# Advanced Graduate Courses 2015/16

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

FALL 2015 |
WINTER 2016 |
SPRING 2016 |

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

S. Akhtari (12:00) | S. Akhtari (12:00) | |

513 Intro to Analysis I | 514 Intro to Analysis II | 515 Intro to Analysis III |

W. He (11:00) | W. He (11:00) | P. Gilkey (11:00) |

521 Differential Equations II | 522 Differential Equations III | 520 Differential Equations I |

J. Isenberg (10:00) | J. Isenberg (10:00) | Addington (10:00) |

531 Intro to Topology I | 532 Intro to Topology II | 533 Intro to Differential Geometry |

B. Elias (9:00) | B. Elias (9:00) | W. He (9:00) |

541 Linear Algebra | ||

B. Young (12:00) | ||

544 Intro to Algebra I | 545 Intro to Algebra II | 546 Intro to Algebra III |

V. Ostrik (13:00) | V. Ostrik (13:00) | V. Ostrik (13:00) |

556 Networks and Combinatorics | 557 Discrete Dynamical Systems | |

N. Proudfoot (15:00) | M. Warren (12:00) | |

558 Introduction to Cryptography | ||

Eischen (15:00) | ||

561 Intro Methods of Statistics I | 562 Intro Methods of Statistics II | 563 Intro Methods of Statistics III |

D. Levin (8:30) | D. Levin (8:30) | D. Levin (8:30) |

567 Stochastic Processes | ||

C. Sinclair (12:00) | ||

607 Mean Curvature Flow | 607 Homological and Representation Stability | 607 Geometric Representation Theory |

M. Warren (10:00) | T. Tran (12:00) | V. Ostrik (12:00) |

607 The Weil Conjectures | 607 Complex Geometry | 607 Homological Algebra |

D. Dugger (14:00) | W. He (14:00) | B. Elias (10:00) |

616 Real Analysis I | 617 Real Analysis II | 618 Real Analysis III |

H. Lin (11:00) | H. Lin (11:00) | C. Phillips (11:00) |

634 Algebraic Topology I | 619 Complex Analysis | |

N. Proudfoot (9:00) | M. Bownik | |

647 Abstract Algebra I | 635 Algebraic Topology II | 636 Algebraic Topology III |

A. Vaintrob (13:00) | N. Proudfoot (9:00) | N. Proudfoot (9:00) |

648 Abstract Algebra II | 649 Abstract Algebra III | |

A. Vaintrob (13:00) | A. Vaintrob (13:00) | |

681 | 682 | 683 |

A. Polishchuk (13:00) | A. Polishchuk (13:00) | A. Polishchuk (13:00) |

684 Stochastic Processes | 685 Stochastic Processes | |

C. Sinclair (11:00) | C. Sinclair (11:00) | |

686 Harmonic and Functional Analysis of Frames | ||

M. Bownik (15:00) | ||

690 | 691 | 692 |

D. Sinha (9:00) | B. Botvinnik (9:00) | Lipshitz (9:00) |

ADVANCED COURSE DESCRIPTIONS:

Chris Sinclair (11:00): **684/685 Stochastic Processes**

This is a continuation of 671-672, and I aim to more-or-less complete the text Probability and Stochastics by Çinlar. We will cover random measures, point processes, and expand our coverage of Levy processes (including Brownian motion) and Markov processes. Upon completion you should have a very strong background in (modern) probability theory.

Marcin Bownik (11:00): ** 686 Harmonic and Functional Analysis of Frames**

A frame is a generalization of the concept of a basis to sets which are overcomplete. That is, frame expansions are in general not unique and instead they satisfy a certain stability condition. Although frames were introduced in 1950’s, this area has experienced a renewed interest in recent years with the advent of wavelets. In this course we plan to explore the following topics depending on the interest of students.

General frames and Riesz bases in Hilbert spaces: dual frames, canonical dual frames, Naimark’s dilation theorem. Frames in finite dimensional spaces, fusion frames, connections with algebraic combinatorics and Littlewood-Richardson tableaux. Frames in infinite dimensional spaces: Kadison’s Pythagorean Theorem, characterization of frame norms with prescribed frame operator and the Schur-Horn theorem. The recent solution of the long standing Kadison-Singer problem and its equivalent formulation in terms of the paving conjecture, the Feichtinger conjecture, and the Bourgain-Tzafriri conjecture. The ramifications of this solution to the frame theory.

Weiyong He (14:00): **607 Complex Geometry**

The course will be based on Griffiths&Harris “Principle of Algebraic Geometry”. We cover basic material such as complex manifolds, sheave cohomology, vector bundle, Hodge theory. We also emphasize the applications of cohomology theory, in particular the geometry of holomorphic line bundles.

Daniel Dugger (14:00) **607 The Weil Conjectures**

In 1949 Andre Weil conjectured a link between certain topological phenomena and formulas counting the number of solutions to arithmetic equations over finite fields. He suggested that this link could be explained by the existence of a certain kind of cohomology theory for algebraic varieties defined over arbitrary fields. In the 1950s Grothendieck and his collaborators undertook a massive project to develop such a theory, which came to be called etale cohomology. The

theory proved most of the Weil conjectures, but the final one resisted attempts for a while. Deligne proved this final conjecture in the 1970s.

The point of this course will be to tell this story and to understand (at least in outline) the proofs by Grothendieck and Deligne. I won’t assume much advanced knowledge of algebraic geometry, although students will benefit from having a basic familiarity with schemes. Much of the story is actually homotopy-theoretic, and I will concentrate on that aspect.

Trithang Tran (12:00) **607 Homological and Representation Stability **

Homological stability results are a classical topic with a rich history in algebraic topology, dating back to Quillen’s definition of algebraic K-theory in the 1970’s. The topic has recently gone through a resurgence, especially after the results of Madsen and Weiss which used homological stability for mapping class groups to solve Mumford’s conjecture on the cohomology of the moduli space of Riemann surfaces.

The course will aim to familiarise students with the ideas of homological stability, broadly speaking. Some buzzwords are: simplicial objects, group homology, classifying spaces, scanning and the group completion theorem. We will see examples in the form of homological stability for symmetric groups, braid groups, configuration spaces, mapping class groups and general linear groups. Where possible, we will discuss applications (such as to K-theory in the case of general linear groups).

We will also look at a related notion called representation stability. This is a very recent development due to Church and Farb and arises when the spaces in question have a symmetric group action, so that their (co)homology groups can be thought of as representations of the symmetric group. This topic will involve studying the category of FI-modules.

Note: This is a large array of topics, and should be thought of as a “possible directions this course can take”. To cover everything mentioned in detail is probably too much for one quarter. Therefore, what we do or do not cover will depend on the tastes of the students (and the lecturer!).

Micah Warren (10:00) **607 Mean Curvature Flow**

Mean Curvature Flow is a concrete introduction to more general geometric evotion equation, so I would hope that this course would be offered in the quarter prior to the course on Ricci Flow which was proposed. For textbook, I plan to use the one by Mantegazza, which is recent and is freely availble online to University of Oregon. (In fact, the interested student should take a look http://link.springer.com/book/10.1007%2F978-3-0348-0145-4) At this point, I see no problem in starting in chapter 1 and proceeding to chapter 5 of this textbook.

Victor Ostrik (12:00) **607 Representation Theory**

This class will be an introduction to geometric methods in representation theory. We will introduce some important geometric objects associated with semisimple groups such as the flag variety, the nilpotent cone, the Springer’s resolution and Grothendieck’s simultaneous resolution and explain how their geometric properties can be used to answer questions about representations of related groups and algebras. The topics will include Deligne-Lusztig theory (describing complex characters of finite groups like GL(n) over finite field), Kazhdan-Lusztig conjecture (describing multiplicities of simple modules in Verma modules), and the theory of Springer representations (describing some geometric actions of Weyl groups). The prerequisites include some familiarity with semisimple Lie groups and Lie algebras, with basic topology of algebraic varieties, and with the theory of sheaves.

Ben Elias (10:00) **607 Homological Algebra**

We will develop the fundamentals of homological algebra, focusing on the category of modules over a ring. Topics will include: projective and injective modules, complexes, derived functors, homological dimension, Grothendieck groups, Morita equivalence, triangulated categories, and homotopy and derived categories. As time permits, we will also discuss: spectral sequences, derived Morita equivalence, stable categories, and hopfological algebra.