Graduate Program

Graduate Course Descriptions Spring 2021

640:502 Theory of Functions of a Real Variable II

The course 502 is a continuation of Fall’s 501, but also offers outlook on applications in other fields of mathematics, and in mathematical physics and engineering. We will pick up where the course 501 ended, so 501 is a pre-requisite. The material is mostly from Folland’s book. We begin with some selected material from section 3 (Signed measures and differentiation), then hop to section 5 (Elements of functional analysis) and continue with sections 6 (L p spaces), 7 (Radon measures), 8 (Elements of Fourier analysis) and 9 (Elements of distribution theory). There is some flexibility as to the material, thus we may also look into some selected material of sections 10 (Probability theory) and 11 (Haar measure, Hausdorf measure) at the end of the course. I plan to occasionally supplement Folland’s book by “hand outs” of typed-up material. Given the pandemic situation, the course is currently planned to be taught in synchronous remote mode; a room in Hill Center has been booked for in-class instruction in case the situation improves significantly.

640:504 Theory of Functions of a Complex Variable II

This is the second semester course of Complex Analysis. We will start with a detailed proof of the Riemann mapping theorem making use of the Ascoli-Arzela theorem. Then we switch to Harmonic Function Theory (including the Poisson integral representation, various mean-value theorems and Schwarz reflection principle), as well as its application to the construction of elliptic functions. After these, we study the Riemann-Zeta function and give a very quick and short proof of the Prime Number Theorem based on the work of Newman. ( Another optional topic to be included in this classical theory part would be the monodromy theory and multiple-valued holomorphic functions.) After these classical topics, depending on students' interest, we will present a detailed discussion to one of the following two  (A). Basic  Riemann surfaces theory, Perron's method on Riemann surfaces and the proof of the uniformization  theorem for simply connected Riemann surfaces.  (B). Basic  Riemann surfaces theory and the proof of the classical Riemann-Roch theorem. -- This  is a foundational course for anyone who likes to know more about pure mathematics and applied mathematics involving some analysis. It fits  perfectly to  undergraduates who have taken a course equivalent to Math 503, also first year and second year graduate students. The instructor has prepared lecture notes (to be distributed during the class) which cover all the materials to be present in the class.

640:508 Functional Analysis II

Course description from Spring 2020 (may change): This course will rapidly review the basics of Sobolev spaces. Then we will study Calderon-Zygmund estimates. Following this we will apply the previous two topics to study elliptic boundary value problems on smooth, bounded domains. The last topic will be to study Strichartz inequalities and some applications to wave and Schrodinger equations.

640:509 Topics in Analysis

We plan to mention recent results about the classification of ancient solutions in the mean curvature flow which was used to prove the Mean convex neighborhood conjecture, which says that if a singularity is cylindrical, then there is a space time neighborhood around a singularity in which the mean curvature is strictly positive.

640:515 Ordinary Differential Equations

This is an introduction to the theory of ordinary differential equations. We will cover the classical results: existence and uniqueness theorems; linear theory including Floquet theory and elementary bifurcations; stable and unstable manifolds; boundary value problems; and a brief introduction to chaotic dynamics. The novelty of the course is that the proofs will be presented in a manner which allows for rigorous computer verification. Using Julia and Matlab we will apply these new techniques to rigorously extract specific solutions and explore the dynamics of explicit nonlinear systems. For a more detailed overview of the philosophy of the course (we will only consider ODEs as opposed to PDEs and FDEs) see (http://www.ams.org/notices/201509/rnoti-p1057.pdf) and the description of the AMS short course delivered at the National Meeting January 2015 (http://www.ams.org/notices/201509/rnoti-p1106.pdf) To have a sense of the cutting edge work see: http://crm.math.ca/camp-nonlinear/

640:518 Partial Differential Equations II

This course will rapidly review the basics of Sobolev spaces. Then we will study Calderon-Zygmund estimates. Following this we will apply the previous two topics to study elliptic boundary value problems on smooth, bounded domains. The last topic will be to study Strichartz inequalities and some applications to wave and Schrodinger equations.

640:533 Introduction to Differential Geometry

Differential geometry is the study of geometric properties of curves, surfaces, and their higher dimensional analogues using the methods of calculus. It has a long and rich history, and, in addition to its intrinsic mathematical value and important connections with various other branches of mathematics, it has many applications in various physical sciences. In this course, we will study differential manifolds, Riemannian metrics, Levi-Civita connections, curvature tensors, geodesics and space forms and possibly comparison theorems in Riemannian geometry.

640:534 Selected Topics Geometry I

Introduction to Symplectic Topology - Symplectic manifolds, Lagrangian submanifolds, examples from Hamiltonian dynamics and algebraic geometry, pseudoholomorphic maps. Further topics may include: symplectic capacities; Floer cohomology, relations with mirror symmetry and tropical geometry.  Please contact the instructor to discuss topic choice.

640:536 Algebraic Geometry II

Depending on student interest, this may either be a standard course on schemes, with a bit of cohomology, following roughly chapter 2 and a bit of chapter 3 in Hartshorne's textbook, or it may be an introduction to complex algebraic geometry with topics from a combination of Griffiths-Harris, Wells, and possibly other sources. Other topics are possible depending on interest. The scheduled time may change to conform to student schedules if needed.

640:537 Select Topics in Geometry II

We will present a topic course that covers on the collapsing theory in Metric Riemannian geometry. We will start with the Gromov’s almost flat manifolds, the nilpotent structure theory of Cheeger-Fukaya-Gromov. We will discuss similar nilpotent structures found on collapsed Riemannian manifolds with local Ricci bounded covering geometry, and on Alexandrov spaces (non-Riemannian spaces) with local bounded covering geometry.
CONTENTS:
I. Collapsed Manifolds with Bounded Sectional Curvature - 1.1. Examples 1.2. The almost flat manifolds 1.3. The nilpotent fiber bundle theorem 1.4. The singular nilpotent fibration theorem 1.5. Applications -
II. Collapsed Manifolds with Local Bounded Ricci Covering Geometry - 2.1. Examples 2.2. The Cheeger-Colding-Naber theory on Ricci limit spaces 2.3. The Margulis lemma 2.4. Maximally collapsed manifolds with local bounded Ricci covering geometry 2.5. The nilpotent fibration theorem 2.6. Applications -
III. Collapsed Alexandrov Spaces - 3.1. Basic Alexandrov spaces 3.2. The fibration theorem 3.3. Maximally collapsed Alexandrov space with local bounded covering geometry 3.4. The nilpotent fibration theorem 3.5. Singular fibration structures 3.6. Applications -

640:541 Introduction to Algebraic Topology II

This is a continuation course of Math 540 Introduction to Algebraic Topology I. Continuing the topic's in the previous semester, the following topics will be covered.
1. Homotopy theory. Homotopy theory studies homotopy classes of maps from spheres to topological spaces, which is a natural generalization of the fundamental group, while their behaviors are very different from each other. The topics we will cover include: Whitehead's theorem, cellular approximation, excision theorem, Hurewicz theorem, long exact sequence of fiber bundles,  connections with cohomology, obstruction theory, Leray-Hirsch theorem and Gysin sequence.
2. Characteristic classes. For any vector bundle over a topological space, a characteristic class associate this vector bundle with a cohomology class of the base space. Characteristic classes are very useful in differential topology and differential geometry. The topics we will cover include: basics on vector bundles, Grassman manifolds, universal bundles, Stiefel-Whitney classes, Euler classes, Chern classes, Pontrjagin classes, Chern numbers, Pontrjagin numbers, the oriented cobordism ring.

640:549 Lie Groups

This will be a highly _individualized_ course covering topics in Lie groups. I will teach this course at two levels, and students can pick the level appropriate to their background, training, and interest. The introductory level will introduce students to basic ideas in Lie groups and Lie algebras, which are useful for application within Lie theory and in other parts of mathematics, such as algebra, geometry, topology, and number theory. This level will be collaborative in nature. The advanced level will be more individualized, where I will help each student choose a topic in Lie theory appropriate to his/her taste. I will explain the main ideas in some recent research work in that area (perhaps, but not necessarily, my own!), suggest generalizations, and possible lines of attack. We will discuss this over the course of the semester, with the aspirational goal of having some *new* mathematics by the end of the semester. In particular, if you took "Lie groups" last time, there need be no overlap! 

640:552 Abstract Algebra II

This is the continuation of Math 551, aimed at a discussion of many fundamental algebraic structures. The course will cover the following topics (and perhaps some others). Basic module theory and introductory homological algebra - most of Chapter 3 and part of Chapter 6 of Basic Algebra II: hom and tensor, projective and injective modules, abelian categories, resolutions, completely reducible modules, the Wedderburn-Artin theorem Commutative ideal theory and Noetherian rings - part of Chapter 7 of Basic Algebra II: rings of polynomials, localization, primary decomposition theorem, Dedekind domains, Noether normalization Galois Theory - Chapter 4 of Basic Algebra I and part of Chapter 8 of Basic Algebra II: algebraic and transcendental extensions, separable and normal extensions, the Galois group, solvability of equations by radicals

640:554 Topics in Algebra

This course will be an introduction to Geometric Group Theory. There are no prerequisites, except for the most basic notions of group theory. In Geometric Group Theory, finitely generated groups are viewed as metric spaces via the path metrics on their Cayley graphs and their large-scale geometry is studied. The topics covered in this course will include: (i) Quasi-isometries and the large-scale geometry of finitely generated groups. (ii) Growth rates of finitely generated groups, including the construction of groups of intermediate growth. (iii) The basic theory of amenable groups.

640:555 Topics in Algebra II

Introduction to and development of selected topics and examples in vertex operator algebra theory, adapted to the interests of the students. Applications of the theory and current research topics will be highlighted. This course will be accessible to students without prior experience in vertex operator algebra theory. More specifically, one of the central themes of vertex operator algebra theory is the hierarchy: Error Correcting Codes --> Lattices --> Vertex Operator Algebras. The most important example of this hierarchy is: The Golay Code --> The Leech Lattice --> The Moonshine Module Vertex Operator Algebra, which has the Monster finite simple group as its automorphism group; which exhibits deep connections with the theory of modular functions in number theory; and which essentially forms an example of a string theory in theoretical physics. We will develop the theories of error correcting codes and lattices, and show how the remarkable properties of the Golay code and Leech lattice lead to correspondingly remarkable properties of the Moonshine Module. Along the way, we will introduce and motivate the relevant basic concepts of vertex operator algebra theory.

640:560 Homological Algebra

Homological algebra is an essential tool in algebraic geometry and algebraic topology and find use in representation theory, Lie theory, and much more. Roughly, it is a first step in applying some category theory to questions of concrete interest. Main topics: chain complexes, abelian and triangulated categories, derived functors, (co)homology. Actual computations will be emphasized!

640:569 Selected Topics in Logic

The Axiom of Determinacy (AD) says that all two player games of perfect information in which players play integers are determined. AD contradicts the Axiom of Choice but has many appealing consequence. Moreover, over the last 60 years it has become increasingly clear that determinacy of definable games is consistent with the usual axioms of set theory. The begining of such results was the original Gale-Steart theorem that all open games are determined, which was then extended by Martin to all Borel games. The Determinacy of more complicated games cannot be shown in ZFC alone. -- In this course, we will study the consequence of the Axiom of Determinacy. In a further course, we will study its connections with areas of mathematics.

640:573 Special Topics Number Theory

Congruences of prime modulus p for polynomials in several variables can be viewed as equations over the field of p elements. This view point is powerful allowing to work in the field extensions. I will present analytic methods, mostly for algebraic curves, using exponential sums and the theory of L-functions. The highlight of the course will be a proof of the Riemann hypothesis for hyperelliptic curves. Numerous advanced topics will be presented in a survey fashion, in particular the L-functions of algebraic varieties

640:601 Mathematics TA Instructional Training

TA Training

640:640 Experimental Mathematics

Experimental Mathematics used to be considered an oxymoron, but the future of mathematics is in that direction. In addition to learning the philosophy and methodology of this budding field, students will become computer-algebra wizards, and that should be very helpful in whatever mathematical specialty they are doing (or will do) research in. We will first learn Maple, and how to program with it. This semester we will learn, from an experimental mathematics point of view, algorithmic enumerative and algebraic combinatorics. The final projects may lead to published papers. People who already took previous editions are welcome to take it again, since except for the basics, there is very little overlap with previous editions.  This class is suitable for graduate students in other departments, and the software development skills learned will be useful for doing any quantitative research. Smart advanced undergraduates are also welcome. In particular, the statistical methods learned for researching enumerative combinatorics should be applicable almost everywhere.

642:528 Methods of Applied Mathematics II

A second semester graduate course primarily intended for students in mechanical and aerospace engineering, biomedical engineering, and other engineering programs. There will be three parts: 1. Complex variable theory, including the differential and integral calculus of functions of a complex variable, conformal mapping, Taylor series, Laurent series and the residue theorem. Introduction to the calculus of variations. 2. Calculus of variation, including the motivation of variational principles from physical laws, derivation of Euler-Lagrange equations, stability criterion, linearization, and brief introduction of the foundation of finite element method. 3. Perturbation methods, including applications to ode systems, examples of boundary layer, multiple-scale problems, and eigenvalue problems. --- Emphasis on applications and calculations which graduate students in engineering may encounter in their courses.

642:564 Statistical Mechanics II: Nonequilibrium

The course will cover both deterministic and stochastic models of nonequilibrium phenomena. Topics will include: kinetic theory, transport processes, approach to equilibrium in closed systems, stationary states of systems in contact with reservoirs at different temperatures, stochastic lattice models of exclusion processes, the contact process (describing spread stochastic lattice models of exclusion processes, the contact process (describing spread of contagious diseases), the voter model, etc. The material will be developed mostly ab initio.

642:575 Numerical Solution of Partial Differential Equations

This course provides an introduction to finite difference and finite element methods for the numerical solution of elliptic, parabolic, and hyperbolic partial differential equations. The course will concentrate on the key ideas underlying the derivation of numerical schemes and a study of their stability and accuracy. Students will have the opportunity to gain computational experience with numerical methods with a minimal of programming by the use of Matlab's PDE Toolbox software. The course is intended for graduate students in applied mathematics, engineering, mathematical finance, and physics. Supplementary textbooks include: (1) "Boundary value problems of mathematical physics" by I. Stakgold, (2) "Numerical partial differential equations: Conservation laws and elliptic equations" by J. W. Thomas, (3) "Numerical partial differential equations: Finite difference methods" by J. W. Thomas, (4) "The mathematical theory of finite element methods" by Brenner and Scott, (5) "Numerical solution of partial differential equations by the finite element method" by C. Johnson

642:581 Graph Theory

This course will serve as a graduate course in graph theory. For a large part of the course we will follow the text by Bela Bollobas on Modern Graph Theory. Some of the topics we will cover include: Matchings, cuts, flows, connectivity, planar graphs, graph colorings, random graphs, extremal graph theory, Ramsey theory, linear algebra methods, and expander graphs.

642:583 Combinatorics II

This is the second part of a two-semester course surveying basic topics in combinatorics. Topics for the full year should (at least) include most of the topics below. Enumeration (basics, generating functions, recurrence relations, inclusion-exclusion, asymptotics) - Matching theory, polyhedral and fractional issues - Partially ordered sets and lattices, Mobius functions - Theory of finite sets, hypergraphs, combinatorial discrepancy, Ramsey theory, correlation inequalities - Probabilistic methods - Algebraic and Fourier methods - Entropy methods

642:588 Arithmetic Combinatorics

Arithmetic combinatorics lies in the intersection of number theory and combinatorics, and studies the behaviour of simple arithmetic operations, such as addition and multiplication, on finite sets of numbers. For example, under what conditions can we guarantee that a set of natural numbers contains a three-term arithmetic progression? How does the size of the sumset A + A differ from the size of A? What if we consider the sumset A+A and the product set A.A simultaneously? Anyone who enjoys either combinatorics and/or number theory should find something of interest in this course. Topics: This course will cover both classical and modern aspects of the subject and we will cover some subset of the following topics: • sumsets and their structure • the sum-product phenomenon and its applications in computer science • Szemeredi's theorem • Fourier analysis on finite abelian groups • geometric tools from incidence geometry • graph theoretic methods, and • algebraic techniques.

642:592 Topics in Probability and Ergodic Theory II

This topics course will focus on two-dimensional phenomena in Probability coming mostly from statistical physics, including tilings, vertex models, percolation, Schramm–Loewner evolution, etc.

642:662 Selected Topics in Mathematical Physics

(Continuation of fall semester course: Intro. to Mathematical Relativity I) - Special- and General-Relativistic Quantum Mechanics: 1. Dirac's equation on curved backgrounds: self-adjointness, spectral analysis - 2. Quantum laws of motion for singularities of spacetime - 3. Interacting photon-electron systems - 3. Interacting photon-electron systems - 4. Least-invasive quantization