LEARNING SEMINAR ON THE FARGUES-FONTAINE CURVE: | Matthew MorrowInstitute of Mathematics of Jussieu |
Review of foundations of perfectoid spaces |

on Monday, October 23, 2017, at 4:00 - 6:00 PM in Science Center 507 |

JOINT DEPARTMENT OF MATHEMATICS AND CMSA RANDOM MATRIX AND PROBABILITY THEORY SEMINAR: | Madhu SudanCS, Harvard |
General Strong Polarization |

on Monday, October 23, 2017, at 12:00 - 1:00 PM in Science Center 232 | ||

A recent discovery (circa 2008) in information theory called Polar Coding has led to a remarkable construction of error-correcting codes and decoding algorithms, resolving one of the fundamental algorithmic challenges in the field. The underlying phenomenon studies the ``polarization'' of a ``bounded'' martingale. A bounded martingale, X_0,...,X_t,... is one where X_t in [0,1]. This martingale is said to polarize if Pr[lim_{t\to infty} X_t \in {0,1}] = 1. The questions of interest to the results in coding are the rate of convergence and proximity: Specifically, given epsilon and tau > 0 what is the smallest t after which it is the case that Pr[X_t in (tau,1-tau)] < epsilon? For the main theorem, it was crucial that t <= min{O(log(1/epsilon)), o(log(1/tau))}. We say that a martingale polarizes strongly if it satisfies this requirement. We give a simple local criterion on the evolution of the martingale that suffices for strong polarization. A consequence to coding theory is that a broad class of constructions of polar codes can be used to resolve the afore-mentioned algorithmic challenge. In this talk I will introduce the concepts of polarization and strong polarization. Depending on the audience interest I can explain why this concept is useful to construct codes and decoding algorithms, or explain the local criteria that help establish strong polarization (and the proof of why it does so). Based on joint work with Jaroslaw Blasiok, Venkatesan Guruswami, Preetum Nakkiran, and Atri Rudra. |

CENTER OF MATHEMATICAL SCIENCES AND APPLICATIONS MATHEMATICAL PHYSICS SEMINAR: | Florian BeckUniversity of Hamburg |
Hitchin systems in terms of Calabi-Yau threefolds |

on Monday, October 23, 2017, at 12:00 PM in CMSA Building, 20 Garden Street, Room G10 | ||

Integrable systems are often constructed from geometric and/or Lie-theoretic data. Two important example classes are Hitchin systems and Calabi-Yau integrable systems. A Hitchin system is constructed from a compact Riemann surface together with a complex Lie group with mild extra conditions. In contrast, Calabi-Yau integrable systems are constructed from a priori purely geometric data, namely certain families of Calabi-Yau threefolds. Despite their different origins there is a non-trivial relation between Hitchin and Calabi-Yau integrable systems. More precisely, we will see in this talk that any Hitchin system for a simply-connected or adjoint simple complex Lie group is isomorphicto a Calabi-Yau integrable system (away from singular fibers). |

JOINT DEPARTMENT OF MATHEMATICS AND CMSA RANDOM MATRIX & PROBABILITY THEORY SEMINAR: | Subhabrata SenMicrosoft and MIT |
Partitioning sparse random graphs: connections with mean-field |

on Wednesday, October 25, 2017, at 2:00 - 3:00 pm in CMSA Building, 20 Garden St, G10 | ||

The study of graph-partition problems such as Maxcut, max-bisection and min-bisection have a long and rich history in combinatorics and theoretical computer science. A recent line of work studies these problems on sparse random graphs, via a connection with mean field spin glasses. In this talk, we will look at this general direction, and derive sharp comparison inequalities between cut- sizes on sparse Erd\ ̋{o}s-R\'{e}nyi and random regular graphs. Based on joint work with Aukosh Jagannath. |

NUMBER THEORY SEMINAR: | Pierre ColmezUPMC and IAS |
On the cohomology of p-adic analytic curves |

on Wednesday, October 25, 2017, at 3:15 pm *sharp in Science Center 507 | ||

I will explain how the various cohomologies of a p-adic analytic curve are related. The case of coverings of the p-adic upper half-plane has applications to the p-adic local Langlands correspondence. |

JOINT DEPARTMENT OF MATHEMATICS AND CMSA RANDOM MATRIX & PROBABILITY THEORY SEMINAR: | Noga AlonTel Aviv University |
Random Cayley Graphs |

on Wednesday, October 25, 2017, at 3:00 - 4:00 PM in CMSA Building, 20 Garden St, G10 | ||

The study of random Cayley graphs of finite groups is related to the investigation of Expanders and to problems in Combinatorial Number Theory and in Information Theory. I will discuss this topic, describing the motivation and focusing on the question of estimating the chromatic number of a random Cayley graph of a given group with a prescribed number of generators. Several intriguing questions that remain open will be mentioned as well. |

CMSA ALGEBRAIC GEOMETRY SEMINAR: | Alexander MollIHES |
Hibert Schemes from Geometric Quantization of Dispersive Periodic Benjamin-Ono Waves |

on Thursday, November 02, 2017, at 3:00 PM in CMSA Building, 20 Garden St, G10 | ||

By Grojnowski and Nakajima, Fock spaces are cohomology rings of Hilbert scheme of points in the plane. On the other hand, by Pressley-Segal, Fock spaces are spaces of J-holomorphic functions on the loop space of the real line that appear in geometric quantization with respect to the Kähler structure determined by the Sobolev regularity s= -1/2 and the Hilbert transform J. First, we show that the classical periodic Benjamin-Ono equation is a Liouville integrable Hamiltonian system with respect to this Kähler structure. Second, we construct an integrable geometric quantization of this system in Fock space following Nazarov-Sklyanin and describe the spectrum explicitly after a non-trivial rewriting of our coefficients of dispersion \ebar = e_1 + e_2 and quantization \hbar = - e_1 e_2 that is invariant under e_2 <-> e_1. As a corollary of Lehn's theorem, our construction gives explicit creation and annihilation operator formulas for multiplication by new explicit universal polynomials in the Chern classes of the tautological bundle in the equivariant cohomology of our Hilbert schemes, in particular identifying \ebar with the deformation parameter of the Maulik-Okounkov Yangian and \hbar with the handle-gluing element. Our key ingredient is a simple formula for the Lax operators as elliptic generalized Toeplitz operators on the circle together with the spectral theory of Boutet de Monvel and Guillemin. As time permits, we discuss the relation of dispersionless \ebar -> 0 and semi-classical \hbar \rightarrow 0 limits to Nekrasov's BPS/CFT Correspondence. |