A SPECIAL TALK IN BASIC NOTIONS: | Jean-Pierre SerreCollège de France |
Linear representations of finite groups |

on Tuesday, April 21, 2015, at 3:00 PM in Science Center Hall D | ||

I shall review the basic results in char. 0, char. p > 0 (Brauer style and homotopy style) and char. 1! |

DIFFERENTIAL GEOMETRY SEMINAR: | Steve RosenbergBOSTON UNIVERSITY |
Diffeomorphism groups of Sasakian manifolds |

on Tuesday, April 21, 2015, at 4:15 PM in Science Center 507 | ||

Using Chern-Simons classes on loop spaces, we show that the diffeomorphism group of a large class of compact Sasakian manifolds has infinite fundamental group. The main technique involves replacing the matrix trace used to define characteristic classes on a finite rank bundle with the Wodzicki residue to get characteristic classes on the tangent bundle to a loop space. |

NUMBER THEORY SEMINAR: | James BorgerAUSTRALIAN NATIONAL UNIVERSITY |
Positivity and periodicity in global Witt vectors |

on Wednesday, April 22, 2015, at 3:00 PM in Science Center 507 | ||

The p-typical Witt vectors of p-adic rings have played an important role in local number theory, and one might hope that multiple-prime Witt vectors of arithmetically global rings will be similarly useful in global number theory. As a step in that direction, I'll give two variations of the Witt vector functor which have a global nature. One deals with periodicity with respect to a modulus of a number field and is related to global class field theory; the other deals with positivity and is closely related to Schoenberg's classical theory of total positivity. |

INFORMAL DYNAMICS AND GEOMETRY SEMINAR: | Holly KriegerMIT |
A case of the dynamical André-Oort conjecture |

on Wednesday, April 22, 2015, at 4:00 - 5:00 PM in Science Center 507 | ||

I will discuss the dynamical André-Oort conjecture proposed by Baker-DeMarco, focusing on a case suggested by Ingram and recently proved by Ghioca, Nguyen, Ye, and myself: if C is an irreducible plane curve containing infinitely many points (a,b) for which z^d+a and z^d+b are both post-critically finite, then C is a fiber of a projection, or the zero locus of h(x) = Ax for some (d-1)st root of unity A. The proof includes an analysis of the existence of linear maps on angles of external rays of the Mandelbrot set. |

CENTER OF MATHEMATICAL SCIENCES AND APPLICATIONS COLLOQUIUM: | Suvrat RajuInternational Centre for Theoretical Sciences |
The Information Paradox and the black hole interior in AdS/CFT |

on Wednesday, April 22, 2015, at 4:00 PM in Science Center Hall A | ||

We describe recent refinements of the information paradox, within the context of the AdS/CFT correspondence. We review a recent proposal that quantum gravity effects may be large enough to modify the structure of the horizon of a large black hole and create a "firewall" there. We test this idea by constructing local probes of the black hole geometry. This leads us to the conclusion that the horizon is smooth --- precisely in line with semi-classical expectations. However, an examination of how the arguments for the firewall are sidestepped points to some interesting aspects of locality in quantum gravity. We also discuss how this construction of operators in the black hole interior relates to the Tomita Takesaki theory of modular isomorphisms of von Neumann algebras. This talk will be addressed to a general mathematical physics audience. So, for a large part, we will not assume any prior familiarity with string theory or the information paradox. |

CENTER OF MATHEMATICAL SCIENCES AND APPLICATIONS RANDOM MATRIX & PROBABILITY THEORY SEMINAR: | Wei-Kuo ChengUNIVERSITY OF CHICAGO |
Self-averaging and universality in spin glasses |

on Wednesday, April 22, 2015, at 11:30 am - 1:00 pm in Science Center 232 | ||

Spin glasses are disordered spin systems originated from the desire of understanding the strange magnetic behaviors of certain alloys in physics. As mathematical objects, they are often cited as examples of complex systems and have provided several fascinating structures and conjectures. In this talk, we will consider several spin glass models and study quenched self-avergaging properties of some physical observables including the magnetization, the Hamiltonian and the cross overlap in the disorder chaos problem. Second, we will discuss some universal properties of the Gibbs measures and conclude in particular that the Parisi ultrametric structure of the Gibbs measure in the mixed p-spin model is universal under relevant moment matching assumptions. Based on the joint works with A. Auffinger and D. Panchenko. |

SPECIAL NUMBER THEORY SEMINAR: | Jean-Pierre SerreCollège de France |
Galois cohomological invariants |

on Monday, April 27, 2015, at 3:00 PM in Science Center Hall A | ||

These invariants are the analogs in Galois cohomology of Chern and Stiefel-Whitney classes in Topology. I shall discuss them mainly for finite groups, e.g. symmetric and alternating group |

NUMBER THEORY SEMINAR: | Alexander MerkurjevUCLA |
Suslin's Conjecture on the reduced Whitehead group of a simple algebra |

on Wednesday, April 29, 2015, at 3:00 PM in Science Center 507 | ||

I will report on the proof of Suslin's Conjecture on the generic non-triviality of the reduced Whitehead group of a simple algebra. |

GAUGE THEORY, TOPOLOGY, AND SYMPLECTIC GEOMETRY SEMINAR: | Yanir RubinsteinUNIVERSITY OF MARYLAND |
The degenerate special Lagrangian equation |

on Friday, May 01, 2015, at 3:30 - 4:30 PM in Science Center 507 | ||

In joint work with J. Solomon, we introduce the {\it degenerate special Lagrangian equation} and develop the basic analytic tools to construct and study its solutions. This equation governs geodesics in the space of positive Lagrangians. It has intriguing relations to uniqueness, existence and stability notions related to special Lagrangians, Lagrangian mean curvature flow, and the topology of Hamiltonian isotopy classes. |