# Advanced Graduate Courses 2014/15

See below for the proposed syllabi for the advanced seminar courses. All the other courses are described in the catalog.

FALL 2014 |
WINTER 2015 |
SPRING 2015 |

513 Intro to Analysis I | 511 Intro to Complex Analysis I | 512 Intro to Complex Analysis II |

B. Siudeja (13:00) | W. He (12:00) | W. He (12:00) |

521 Differential Equations II | 514 Intro to Analysis II | 515 Intro to Analysis III |

J. Vaaler (10:00) | B. Siudeja (13:00) | P. Gilkey (13:00) |

531 Intro to Topology I | 523 Differential Equations III | 520 Differential Equations I |

A. Vaintrob (11:00) | J. Vaaler (10:00) | P. Lu (10:00) |

544 Intro to Algebra I | 532 Intro to Topology II | 533 Intro to Differential Geometry |

B. Young (9:00) | A. Vaintrob (11:00) | P. Lu (11:00) |

561 Intro Methods of Statistics I | 545 Intro to Algebra II | 541 Linear Algebra |

D. Levin (16:00) | B. Young (9:00) | S. Akhtari (12:00) |

607 Homological Alg-Sheaf Theory | 556 Networks and Combinatorics | 546 Intro to Algebra III |

N. Proudfoot (10:00) | A. Ellis (14:00) | B. Young (9:00) |

607 Algebraic Number Theory | 562 Intro Methods of Statistics II | 557 Discrete Dynamical Systems |

S. Akhtari (12:00) | D. Levin (16:00) | TBA (14:00) |

607 Geometric Analysis | 567 Stochastic Processes | 558 Introduction to Cryptography |

W. He (2:00) | C. Sinclair (12:00) | V. Ostrik (15:00) |

616 Real Analysis I | 607 Categorification | 563 Intro Methods of Statistics III |

M. Bownik (13:00) | A. Ellis (12:00) | D. Levin (16:00) |

634 Algebraic Topology I | 607 Mirror Symmetry | 607 Derived Categ in Alg Geom |

V. Vologodsky (11:00) | V. Vologodsky (14:00) | A. Polishchuk (14:00) |

637 Differential Geometry I | 617 Real Analysis II | 607 Diophantine Analysis |

M. Warren (10:00) | M. Bownik (13:00) | S. Akhtari (12:00) |

647 Abstract Algebra I | 635 Algebraic Topology II | 618 Real Analysis III |

A. Polishchuk (9:00) | V. Vologodsky (11:00) | M. Bownik (13:00) |

681 Top Lie Algebras | 638 Differential Geometry II | 636 Algebraic Topology III |

J. Brundan (9:00) | M. Warren (10:00) | V. Vologodsky (11:00) |

684 Top Harmonic Analysis | 648 Abstract Algebra II | 649 Abstract Algebra III |

C. Phillips (13:00) | A. Polishchuk (14:00) | A. Polishchuk (14:00) |

690 Top Characteristic Classes | 672 Theory of Probability I & II | 673 Theory of Probability II & III |

D. Sinha (11:00) | C. Sinclair (14:00) | C. Sinclair (14:00) |

682 Lie Algebras & Representations | 683 Lie Groups | |

J. Brundan (9:00) | A. Berenstein (9:00) | |

685 Func Analysis/Banach Algebras | 686 Banach Alg/C*-algebras | |

C. Phillips (13:00) | C. Phillips (13:00) | |

691 Cobordism/Morse Theory | 692 WETSK | |

B. Botvinnik (11:00) | D. Sinha (11:00) |

ADVANCED COURSE DESCRIPTIONS:

C. Phillips (13:00): **684,685,686 Functional Analysis, Banach algebras and C*-algebras**

**Math 684**

- 1. A little review of Hilbert space
- 2. Basic definitions of linear operators, mostly on Hilbert space; examples.
- 3. Compact operators on Hilbert Spaces
- 4. Definition of a Banach algebra; examples.
- 5. Spectrum, spectral radius, holomorphic functional calculus. (The proofs for holomorphic functional calculus will be omitted.)
- 6. The weak and weak* topologies, Alaoglu’s Theorem, Krein-Milman Theorem. (Some proofs omitted, and general theory of topological vector spaces omitted.)
- 7. Maximal ideal space of a commutative Banach algebra, Gelfand transform. (Nonunital case left as an exercise.)
- 8. Definition of a C*-algebra; examples.
- 9. Foundations of the general theory of C*-algebras: continuous functional calculus, positive elements, ideals, quotients, approximate identities.
- 10. States and representations: GNS construction. Existence of irreducible representations (parts left as exercises)

**Math 685**

- 1. Double commutant and Kadison transitivity theorems.
- 2. Examples of simple C*-algebras: UHF algebras; crossed products by minimal homeomorphisms; Cuntz algebras; the Calkin algebra. Without proof: The reduced C*-algebra of a free group. This topic includes enough about crossed products by Z to do things like proving simplicity of specific cases.
- 3. General theory of purely infinite simple C*-algebras.
- 4. An outline of nuclearity, mostly without proofs. Tensor products of C*-algebras.

**Math 686**

- 1. Outlines of the proofs of Kirchberg’s stability theorems, leaving some details to be filled in.
- 2. Asymptotic morphisms and a sketch of E-theory.
- 3. Asymptotic morphisms between Kirchberg algebras: homotopy implies asymptotic unitary equivalence.
- 4. What is the UCT (Universal Coefficient Theorem)?
- 5. Sketch of the proof of the classification theorem for UCT Kirchberg algebras.

Pre-requisites: Math 616, 617, 618

Shabnam Akhtari (12:00): **607 Algebraic Number Theory**

TEXTS:

- Number Fields, Daniel A. Marcus, Springer-Verlag
- Algebraic Number Fields, Gerald Janusz, AMS
- J.S. Milne Course Notes: http://www.jmilne.org/math/CourseNotes/ant.html.

This is a graduate-level course on algebraic number theory. The goal is to learn about some objects of algebraic number theory, such as number fields, function fields, local fields, class fields, and elliptic curves. We will learn the proofs of important theorems, such as unique factorization of ideals in Dedekind domains, structure of factorization of rational primes (decomposition and inertia groups), finiteness of the class group and unit group.

Pre-requisites: 500 Algebra

Alexander Ellis (12:00) **607 Categorification**

Categorification is the study of “higher structures” which control more familiar algebraic objects. A basic example: singular (co)homology categorifies the Euler characteristic of a space. While categorification is an ad hoc process, decategorification usually takes the form of an Euler characteristic, Grothendieck group, or similar invariant. A categorification often has a richer structure than the original object (e.g. cup product, Poincare duality in the example of singular cohomology).

More recent instances of categorification involve higher category theory, derived and other triangulated categories, and diagrammatic algebras. In this course, our examples will come from topology and representation theory. Polynomial invariants of links such as the Jones polynomial can be categorified by link homology theories. We will start with a quick review of category theory and homological algebra, followed by a discussion of TQFTs in low dimensions. This will allow us to construct Khovanov homology, a bigraded homology theory for links which categorifies the Jones polynomials. We will then turn to examples coming from “towers of algebras”, in which induction and restriction functors categorify the multiplication and comultiplication of a Hopf algebra (work of Geissinger, Zelevinsky, Khovanov-Lauda). Finally, we will bring derived categories, higher categories, and diagrammatic algebra into the picture and study the 2-representation theory of Lie algebras and their quantized enveloping algebras (work of Chuang-Rouquier, Khovanov-Lauda, Rouquier, Cautis-Kamnitzer-Licata, and others). If time permits, we may discuss other topics as well.

By the end of this course, students should expect to have a “category number” of at least 2 (see: http://en.wikipedia.org/wiki/N-category_number).

Vadim Vologodsky (14:00) **607 Mirror Symetry**

Description: The goal of the course is to understand the statement and a proof of “the mirror identity” for hypersurfaces of dimension 3. Roughly, this identity computes the number of rational curves of given degree on quintic 3-folds. Along the way we will discuss moduli spaces of algebraic curves, the quantum cohomology, and the Atiyah-Bott localization formula.

Pre-requisites: 600 topology, 600 algebra, and 2 quarters of algebraic geometry.

Weiyong He (2:00) **607 Geometric Analysis**

Why does a cat curl into a ball when it is cold? Why drums are almost always round? Both questions are related to the eigenvalues of the Laplace operator. We will calculate explicitly almost all known cases. Since just a few are actually known, we will explore possible ways of estimating Laplace eigenvalues using geometric properties of the domains (e.g. area, perimeter, diameter).

The class requires basic knowledge of undergraduate analysis, matrix algebra and ode. We will try to use computers to visualize the topic.

- 1. Bochner technique (Bochner formula)
- 2. Bishop-Gromov volume comparison
- 3. Laplacian comparison
- 4. Gradient estimate of harmonic function (on manifold w/ Ricci curvature bound)-Cheng-Yau estimate
- 5. Maximum principle
- 6. Gromov-Hausdorff topology
- 7. Cheeger-Gromoll splitting theorem and almost rigidity.

Prerequisite: Riemannian geometry

Alexander Polishchuk (14:00) **607 Derived Categories in Algebraic Geometry**

I’d like to present some important recent developments in algebraic geometry involving derived categories of coherent sheaves. I will focus on topics that are relevant for Kontsevich’s homological mirror conjecture (and its generalizations) in which these derived categories play a key role.

The tentative plan is to study:

- 1. Fourier-Mukai functors
- 2. Exceptional collections
- 3. Matrix factorizations (these are very useful when discussing derived categories of hypersurfaces in projective spaces).
- 4. Bridgeland stability conditions

Prerequisite: A course in Algebraic geometry.

Shabnam Akhtari (12:00) **607 Diophantine Analysis**

Summary: We will introduce the concept of irrationality measure and study the approximation of irrational numbers by rational numbers. Although almost all real numbers are irrational, it is very difficult to establish the irrationality of a given number. We will study a variety of techniques which can examine the irrationality of different numbers and functions. Later we will focus on the transcendence theory.

Some specific topics to be covered:

- 1. Heights and absolute values in number fields – Diophantine approximation
- 2. Theory of continued fractions
- 3. Diophantine equations and inequalities
- 4. Theory of linear forms in logarithms
- 5. Irrationality of Riemann zeta function at special values – uniform distribution
- 6. Normal numbers
- 7. Transcendence measure

Prerequisite: 500 Algebra and 500 Real Analysis.