# Upcoming Seminars & Events

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##### Rare Region Effects and Many-Body Localization

Certain strongly disordered many-body quantum systems are incapable of reaching thermal equilibrium. The nature of this so-called many-body localized (MBL) phase has recently been an active area of research. The phenomenon can be understood through perturbative approximations, but rare regions with weak disorder (Griffiths regions) have the potential to bypass barriers to thermalization. I show that these effects do not destroy the MBL phase in one dimension, under a natural assumption on eigenvalue statistics.

##### The Hodge decomposition for some non-Kahler threefolds with trivial canonical bundle.

We show that the \partial\bar{\partial}-lemma holds for the non-Kahler compact complex manifolds of dimension three with trivial canonical bundle constructed by Clemens as deformations of Calabi-Yau threefolds contracted along smooth rational curves with normal bundle of type (-1, -1), at least on an open dense set in moduli. The proof uses the mixed Hodge structure on the singular fibers and an analysis of the variation of the Hodge filtration for the smooth fibers.

##### Iron Age Hebrew Epigraphy in the Silicon Age - An Algorithmic Approach To Study Paleo-Hebrew Inscriptions

Handwriting comparison and identification, e.g. in the setting of forensics, has been widely addressed over the years. However, even in the case of modern documents, the proposed computerized solutions are quite unsatisfactory. For historical documents, such problems are worsened, due to the inscriptions’ preservation conditions. In the following lecture, we will present an attempt at addressing such a problem in the setting of First Temple Period inscriptions, stemming from the isolated military outpost of Arad (ca. 600 BCE).

##### Cover time of trees and of the two dimensional sphere

I will begin by reviewing the general relations that exist between the cover time of graphs by random walk and the Gaussian free field on the graph, and explain the strength and limitations of these general methods. I will then discuss recent results concerning the cover time of the binary tree of depth $n$ by simple random walk, and in particular sharp fluctuation results for the cover time, mirroring those for the maximal displacement of branching random walk; certain barrier estimates for Bessel processes play a crucial role.

##### New Faculty Talks, III

4:30 p.m. | Yueh-Ju Lin, Instructor |

4:50 p.m. | Casey Kelleher, Postdoctoral Research Fellow |

5:10 p.m. | Huy Nguyen, Postdoctoral Research Associate |

5:30 p.m. | Joe Waldron, Instructor |

##### TBA-Shaoyun Bai

##### Hadamard well-posedness of the gravity water waves equations

The gravity water waves equations consist of the incompressible Euler equations and an evolution equation for the free boundary of the fluid domain. Assuming the flow is irrotational, Alazard-Burq-Zuily (Invent. Math, 2014) proved that for any initial data in Sobolev space $H^s$, the problem has a unique solution lying in the same space, here s is the smallest index required to ensure that the fluid velocity is spatially Lipschitz. We will discuss the strategy of a proof of the fact that the flow map is continuous in the strong topology of H^s.

##### Homotopy Group Actions and Group Cohomology

Understanding the symmetries of a topological space is a classical problem in mathematics. In this talk we will consider the somewhat more flexible notion of a group action up to homotopy. This leads to interesting interactions between topology, group theory and representation theory. This is joint work with Jesper Grodal.

##### TBA-Boris Bukh

##### The arithmetic intersection conjecture

The Gan-Gross-Prasad conjecture relates the non-vanishing of a special value of the derivative of an L-function to the non-triviality of a certain functional on the Chow group of a Shimura variety. Beyond the one-dimensional case, there is little hope for proving this conjecture. I will explain a variant of this conjecture (suggested by Wei Zhang) which seems more accessible and report on progress on it. This is joint work with B. Smithling and W. Zhang.

##### Solutions after blowup in ODEs and PDEs: spontaneous stochasticity

We discuss the extension of solutions beyond a finite blowup time, i.e., the time at which the system ceases to be Lipschitz continuous. For larger times solutions are defined first by using a (physically motivated) regularization of equations and then taking the limit of a vanishing regularization parameter. We report on several generic situations when such a limit leads to stochastic solutions defining uniquely a probability to choose one or the other (non-unique) path.

##### Mayer-Vietoris sequence for relative symplectic cohomology

I will first recall the definition of an invariant that assigns to any compact subset K of a closed symplectic manifold M a module SH_M(K) over the Novikov ring. I will go over the case of M=two sphere to illustrate various points about the invariant. Finally I will state the Mayer-Vietoris property and explain under what conditions it holds.

##### Willmore Stability of Minimal Surfaces in Spheres

Minimal surfaces in the round n-sphere are prominent examples of surfaces critical for the Willmore bending energy W; those of low area provide candidates for W-minimizers. To understand when such surfaces are W-stable, we study the interplay between the spectra of their Laplace-Beltrami, area-Jacobi and W-Jacobi operators. We use this to prove: 1) the square Clifford torus in the 3-sphere is the only W-minimizer among 2-tori in the n-sphere; 2) the hexagonal Itoh-Montiel-Ros torus in the 5-sphere is the only other W-stable minimal 2-torus in the n-sphere, for all n; 3) the Itoh-Montiel-R

##### High frequency back reaction for the Einstein equations

It has been observed by physicists (Isaacson, Burnett, Green-Wald) that metric perturbations of a background solution, which are small amplitude but with high frequency, yield at the limit to a non trivial contribution which corresponds to the presence of a stress-energy tensor in the equation for the background metric. This non trivial contribution is due to the nonlinearities in Einstein equations, which involve products of derivatives of the metric.

##### Mean estimation: median-of-means tournaments

One of the most basic problems in statistics is how to estimate the expected value of a distribution, based on a sample of independent random draws. When the goal is to minimize the length of a confidence interval, the usual empirical mean has a sub-optimal performance, especially for heavy-tailed distributions. In this talk we discuss some estimators that achieve a sub-Gaussian performance under general conditions. The multivariate scenario turns out to be more challenging. We present an estimator with near-optimal performance.

##### TBA-Yuan Gao

##### Convergence of percolation-decorated triangulations to SLE and LQG

The Schramm-Loewner evolution (SLE) is a family of random fractal curves, which is the proven or conjectured scaling limit of a variety of two-dimensional lattice models in statistical mechanics. Liouville quantum gravity (LQG) is a model for a random surface which is the proven or conjectured scaling limit of discrete surfaces known as random planar maps (RPM). We prove that a percolation-decorated RPM converges in law to SLE-decorated LQG in a certain topology. This is joint work with Bernardi and Sun.

##### Transcendence of period maps

Period domains D can be described as certain analytic open sets of flag varieties; due to the presence of monodromy, however, the period map of a family of algebraic varieties lands in a quotient D/\Gamma by an arithmetic group. In the very special case when D/\Gamma is itself algebraic, understanding the interaction between algebraic structures on the source and target of the uniformization D\rightarrow D/\Gamma is a crucial component of the modern approach to the André-Oort conjecture. We prove a version of the Ax-Schanuel conjecture for general period maps X\rightarrow D/\Gamma which s

##### Quantum Markov Semigroups with detailed balance as gradient flow for relative entropy and entropy production inequalities

Semigroups of completely positive trace preserving maps satisfying a certain detailed balance condition are gradient flow driven by dissipation of the quantum relative entropy with respect to a non-commutative analog of the 2-Wasserstein metric on the space of probability densities on Euclidean space. As in the classical case, this way of viewing the evolution equations solved by these semigroups leads to sharp entropy production inequalities. This perspective has resolved some recent conjectures in quantum information theory. This is joint work with Jan Maas.