Thursday, October 11, 2018
Time  Items 

All day 

3pm 
10/11/2018  3:00pm Birational superrigidity and Kstability are properties of Fano varieties that have many interesting geometric implications. For instance, birational superrigidity implies nonrationality and Kstability is related to the existence of KählerEinstein metrics. Nonetheless, both properties are hard to verify in general. In this talk, I will first explain how to relate birational superrigidity to Kstability using alpha invariants; I will then outline a method of proving birational superrigidity that works quite well with most families of index one Fano complete intersections and thereby also verify their Kstability. This is partly based on joint work with Charlie Stibitz and Yuchen Liu. Location:
HLH17 03
17 Hillhouse Ave, Basement level, room 3
New Haven, CT
06511

4pm 
10/11/2018  4:00pm Is there a single infinite sequence of directions that “solves” every maze in the infinite square grid? I will speak about various problems that arise from this seemingly innocuous question, focusing on connections to various “resilience properties” of the simple random walk. Location:
DL 431
10/11/2018  4:15pm In 1979, O.Heilmann and E.H. Lieb introduced an interacting dimer model with the goal of proving the emergence of a nematic liquid crystal phase in it. In such a phase, dimers spontaneously align, but there is no long range translational order. Heilmann and Lieb proved that dimers do, indeed, align, and conjectured that there is no translational order. I will discuss a recent proof of this conjecture. This is joint work with Elliott H. Lieb. Location:
LOM 215
10/11/2018  4:15pm Let $X$ be a smooth curve over a finitely generated field $k$, and let $\ell$ be a prime different from the characteristic of $k$. We analyze the dynamics of the Galois action on the deformation rings of mod $\ell$ representations of the geometric fundamental group of $X$. Using this analysis, we prove analogues of the Shafarevich and FontaineMazur finiteness conjectures for function fields over algebraically closed fields in arbitrary characteristic, and a weak variant of the FreyMazur conjecture for function fields in characteristic zero. For example, we show that if $X$ is a normal, connected variety over $\mathbb{C}$, the (typically infinite) set of representations of $\pi_1(X^{\text{an}})$ into $GL_n(\overline{\mathbb{Q}_\ell})$, which come from geometry, has no limit points. As a corollary, we deduce that if $L$ is a finite extension of $\mathbb{Q}_\ell$, then the set of representations of $\pi_1(X^{\text{an}})$ into $GL_n(L)$, which arise from geometry, is finite. Location:
LOM 214
10/11/2018  4:15pm Given two permutations A and B which “almost” commute, are they “close” to permutations A’ and B’ which really commute? Arzhantseva and Paunescu (2015) formalized this question and answered affirmatively. This can be viewed as a property of the equation XY=YX, and turns out to be equivalent to the following property of the group Z^2 = < X,Y  XY=YX >: Every “almost action” of Z^2 on a finite set is close to a genuine action of Z^2. This leads to the notion of stable groups.
We will describe a relationship between stability, invariant random subgroups and sofic groups, giving, in particular, a characterization of stability among amenable groups. We will then show how to apply the above in concrete cases to prove and refute stability of some classes of groups. Finally, we will discuss stability of groups with Kazhdan’s property (T), and some results on the quantitative aspect of stability.
Based on joint works with Alex Lubotzky, Andreas Thom and Jonathan Mosheiff.
Location:
LOM 205
10/11/2018  4:30pm Vera Serganova and I have recently introduced an analog of category O for the three finitary
Lie algebras sl(infty), o(infty), sp(infty). The definition of category O is not obvious in this setting,
and our category consists of “smallest possible” modules. An interesting price to pay is that the Borel
subalgebra involved in the construction is not of Dynkin type. In the talk I will explain that our category is a highest
weight category and will also compute the annihilators in the enveloping algebra of the simple
objects of the category.
Location:
LOM 206
