Prerequisites: No prerequisites for Math Ph.D. students; permission of instructor otherwies
Description: Basic real variable function theory, measure and integration theory prerequisite to pure and applied analysis.
Topics: Riemann and Lebesgue-Stieltjes integration; measure spaces, measurable functions and measure; Lebesgue measure and integration; convergence theorems for integrals; Lusin and Egorov theorems; product measures and Fubini-Tonelli theorem; signed measures, Radon-Nikodym theorem, and Lebesgue's differentiation theorem.
Note: A Problem Session for 501 will be held on Wednesdays, 10:00am-11:40am in room 425 for the Fall semester. There is no need to register for the Problem Sessions.
Text: Complex Analysis, Elias M. Stein and Rami Shakarchi, Princeton Lectures i n Analysis II, Princeton University Press (April 7, 2003). ISBN-10: 0691113858
Prerequisites: Acquaintance with analytic arguments at the level of Rudin's Principles of Mathematical Analysis is necessary. Some knowledge of algebra and point-set topology is useful.
Description: The study of differentiable functions of one complex variable has applications and extensions to many other areas of mathematics, so that it is a basic tool in diverse situations. Topics to be covered include: differentiability of complex functions, complex integration and Cauchy's theorem, series expansions, calculus of residues, maximal principle, conformal mapping, analytic continuation and time permitting, the Prime Number Theorem.
Note: A Problem Session for 503 will be held on Wednesdays, 10:00am-11:40am in room 425 for the Fall semester. There is no need to register for the Problem Sessions.
Description: This is the first half of a year-long introductory graduate course on PDEs, and should be useful for students with a variety of research interests: physics and mathematical physics, applied analysis, numerical analysis, differential geometry, complex analysis, and, of course, partial differential equations. The beginning weeks of the course aim to develop enough familiarity and experience with the basic phenomena, approaches, and methods in solving initial/boundary value problems in the contexts of the classical prototype linear PDEs of constant coefficients: the Laplace equation, the D'Alembert wave equation, the heat equation and the Schroedinger equation. A variety of tools and methods, such as Fourier series/eigenfunction expansions, Fourier transforms, energy methods, and maximum principles will be introduced. More importantly, appropriate methods are introduced for the purpose of establishing quantitative as well as qualitative characteris tic properties of solutions to each class of equations. It is these properties that we will focus on later in extending our beginning theories to more general situations, such as variable coefficient equations and nonlinear equations.
Next we will discuss some notions and results that are relevant in treating general PDEs: characteristics, non-characteristic Cauchy problems and Cauchy-Kowalevski theorem, Hadamard-Petrovsky wellposedness criteria.
Towards the end of the semester, we will begin some introductory discussion on the extension of the energy methods to variable coefficient wave/heat equations, and/or the Dirichlet principle in the calculus of variations.
The purpose here is to motivate and introduce the notion of weak solutions and Sobolev spaces, which will be more fully developed in the second semester.
Subtitle: Some partial differential equations in conformal geometry
Prerequisites: Math517 or permission by instructor
Description: In this course, We will discuss some partial differential equations arising from the study of conformally invariant quantities in Riemannian geometry.
A classical problem in this field is the Yamabe problem, that is to find on a compact Riemannian manifold a conformal metric which has constant scalar curvature. This is equivalent to solving a conformally invariant semi-linear partial different equation of critical exponent.
We will introduce the problem and give a review of the solution to this problem as well as results on compactness of the solution set. We will outline the proofs of these results.
We will then focus on a fully nonlinear version of the Yamabe problem, starting from an introduction of the problem and presenting up-to-date results on the problem. We will provide details of proofs to key results.
Liouville theorems for conformally invariant elliptic and degenerate elliptic fully nonlinear equations in Euclidean equations, local gradient estimates and second derivatives estimates for the fully nonlinear Yamabe problem, Evans-Krylov estimates for second order uniformly fully nonlinear elliptic equations and open problems, the existence and compactness results for the fully nonlinear Yamabe problem on locally conformally flat Riemannian manifolds, the existence results for the problem on general Riemannian manifolds for second elementary symmetric functions case making use of the variational
structure of the problem, the existence and compactness results for the problem for the $k-$elementary symmetric functions cases when $k$ is not less than half of the dimension of the Riemannian manifolds (no available variational structure).
In the course, we will present a number of open problems which should be accessible to graduate students and younger researchers.
Description: In this Harmonic Analysis course, we will deal mainly with Calderon-Zygmund theory. We will also study Littlewood-Paley theory, Strichartz estimates for Wave and Schrodinger equations. Other topics we will cover are Hardy-Littlewood-Sobolev fractional integral operators, interpolation theorems of Marcinkiewicz and Stein and the spaces of Bounded Mean Oscillation.
Description: We will cover a wide range of topics in geometry. The main theme will be Riemann surfaces, Teichmuller spaces and hyperbolic structures. Discretization of Riemann surfaces and uniformization theorem will be introduced. Other related topics to be covered include: integral geometry, convexity, ergodicity, Cauchy rigidity theorem, Hadwiger theorem, Steiner’s theorem, Crofton formulae, circle packing and their rigidity, Steinitz realization theorem, Brunn-Minkowski inequality, billiards and geodesic flows.
This is intended to be a self-contained course for graduate and advanced undergraduate students.
Prerequisites: Introduction to differential geometry
Description: We will study various topics in Riemannian geometry such as comparison theorems, gradient estimates, epsilon-regularity theorems, and compactness theorems for Einstein manifolds of dimension 4.
Prerequisites: Standard course in Abstract Algebra for undergraduate students at the level of our Math 451.
Description: This is a standard course for beginning graduate students. It covers Group Theory, basic Ring & Module theory, and bilinear forms. Group Theory: Basic concepts, isomorphism theorems, normal subgroups, Sylow theorems, direct products and free products of groups. Groups acting on sets: orbits, cosets, stabilizers. Alternating/Symmetric groups. Basic Ring Theory: Fields, Principal Ideal Domains (PIDs), matrix rings, division algebras, field of fractions. Modules over a PID: Fundamental Theorem for abelian groups, application to linear algebra: rational and Jordan canonical form. Bilinear Forms: Alternating and symmetric forms, determinants. Spectral theorem for normal matrices, classification over R and C. (Class supplement provided) Modules: Artinian and Noetherian modules. Krull-Schmidt Theorem for modules of finite length. Simple modules and Schur's Lemma, semisimple modules. (from Basic Algebra II) Finite-dimensional algebras: Simple and sem isimple algebras, Artin-Wedderburn Theorem, group rings, Maschke's Theorem.
Note: A Problem Session for 551 will be held on Wednesdays, 5:00pm-6:20pm in room 423 for the Fall semester. There is no need to register for the Problem Sessions.
Text: No textbook. Lectures are based on papers available online.
Prerequisites: First year graduate algebra and analysis courses. Basic knowledge in vertex operator algebras will be very helpful but is not needed.
Description: Intertwining operator algebras are a class of algebras underlying a number of important algebraic and analytic structures in mathematics and physics. The main axioms of intertwining operator algebras are commutativity and associativity corresponding to operator product expansion and analytic extension property in physics. In the theory of vertex operator algebras and two-dimensional conformal field theory, the main objects to study are in fact intertwining operator algebras. They are equivalent to vertex tensor categories which in turn give braided tensor categories. They are precisely the algebras describing nonabelian anyons in physics. They are also exactly genus-zero chiral two-dimensional conformal field theories. When they are modular invariant, they also give genus-one chiral two-dimensional conformal field theories. They can be constructed naturally using the representation theory of vertex operator algebras.
In this course, I will introduce intertwining operator algebras and study the theory and applications of intertwining operator algebras without assuming that the students have any knowledge in vertex operator algebras or two-dimensional conformal field theory. Vertex operator algebras will be given as special examples of intertwining operator algebras. Below are the detailed topics to be covered in this course:
1. The representation theory of vertex operator algebras and a construction of intertwining operator algebras.
2. Examples of intertwining operator algebras from representations of affine Lie algebras and Virasoro algebras.
3. The construction of the moonshine module vertex operator algebra as an example of the construction of intertwining operator algebra.
4. Modular invariance of intertwining operator algebras.
Text: An introduction to homological algebra, by C. Weibel, Cambridge U. Press, paperback edition (1995).
Prerequisites: First-year knowledge of groups and modules.
Description: This will be an introduction to the subject of Homological Algebra, which is a tool used in many branches of mathematics, especially in Algebra, Topology and Algebraic Geometry. The first part of the course will cover Chain Complexes, Projective and Injective Modules, Derived Functors, Ext and Tor, Universal Coefficients. In addition, some basic notions of Category Theory will be presented:
adjoint functors, abelian categories, natural transformations, limits and colimits.
The second part of the course will cover topics determined by the interests of the students in the class. Possible topics are: Spectral Sequences, Homology of Groups and Lie Algebras, Derived Categories, Model Categories, Sheaf cohomology.
Prerequisites: Familiarity with the basic theory of complete separable metric spaces and their Borel subsets
Description: This course will be an introduction to countable Borel equivalence relations, a very active area of classical descriptive set theory which interacts nontrivially with such diverse areas of mathematics as model theory, computability theory, group theory and ergodic theory. The topics to be covered will include applications of superrigidity theory to countable Borel equivalence relations, as well as some recent applications of Borel determinacy. No prior knowledge of superrigidity or determinacy will be assumed.
Prerequisites: Open to all mathematics Ph.D. students; other students require permission of instructor.
Description: This will be an introductory course in number theory. I plan to cover some aspects of Dirichlet characters, Dirichlet L-functions, quadratic forms, p-adic numbers, and Diophantine approximation, plus other topics depending on the interest of the audience.
Prerequisites: Students should know multivariable calculus, elementary differential equations, and some linear algebra.
Description: This is a first-semester graduate course, intended primarily for students in mechanical and aerospace engineering, biomedical engineering, and other engineering programs. Topics include power series and the method of Frobenius for solving differential equations; Laplace transforms; nonlinear differential equations and phase plane methods; vector spaces of functions and orthonormal bases; Fourier series and Sturm-Liouville theory; Fourier transforms; and separation of variables and other elementary solution methods for linear differential equations of physics, including the heat, wave, and Laplace equations.
Description: This is the second part of the two-part course Introduction to Mathematical Physics. It introduces the student to the mathematically well-developed parts of Quantum Physics and is taught in the spirit of part I (Classical Physics) as an theory about an objectively existing world made of material points that move. Physics topics covered are (a) Non-relativistic quantum mechanics: deBroglie-Bohm theory, Schroedinger's equation, motion of point particles, electrical Coulomb and gravitational Newton pair interactions, "external" magnetic fields, particle spin and Pauli equation, stability of everyday matter; (b) Relativistic quantum mechanics: electrons and photons, Dirac's equation, Chandrasekhar's theory of white dwarf stars. Mathematical key words: Hilbert space, self-adjoint operators, unitary operators.
Description: The course will cover traditional areas of statistical mechanics with a mathematical flavor. It will describe exact results where available and heuristic physical arguments where applicable. A rough outline is given below:
I. Overview: microscopic vs. macroscopic descriptions; microscopic dynamics and thermodynamics.
II. Energy surface; microcanonical ensemble; ideal gases; Boltzmann’s entropy, typicality.
III. Alternate equilibrium ensembles; canonical, grand-canonical, pressure, etc. Partition functions and thermodynamics.
IV. Thermodynamic limit; existence; equivalence of ensembles; Gibbs measures.
V. Cooperative phenomena: phase diagrams and phase transitions; probabilities, correlations and partition functions. Law of large numbers, fluctuations, large deviations.
VI. Ising model, exact solutions. Griffith’s, FKG and other inequalities; Peierle’s argument; Lee-Yang theorems.
VII. High temperature; low temperature expansions; Pirogov-Sinai theory.
VIII. Fugacity and density expansions.
IX. Mean field theory and long range potentials.
X. Approximate theories: integral equations, Percus-Yevick, hypernetted chain. Debye-Hückel theory.
XI. Critical phenomena: universality, renormalization group.
XII. Percolation and stochastic Loewner evolution.
If you have any questions about the course please email me: email@example.com. We can then set up a time to meet.
Prerequisites: Some algebra (especially linear algebra), some discrete probability, and mathematical maturity.
Description: This will be a basic introduction to combinatorics at the graduate level.
We will cover topics such as enumeration, symmetry, partial orders, set systems, Ramsey theory, discrepancy, additive combinatorics and quasirandomness. There will be emphasis on general techniques, including probabilistic methods, linear-algebra methods, analytic methods, topological methods and geometric methods.
Prerequisites: There are no formal prerequisites but it will help to have at least seen the combinatorics sequence, 642.582-3. I'll try to fill in background as we go along. Check with me if in doubt.
Description: The term "Extremal Combinatorics" covers many of the most significant discrete developments of recent (and less recent) years, and many of the most interesting open problems. We'll sample some of these, trying to emphasize the wide range of ideas, methods and extra-combinatorial machinery that come into play.