NUMBER THEORY SEMINAR: | Aaron PollackPrinceton University |
Rankin-Selberg integrals associated to nonunique models |

on Wednesday, April 23, 2014, at 3:00 PM in Sci Center 507 | ||

Most known Rankin-Selberg integral representations derive their Euler product structure from the uniqueness of some model, such as the Whittaker model. However, there are a few examples of global Rankin-Selberg convolutions which are known to represent L functions, yet don't unfold to a unique model. Using an extension of a method devised by Piatetski-Shapiro and Rallis, we will give two new interesting examples of such integrals: one for the exterior square L function on GU(2,2), the other for the Spin L function on GSp(6). As I will explain, these integrals deserve to be considered analogues of the integral representation of Kohnen-Skoruppa for the Spin L function on GSp(4). This is joint work with Shrenik Shah. |

FRIENDS OF THE MATHEMATICS DEPARTMENT AFTERNOON LECTURE: | Barry Mazur, speaking from the MSRI in BerkeleyHARVARD UNIVERSITY |
PRIMES - based on the book co-authored with William Stein |

on Friday, April 25, 2014, at 2:00 PM in Science Center 507 | ||

The Riemann Hypothesis is one of the great unsolved problems of mathematics and the reward of $1,000,000 of Clay Mathematics Institute prize money awaits the person who solves it. But, with or without money, its resolution is crucial for our understanding of the nature of numbers. There are at least four full-length books recently published, written for a general audience, that have the Riemann Hypothesis as their main topic. A reader of these books will get a fairly rich picture of the personalities engaged in the pursuit, and of related mathematical and historical issues. This is not the mission of this book. We aim, instead, to explain in as direct a manner as possible and with the least mathematical background required, what this problem is all about and why it is so important. For even before anyone proves this hypothesis to be true (or false!), just getting familiar with it and with some of the the ideas behind it, is exciting. Moreover, this hypothesis is of crucial importance in a wide range of mathematical fields; for example, it is a confidence-booster for computational mathematics: even if the Riemann Hypothesis is never proved, its truth gives us an excellent sense of how long certain computer programs will take to run, which, in some cases, gives us the assurance we need to initiate a computation that might take weeks or even months to complete. |

FRIENDS OF THE MATHEMATICS DEPARTMENT AFTERNOON LECTURE: | Martin NowakHARVARD UNIVERSITY |
Evolutionary Dynamics and Treatment of Cancer |

on Friday, April 25, 2014, at 3:00 PM in Science Center 507 | ||

Cancer is an evolutionary process. Cancer initiation and progression are caused by somatic mutation and selection of dividing cells. The mathematical theory of evolution can therefore provide quantitative insights into human cancer. I will discuss the role of chromosomal instability (CIN) and the accumulation of drivers and passengers in growing tumors. I will study success and failure of targeted therapy including combination of different drugs and evolution of resistance. A simple conclusion is that combination treatment can succeed, if the cancer requires at least two point mutations to gain resistance. From the perspective of preventing resistance, simultaneous therapy is highly recommended whereas sequential therapy is a recipe for almost certain treatment failure. |

GAUGE THEORY & TOPOLOGY SEMINAR: | Robert LipshitzCOLUMBIA UNIVERSITY |
Hochschild localization and bordered Heegaard Floer homology |

on Friday, April 25, 2014, at 3:30 PM in Science Center 309 | ||

We will discuss an analogue in Hochschild homology of the Smith inequality for a Z/2 action on a topological space. We will then apply this theorem to bordered Heegaard Floer homology, to obtain results about the Heegaard Floer homology of double covers, and speculate wildly about other possible extensions and applications. This is joint work with David Treumann. |

NUMBER THEORY SEMINAR: | Brandon LevinIAS |
Moduli of finite flat group schemes and local models |

on Wednesday, April 30, 2014, at 3:00 PM in Sci Center 507 | ||

In Kisin’s work on modularity lifting, he studies flat deformation rings using a moduli space of finite flat group schemes. In turn, the geometry of this space can be related to local models of Shimura varieties. We discuss generalizations of the moduli space, the local models, and the relationship between them. |

NUMBER THEORY SEMINAR: | Gopal PrasadUNIVERSITY OF MICHIGAN |
Fake projective spaces |

on Wednesday, May 07, 2014, at 3:00 PM in Science Center 507 | ||

A fake projective plane (fpp) is a smooth projective complex algebraic surface which has same Betti numbers as the complex projective plane. The first example of an fpp was constructed by David Mumford using p-adic uniformization. It was known that there are only finitely many fpp's and their fundamental groups are arithmetic subgroups of PU(2,1). In a joint work with Sai-Kee Yeung, using number theory, Bruhat-Tits theory, and my volume formula for the volumes of symmetric spaces modulo arithmetic groups, we have classified fpp's in 28 nonempty classes. This classification, with extensive computer-assisted computations by Steger and Cartwright, has led to a complete determination of all the fpp's--there are exactly 100 of them. This work has also given us an unexpected algebraic surface. In my talk I will describe the work, and its generalization to higher dimensions. For example, we have found fake analogues of complex projective 4-space. Our results have interesting consequences for the cohomology of arithmetic groups and automorphic forms. |