MATHEMATICAL PHYSICS SEMINAR: | Masahito YamazakiUniversity of Tokyo |
Integrable Lattice Models from Gauge Theory |

on Tuesday, February 21, 2017, at 2:45 pm in Jefferson 453 | ||

In a celebrated paper in 1989, E. Witten discovered a beautiful connection between knot invariants (such as the Jones polynomial) and three-dimensional Chern-Simons theory. Since there are similarities between knot theory and integrable models, it is natural to ask if there is also a gauge theory explanation for integrable models. The answer to this question was recently given by K. Costello in 2013. In this talk I will describe my ongoing work, which explains many results in integrable models from the standard quantum field theory analysis of Costello's theory. |

HARVARD/MIT ALGEBRAIC GEOMETRY SEMINAR: | David HyeonSeoul National University |
Commuting nilpotents modulo simultaneous conjugation and Hilbert scheme |

on Tuesday, February 21, 2017, at 3:00 pm in Science Center 507 | ||

Pairs of commuting nilpotent matrices have been extensively studied, especially from the view point of quivers. But the space of commuting nilpotents modulo simultaneous conjugation has not received any attention at all although it has a definite moduli theory flavor. Unlike the case of commuting nilpotents paired with a cyclic vector, the GIT is not well behaved in this case. I will explain how a 'moduli space' can be constructed as a homogeneous space, and show that it is isomorphic to an open subscheme of a punctual Hilbert scheme. Over the field of complex numbers, thus constructed space is diffeomorphic to a direct sum of twisted tangent bundles over a projective space. This is a joint work with W. Haboush. |

DIFFERENTIAL GEOMETRY SEMINAR: | Alex WaldronSimons Center at Stony Brook |
Long-time existence for Yang-Mills flow |

on Tuesday, February 21, 2017, at 3:15 - 4:15 PM in CMSA Building, 20 Garden St, G10 | ||

I'll describe the general picture of Yang-Mills flow on a four-dimensional Riemannian manifold, where curvature concentration is a subtle problem. Time permitting, I'll give an indication of my recent proof that bubbling occurs only at infinite time, which was conjectured in 1997. |

CENTER OF MATHEMATICAL SCIENCES AND APPLICATIONS SOCIAL SCIENCE APPLICATIONS FORUM: | Ravi JagadeesanHarvard University |
Complementary inputs and the existence of stable outcomes in large trading networks |

on Tuesday, February 21, 2017, at 4:30 - 5:30 PM in CMSA Building, 20 Garden St, G02 | ||

This paper studies a model of large trading networks with bilateral contracts. The model allows income effects, unlike in parts of the matching literature, and imperfectly tradeable goods, unlike in the general equilibrium literature. In our setting, under standard continuity and convexity conditions, a stable outcome is guaranteed to exist in any acyclic network, as long as all firms regard sales as substitutes and the market is large. Thus, complementarities between inputs do not preclude the existence of stable outcomes in large markets. Even when there are complementarities between sales, this paper shows that tree stable outcomes are guaranteed to exist in large markets, under continuity and convexity conditions. The model presented in this paper generalizes and unifies versions of general equilibrium models with divisible and indivisible goods, matching models with continuously divisible contracts, models of large (two-sided) matching with complementarities, and club formation models. Additional results provide intuition for the role of uni-directional substitutability conditions and acyclicity in the main existence results, and explain what kinds of equilibria are guaranteed to exist even when these conditions are relaxed. Unlike in two-sided large-market settings, the sufficient conditions described in this paper pin down maximal domains for the existence of equilibria. |

JOINT DEPARTMENT OF MATHEMATICS AND CENTER OF MATHEMATICAL SCIENCES AND APPLICATIONS RANDOM MATRIX AND PROBABILITY THEORY SEMINAR: | Bob HoughStony Brook University |
Random walk on unipotent groups |

on Wednesday, February 22, 2017, at 3:00 pm in CMSA Building, 20 Garden St, G10 | ||

I will describe results of two recent papers from random walk on unipotent groups. In joint work with Diaconis (Stanford), we obtain a new local limit theorem on the real Heisenberg group, and determine the mixing time of coordinates for some random walks on finite unipotent groups. In joint work with Jerison and Levine (Cornell) we prove a cut-off phenomenon in sandpile dynamics on the torus $(\mathbb{Z}/m\mathbb{Z})^2$ and obtain a new upper bound on the critical exponent of sandpiles on $\mathbb{Z}^2$. |

NUMBER THEORY SEMINAR: | Yiwei SheIAS and Columbia University |
The (unpolarized) Shafarevich conjecture for K3 surfaces |

on Wednesday, February 22, 2017, at 3:00 pm in Science Center 507 | ||

Let K be a number field, S a finite set of places of K, and g a positive integer. Shafarevich made the following conjecture for higher genus curves: the set of isomorphism classes of genus g curves defined over K and with good reduction outside of S is finite. Faltings proved this conjecture for curves and the analogous conjecture for polarized abelian surfaces and Zarhin removed the necessity of specifying a polarization. Building on the work of Faltings and Andre and using technical advances by Madapusi Pera, we prove the unpolarized Shafarevich conjecture for K3 surfaces. |

CENTER OF MATHEMATICAL SCIENCES AND APPLICATIONS COLLOQUIUM: | Steven RayanUniversity of Saskatchewan |
Higgs bundles and the Hitchin system |

on Wednesday, February 22, 2017, at 4:30 pm in CMSA Building, 20 Garden St, G10 | ||

I will give an informal introduction to the Hitchin system, an object lying at the crossroads of geometry and physics. As a moduli space, the Hitchin system parametrizes semistable Higgs bundles on a Riemann surface up to equivalence. From this point of view, the Hitchin map and spectral curves emerge. We'll use these to form an impression of what the moduli space "looks like". I will also outline the appearances of the Hitchin system in dynamics, hyperkaehler geometry, and mirror symmetry. |

HARVARD MIT ALGEBRAIC GEOMETRY SEMINAR: | Daniel LittColumbia University |
Arithmetic Restrictions on Geometric Monodromy |

on Tuesday, February 28, 2017, at 3:00 pm in MIT 4-153 | ||

Let X be an algebraic variety over a field k. Which representations of pi_1(X) arise from geometry, e.g. as monodromy representations on the cohomology of a family of varieties over X? We study this question by analyzing the action of the Galois group of k on the fundamental group of X. As a sample application of our techniques, we show that if X is a normal variety over a field of characteristic zero, and p is a prime, then there exists an integer N=N(X,p) satisfying the following: any irreducible, non-trivial p-adic representation of the fundamental group of X, which arises from geometry, is non-trivial mod p^N. |

NUMBER THEORY SEMINAR: | David HansenColumbia University |
Some remarks on local Shimura varieties |

on Wednesday, March 01, 2017, at 3:00 pm in Science Center 507 | ||

I'll give an introduction to local Shimura varieties and (more generally) moduli spaces of mixed-characteristic local shtukas as defined by Scholze. I'll also discuss some recent results and conjectures on their geometry and cohomology. This is partially joint work with Jared Weinstein. |

JOINT DEPARTMENT OF MATHEMATICS AND CENTER OF MATHEMATICAL SCIENCES AND APPLICATIONS RANDOM MATRIX AND PROBABILITY THEORY SEMINAR: | Subhajit GoswamiUniversity of Chicago |
Liouville first-passage percolation and Watabiki's prediction |

on Wednesday, April 12, 2017, at 3:00 pm in CMSA Building, 20 Garden St, G10 | ||

In this talk I will give a brief introduction to Liouville first-passage percolation (LFPP) which is a model of random metric on a finite planar grid graph. It was studied primarily as a way to make sense of the random metric associated with Liouville quantum gravity (LQG), one of the major open problems in contemporary probability theory. I will discuss some recent results on this metric and the main focus will be on estimates of the typical distance between two points. I will also discuss about the apparent disagreement of these estimates with a prediction made in the physics literature about LQG metric. The talk is based on a joint work with Jian Ding. |