebra and the Poincare-Birkhoff-Witt theorem. We will then cover highest-weight representations of Lie algebras, including Verma modules and finite-dimensional irreducible representations. We will also discuss the relationship between the representations of a semisimple algebraic group and the representations of its Lie algebra.
- MATH 7311: Real Analysis I.
- Instructor: Prof. Davidson.
- Prerequisite: Math 4032 or 4035 or the equivalent.
- Required Text: L. Richardson, Measure and Integration: A Concise Introduction to Real Analysis, John Wiley & Sons, June 2009. ISBN: 978-0-470-25954-2 ;
- Recommended Reference: H. L. Royden, Real Analysis, 3rd ed., Macmillan, ISBN 0024041513.
- We will treat measure theory and integration on measure spaces. The examples of the real line and of Euclidean space will be emphasized throughout. Topics will include the Hopf extension theorem, completion of the Borel measure space, Egoroff's theorem, Lusin's theorem, Lebesgue dominated convergence, Fatou's lemma, product measures, Fubini's theorem, absolute continuity, bounded variation, Vitali's covering theorem, Lebesgue differentiation theorems, and the Radon-Nykodim theorem. Applications to Lp and its dual, and the Riesz-Markov-Saks-Kakutani theorem may be presented if there is sufficient time.
- MATH 7350: Complex Analysis.
- Instructor: Prof. Shipman.
- 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, power series, singularities, mapping properties, normal families, other topics.
- MATH 7360: Probability Theory.
- Instructor: Prof. Kuo.
- Prerequisite: Math 7311 (Real Analysis I) or the equivalent.
- Text: John W. Lamperti: Probability, second edition, John Wiley and Sons, Inc., 1996
- Coverage:
In the first two weeks I will give a brief review of elementary probability
theory and measure theory. Topics to be covered include the following:
- Kolmogorov's extension theorem
- Various types of convergence
- Laws of large numbers
- Convergence of random series
- Law of iterated logarithm
- Characteristic functions
- Bochner theorem
- Levy's continuity theorem
- Levy's equivalence theorem
- Central limit theorem
- Stable and infinitely divisible laws
- MATH 7370: Lie Groups and Representation Theory.
- Instructor: Prof. Olafsson.
- Prerequisites: Math 7311. A basic knowledge of differential geometry is also helpful.
- Text: We will not use any fixed textbook but mainly use our own lecture
notes. A preliminary version can be found at
http://www.math.lsu.edu/~olafsson/pdf_files/ln7370.pdf
Good introductory books on Lie groups include- J. Hilgert and K-H. Neeb: Structure and Geometry of Lie Groups
- S. Helgason: Differential Geometry, Lie Groups, and Symmetric Spaces
- V. S. Varadarajan: Lie Groups, Lie Algebras, and their Representations
- N. R. Wallach: Harmonic Analysis on Homogeneous Spaces.
- This is an introductory course in Lie groups and homogeneous spaces needed for further research in harmonic analysis and representation theory related to finite dimensional or infinite dimensional Lie groups. We will mostly consider linear Lie groups. The course starts with basic definitions and examples. Then we will discuss the exponential map, closed subgroups, the Lie algebra of a closed subgroup, group actions on manifolds, homogeneous spaces, and homogeneous vector bundles. Further material will depend on time and interests of the participants. The course will continue during the spring semester 2014. There we will cover more advanced topics in representation theory and harmonic analysis related to homogeneous spaces.
- MATH 7384: Material Science
- 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:
- Asymptotic evaluation of integrals: Watson's lemma and Laplace's method, the steepest descent method, the method of stationary phase.
- The WKB method
- Singular perturbation theory
- MATH 7386: Theory of Partial Differential Equations.
- Instructor: Prof. Antipov.
- Prerequisites: 7311 (Real Analysis I) or the equivalent.
- Text: Methods of Mathematical Physics. Vol. 2, Partial Differential Equations by Richard Courant.
Additional texts: Partial Differential Equations by Paul R. Garabedian, Partial Differential Equations by Lawrence C. Evans, and lecture notes. - Topics to be covered include
- General theory of first order PDEs including the Hamilton-Jacobi theory.
- Differential equation of higher order, their classification and methods of solution.
- Elliptic equations and potential theory. Boundary value problems and the method of integral equations. Introduction into the theory of pseudoanalytic functions.
- Hyperbolic equations in two independent variables. Method of characteristics. Applications to dynamics of compressible fluids. Discontinuities of solutions and shocks. Equations in more than two independent variables.
- Representation of solutions. Method of descent. Method of spherical means.
- MATH 7390-2: Topics in Optimization.
- Instructor: Prof. Zhang.
- Prerequisites: Math 2085 (Linear Algebra), Math 2057/2058 (Multidimensional Calculus ), Math 4032 (Advanced Calculus)
- Text: No textbook. Course will be based on class notes.
- Convex optimization has found a rich theory and wide applications in control systems, signal processing, data analysis and modeling, etc. The main goal of this course is to introduce some basic concepts in convex optimization and help the students to understand the complexity of the convex optimization problems, and study some provable efficiency of numerical algorithms supported by complexity bounds. Depending on the time, this course will cover some basic concepts and algorithms in nonlinear optimization, smooth and nonsmooth convex optimization, and some structural optimization.
- MATH 7390-3: Harmonic Analysis.
- Instructor: Prof. Rubin.
- Prerequisites: Math 7311
- Texts:
- Javier Duoandikoetxea, Fourier Analysis, Graduate Studies in Mathematics Vol. 29, AMS, 2001 978-0-8218-2172-5 (Required)
- L. Grafakos, Classical Fourier Analysis, 2008 (Recommended).
- L. Grafakos, Modern Fourier Analysis, 2008 (Recommended).
- Stein, E.M., and Weiss, G., Introduction to Fourier analysis on Euclidean spaces, Princeton Univ. Press, Princeton, NJ, 1971, ISBN-10: 069108078X, ISBN-13: 978-0691080789. (Recommended)
- This is a basic introduction to Fourier analysis on R^{n}. It includes the theory of the Fourier transform, Fourier series, and related topics. We will also discuss the theory of distributions, Fourier multipliers, Calderon-Zygmund singular integrals, interpolation theory, the Hardy-Littlewood maximal operator, and more.
- MATH 7490: Matroid Theory
- Instructor: Prof. Vertigan.
- 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. Dasbach.
- 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. We will also introduce simplicial complexes and manifolds, using them often as examples. A good supplementary reference is Chapter 1 of Algebraic Topology by Allen Hatcher, available online .
- MATH 7520: Algebraic Topology.
- Instructor: Prof. Cohen.
- Prerequisite: 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 in MATH 7510, the homology groups in MATH 7512) 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 in MATH 7512. 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.
- MATH 7590: Contact Topology.
- Instructor: Prof. Vela-Vick.
- Prerequisites: 7510 recommended, 7550 recommended
- Text: An Introduction to Contact Topology, by Hansjörg Geiges, Cambridge University Press, 2008, ISBN: 9780521865852.
- This course will introduce the tools and techniques of modern contact geometry. This includes foundational results like Darboux's theorem, Gray's theorem, and the existence of contact structures on all 3-manifolds. We will also discuss the tight/overtwisted dichotomy and Legendrian and transverse knots. With basic tools in hand, we will focus on exploring known construction and classification results using characteristic foliations, convex surfaces, symplectic fillings, contact surgery and open book decompositions.
Spring 2014
- MATH 4997-1: Vertically Integrated Research: Diagrammatic Algebra and Representation Theory
- Instructor: Profs. Sage and Achar.
- Prerequisites: Familiarity with basic group theory and linear algebra, such as from Math 4200 and Math 4153. Also, Math 4997-6 in Fall 2013 or permission of the instructors
- Text:
- Ordinary "algebra" is essentially a language for keeping track of compositions of functions. (Operations like addition and multiplication are, after all, themselves functions, and a complicated expression involving many of those operations is really a composition.) "Diagrammatic algebra" is a general term that refers to various ways of working with function-like objects that can compose not just sequentially, but in a 2-dimensional way. Diagrammatic algebra has its origins in category theory and has connections to and applications in representation theory, topology, mathematical physics, and other areas. In this course, we will look at many concrete, down-to-earth examples, which will involve drawing lots of pictures. We will then concentrate on how diagrammatic algebra can be used in representation theory to study reflection groups and quantum groups.
- MATH 4997-2: Vertically Integrated Research: The A-Polynomial and its Relatives
- Instructor: Profs. Dasbach and Stoltzfus, Dr. Kearney.
- Prerequisites: Topology I (MATH 7510) or equivalent, or permission of instructors.
- Text:
- The A-polynomial of a knot roughly measures the eigenvalues of those representations of the fundamental group of the torus tube about a knot which extend to the entire knot complement. We will study different known ways to effectively compute using the various presentations of fundamental groups of knot complements. We will also have a visiting expert in the field during the semester.
- 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: Algebra II.
- Instructor: Prof. Yakimov.
- Prerequisites: Math 7210 Algebra I.
- Text: Abstract Algebra, 3rd ed; Dummit and Foote, Wiley 2003.
- 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 7230: Topics in Number Theory: Analytic Number Theory.
- Instructor: Prof. Mahlburg.
- Prerequisites: MATH 7210 (Algebra I) and MATH 4181 (Elementary Number Theory); some background in Fourier Transforms (at the level of MATH 4325) and Complex Analysis (MATH 4036) is helpful but not required.
- Text: Analytic Number Theory by Iwaniec and Kowalski.
- This is an introductory graduate course in Analytic Number Theory, which is a quantitative study of the arithmetic property of the integers. Topics will include arithmetic functions; the Prime Number Theorem; primes in arithmetic progression and Dirichlet's theorem; Dirichlet characters; L-functions, zeta functions, and the Riemann Hypothesis; Sieve techniques; quadratic forms; Tauberian theorems; combinatorial applications, Hardy-Ramanujan's formula for integer partitions.
- MATH 7240: Topics in Algebraic Geometry: Toric Geometry
- Instructor: Prof. Hoffman.
- Prerequisites: Familiarity with abstract algebra (groups, rings, ideals, modules) and basic topology (topological spaces). These are typically taught in Math 7210 and Math 7510, respectively.
- Text: No required text. Some recommended books:
- Toric Varieties, Cox/Little/Schenck, Graduate Studies in Mathematics, AMS.
- Introduction to Toric Varieties, William Fulton, Princeton U. Press.
- Combinatorial Commutative Algebra, Miller/Sturmfels, Graduate Texts in Mathematics, 227, Springer.
- Lectures on Toric Varieties, and Minicourse on Toric Varieties, available at David Cox’s webpage, http://www3.amherst.edu/~dacox
- Sage for Power Users, William Stein, available at his webpage http://modular.math.washington.edu
- Toric Geometry is a branch of Algebraic Geometry that connects combinatorial objects such as fans (= a collection of cones) and polytopes, with algebraic varieties. Toric varieties occur in many contexts, e.g., Invariant theory, Moduli Spaces, Calabi-Yau manifolds. This course does not presuppose any previous exposure to Algebraic Geometry, and will develop the necessary language (ideals, varieties, etc.) as needed. We will show how the combinatorics of the fans/polytopes is reflected in the geometry of the toric variety, and conversely. Depending on background and interest of students, advanced topics such as the theory of A-resultants, or the geometric of toric hypersurfaces will be covered. There will be an introduction to the use of software such as Sage and Magma for use in toric geometry. Each student will have a course project to do for the grade.
- MATH 7260: Homological Algebra
- Instructor: Prof. Achar.
- Prerequisites: Math 7211 and 7512
- Text: Charles Weibel, An Introduction to Homological Algebra, Cambridge Studies in Advanced Mathematics 38
- 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 7320: Ordinary Differential Equations.
- Instructor: Prof. Wolenski.
- Prerequisites:
- Text:
- 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 7366: Stochastic Analysis.
- Instructor: Prof. Kuo.
- Prerequisites: Math 7311 (Real Analysis I) or equivalent
- Texts:
- Kuo, H.-H.: Introductory Stochastic Integration. Universitext, Springer 2006
- Kuo, H.-H.: Gaussian Measures in Banach Spaces. Lecture Notes in Math., Vol. 463, Springer-Verlag, 1975 (New Printing by BookSurge, Oct 2006.)
- In this course we will cover the following topics:
- The Ito theory of stochastic integration
- Applications to mathematical finance.
- Infinite dimensional analysis
- White noise theory.
- A new theory of stochastic integration.
- MATH 7370: Lie Groups and Representation Theory
- Instructor: Prof. Olafsson.
- Prerequisites: 7311 and basic theory of Lie groups
- Text: Own lecture notes. References will be provided during the course.
- Lie groups, homogeneous spaces and representation theory plays a central role in several fields of mathematics including geometric analysis, algebraic geometry and number theory. We start by giving a short overview over basic theory of Lie groups and homogeneous spaces. We then introduce the notion of symmetric spaces and derive the basic structure of those spaces. We then concentrate on harmonic analysis on compact symmetric spaces like the sphere or the Grassmann manifold of p-dimensional subspaces of Euclidean space. In particular we give a classification of spherical representations of compact Lie groups. We give a short discussion on how Fourier analysis is related to representation theory and use that as a motivation for harmonic analysis on compact symmetric spaces. This course is suitable for everyone interested in Lie groups, representation theory and harmonic analysis.
- MATH 7384: Material Science: Mathematics of Resonance
- Instructor: Prof. Shipman.
- Prerequisites: Complex analysis, real analysis
- Text: Class notes and references will be provided
- The topic is the mathematics of resonance. Resonance occurs when waves in a spatially extended system excite the vibrational modes of an oscillator. This results in field amplification in the oscillator at critical frequencies and wave scattering that is hyper-sensitive to frequency detuning. A precise analysis of resonance centers on the complex poles of the resolvent of an underlying operator, or, equivalently, the poles of a "scattering matrix". When a pole is on the real axis and is embedded in the continuous spectrum of the operator, delicate resonance phenomena are revealed through perturbation analysis of the scattering matrix. The mathematics of resonance intimately connects complex analysis, spectral theory, Fourier analysis, and perturbation of linear operators.
- MATH 7390-1: Control Theory.
- Instructor: Prof. Malisoff.
- Prerequisites: Maths 2090 and 4032, or their equivalents. Please contact the instructor to confirm that you have the necessary prerequisites if you have not taken Maths 2090 and 4032.
- Text: Students will not need to purchase any text for this course. Lecture notes or online readings will be assigned by the instructor.
- Control theory is a central area at the interface of applied mathematics and engineering. The field studies mathematical methods that influence the behavior of complicated dynamical systems, by finding feedbacks. Feedback refers to automatic adjustments in actions of a system in response to information about the system's own state and its environment. Feedback is essential for robust performance of self-regulated dynamical systems in many biological and engineering applications. This course will involve control theory for time-varying ordinary differential equations. The first part of the course will review the necessary background on nonlinear ordinary differential equations, including existence and uniqueness of solutions and continuous dependence of trajectories on initial conditions. The second part will cover key foundational topics on control theory, including Lyapunov functions and robustness properties of nonlinear dynamics. The final part of the course will cover feedback design, which involves choosing state dependent parameters of the equations that ensure that certain control objectives are met, and will include robustness of feedback controlled systems under the influence of control or modeling uncertainty. This is an important course that should be taken by all PhD students interested in specializing in applied mathematics, and it can be taken as a substitute for MATH 7320. The course is also suitable for undergraduates and engineering majors who satisfied the prerequisites.
- MATH 7410: Graph Theory.
- 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: Combinatorial Optimization.
- Instructor: Prof. Ding.
- Prerequisites: Math 4171 or equivalent.
- Text: None
- First we cover classical min-max results like Menger theorem, max-flow-min-cut theorem, and Konig theorem. Then we establish a connection between these results and Integer Programming. Under this general framework, we discuss more min-max results concerning packing and covering of various combinatorial objects.
- MATH 7512: Topology II.
- Instructor: Prof. Stoltzfus.
- Prerequisites: Math 7510
- Text: Algebraic Topology by Allen Hatcher.
- This course will introduce the homology theory of topological spaces. To each space there is assigned a sequence of abelian groups, its homology. We will develope techniques for computation, particularly the Mayer-Vietoris sequence of "cut and paste." These calculations will then be applied to prove results such as the Brouwer fixed-point theorem (in all dimensions) and generalizations of the Jordan curve theorem.
- MATH 7550: Differential Geometry.
- Instructor: Prof. Kearney.
- 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: Geometric Topology: Hyperbolic Geometry.
- Instructor: Prof. Litherland.
- Prerequisites: Math 7550
- Text: Notes, to be put online later.
- Hyperbolic geometry has been an important tool for the study of
three-manifolds since the groundbreaking work of Thurston in the
1970s. In particular, Thurston's geometrization conjecture, proved by
Perelman in 2003, is a far-reaching generalization of the Poincare
conjecture. This course will discuss the geometry of hyperbolic space
and the definition of a hyperbolic structure on a manifold. Examples
of hyperbolic structures on 2- and 3-manifolds will be given, and we
will discuss the idea of hyperbolic Dehn surgery.
A good source for this material and more (though not a required text)
is:
John G. Ratcliffe, Foundations of Hyperbolic Manifolds (2nd ed.), Springer GTM 149.
Also of interest:
William P. Thurston, Three-Dimensional Geometry and Topology (ed. Silvio Levy), Princeton Mathematical Series 35. This is an expanded version of the first few chapters of Thurston's lecture notes from 1980. The original is available online at: Thurston, 1980
- MATH 7590-2: Geometric Group Theory.
- Instructor: Prof. Dani.
- Prerequisites: Math 7210 and 7510
- Text: None. Suggested references will be provided during the course.
Geometric group theory is a field which sits at the juncture of geometry, topology and group theory. The basic idea is to glean algebraic information about finitely generated groups via their actions on geometric or topological spaces, and we will see many examples of this in the course. We will emphasize the questions and techniques which highlight the interplay with low dimensional topology.
After starting with the basics (Cayley graphs, geometric group actions, quasi-isometries and various quasi-isometry invariants) we will study delta-hyperbolic spaces (generalizations of fundamental groups of closed hyperbolic manifolds) and non-positively curved spaces in detail. Then we will make brief forays into a few other topics, tailored to the interests of the class. Possibilities include the Mostow Rigidity theorem, the Gromov polynomial growth theorem, PL Morse theory and right-angled Artin groups, or special cube complexes and the virtual Haken conjecture.
- MATH 7710: Advanced Numerical Linear Algebra.
- Instructor: Prof. Sung.
- Prerequisites: linear algebra, advanced calculus and some programming experience
- 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).