# Advanced Graduate Courses 2013/14

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

FALL 2013 |
WINTER 2014 |
SPRING 2014 |

511 Intro to complex analysis I | 511 Intro to complex analysis I | 512 Intro to complex analysis II |

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

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

H. Lin (9:00) | H. Lin (9:00) | P. Gilkey (9:00) |

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

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

531 Intro to topology I | 532 Intro to topology II | 533 Intro to diff’l geometry |

D. Sinha (13:00) | D. Sinha (13:00) | W. He (13:00) |

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

V. Vologodsky (11:00) | V. Vologodsky (11:00) | V. Vologodsky (11:00) |

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

H. Wang (14:00) | H. Wang (14:00) | H. Wang (14:00) |

556 Networks and combinatorics | 557 Discrete dynamical systems | |

N. Proudfoot (14:00) | A. Vaintrob (12:00) | |

541 Linear algebra | ||

S. Yuzvinsky (12:00) | ||

567 Stochastic processes | 558 Cryptography | |

C. Sinclair (12:00) | TBA (15:00) | |

607 Symplectic geometry | 607 Infinite dim’l Lie algebras | 607 Hypergeometric series |

N. Proudfoot (12:00) | V. Ostrik (12:00) | B. Young (10:00) |

607 Quantum groups | 607 Spectral theory of Laplace | 607 PDE |

J. Brundan (10:00) | B. Siudeja (15:00) | M. Warren (12:00) |

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

D. Levin (9:00) | P. Lu (9:00) | C. Phillips (9:00) |

619 Complex analysis | ||

TBA (15:00) | ||

634 Algebraic topology I | 635 Algebraic topology II | 636 Algebraic topology III |

D. Dugger (13:00) | D. Dugger (13:00) | D. Dugger (13:00) |

647 Abstract algebra I | 648 Abstract algebra II | 649 Abstract algebra III |

A. Kleshchev (11:00) | A. Kleshchev (11:00) | A. Kleshchev (11:00) |

672 Probability theory I/II | 673 Probability theory II/III | |

D. Levin (14:00) | D. Levin (14:00) | |

681 Commutative algebra | 682 Algebraic geometry | 683 Algebraic geometry |

A. Polishchuk (14:00) | A. Polishchuk (14:00) | A. Polishchuk (14:00) |

684 Harmonic analysis | 685 Harmonic analysis | 686 Further topics in analysis |

M. Bownik (15:00) | M. Bownik (15:00) | C. Phillips (15:00) |

690 Characteristic classes | 691 Spectral sequences | 692 WETSK |

V. Vologodsky (13:00) | H. Sadofsky (13:00) | B. Botvinnik (13:00) |

ADVANCED COURSE DESCRIPTIONS:

Marcin Bownik, 684/5: **Harmonic analysis and wavelets**

The course is an introduction to harmonic analysis on Euclidean spaces (first term) and wavelets (second term). In the first term I am planning to introduce students to core techniques of harmonic analysis such as:

- 1. Fourier transform, Schwartz class, and tempered distributions
- 2. Marcinkiewicz and Riesz-Thorin Interpolation Theorems
- 3. Hardy-Littlewood Maximal Function
- 4. Hilbert transform, Riesz transforms, and Calderon-Zygmund singular operators
- 5. Littlewood-Paley theory and multiplier theorems
- 6. Hardy H^p spaces, atomic decompositions
- 7. H^1-BMO duality, John-Nirenberg theorem
- 8. Weighted inequalities and Muckenhoupt A_p weights
- 9. T1 theorem of David-Journe

The last three topics will be covered very lightly due to time constraints.

In the second term I am planning to cover constructions of wavelet bases and their applications in the study of function spaces:

- 1. Multiresolution analysis
- 2. Construction of 1D Meyer and Daubechies wavelets
- 3. Higher dimensional constructions of wavelets
- 4. Besov and Triebel-Lizorkin spaces
- 5. Characterization of function spaces by wavelet coefficients

Pre-requisites: 600 analysis.

Jon Brundan, 607: **Quantum groups**

This will be a one term course in quantum groups. We will focus on the quantized enveloping algebras of Drinfeld and Jimbo, which are deformations of the universal enveloping algebras of semisimple Lie algebras.

One of the most interesting features of these non-commutative non-cocommutative Hopf algebras is that they come equipped with an R-matrix making their module category into a braided tensor category. This has consequences to knot theory: the Reshetikhin-Turaev invariants of a knot or link are constructed from the R-matrix of a quantum group. A special case for the quantum group corresponding to the Lie algebra sl_2 yields the famous Jones polynomial.

Another aspect I hope to discuss is the existence of remarkable bases—Lusztig’s canonical and Kashiwara’s crystal bases—which have revolutionized the combinatorial representation theory of semisimple Lie algebras. The definition of these bases is impossible without quantum groups.

This course is aimed at people who have been exposed to a little bit of the theory of semisimple Lie algebras, for example from Victor’s course this year. But it should be possible for a student willing to do a little study on the side to follow the course without this background.

Pre-requisites: 600 algebra plus—ideally—a bit of background about semisimple Lie algebras as in for example Humphreys’ book.

Victor Ostrik, 607: **Infinite dimensional Lie algebras**

In this class we will discuss some interesting representations of most interesting infinite dimensional Lie algebras: Heisenberg algebra, Virasoro algebra, and affine Lie algebras. Topics will include the highest weight representations of Virasoro algebra and affine Lie algebras, their characters and applications to Macdonald identities.

Pre-requisites: 600 algebra plus—ideally—a bit of background about semisimple Lie algebras as in for example Humphreys’ book.

Ben Young, 607: **Hypergeometric series**

This is a course from the enjoyable and well-written textbook “A=B” by Marko Petkovsek, Herbert Wilf and Doron Zeilberger, available for free online. The subject of the course a collection of methods which allow computers to discover proofs of hypergeometric function identities (a broad class of identities which occur all the time in combinatorics and elsewhere). Amazingly, the proof that is generated is always human-readable: indeed the entire proof is routine, except that it hinges upon a “brilliant idea” which the computer thinks of.

Prerequisite: Willingness to use a computer to do mathematics. If you’ve never used computer algebra software before, this could be a good place to start.

Nicholas Proudfoot, 607: **Symplectic geometry**

Symplectic geometry is a beautiful field with strong connections to algebraic geometry and representation theory. This will be an introduction to the subject, focusing on actions of Lie groups on symplectic manifolds. Along the way, we will do some Morse theory, learn about equivariant cohomology, and become familiar with toric varieties.

To take this class, you should be comfortable with the basics of smooth manifolds and differential forms. Taking differential geometry (previously or concurrently) would be helpful, but not strictly necessary.

Bartlomiej Siudeja, 607: **Spectral theory of the Laplace operator**

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.