# Upcoming Seminars & Events

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##### Moment map formalism, DUY theorem and beyond

In this expository talk, I will introduce the basic ideas about identifying symplectic quotient and good quotient in the sense of geometric invariant theory. After presenting some finite-dimensional examples, I will discuss the renowned Donaldson-Uhlenbeck-Yau theorem relating slope-stable holomorphic bundle with Hermitian-Yang-Mills connections. If time permits, I will present more applications of the moment map formalism, such as Hitchin's equations and nonabelian Hodge theory.

##### 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.

##### Ranks of matrices with few distinct entries

Many applications of linear algebra method to combinatorics rely on the bounds on ranks of matrices with few distinct entries and constant diagonal. In this talk, I will explain some of these application. I will also present a classification of sets L for which no low-rank matrix with entries in L exists.

##### 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.

##### 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

##### 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.

##### 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.

##### 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.

##### 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.

##### Wrapped Fukaya categories and Functors

Inspired by homological mirror symmetry for non-compact manifolds, one wonders what functorial properties wrapped Fukaya categories have as mirror to those for the derived categories of the mirror varieties, and also whether homological mirror symmetry is functorial. Comparing to the theory of Lagrangian correspondences for compact manifolds, some subtleties are seen in view of the fact that modules over non-proper categories are complicated.

##### 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.

##### 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.

##### Taylor Proudman columns: "rotation stabilizes the fluid around the 2D flow..."

##### 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.

##### Elliptic curves of rank two and generalised Kato classes

The generalised Kato classes of Darmon-Rotger arise as p-adic limits of diagonal cycles on triple products of modular curves, and in some cases, they are predicted to have a bearing on the arithmetic of elliptic curves over Q of rank two. In this talk, we will report on a joint work in progress with Ming-Lun Hsieh concerning a special case of the conjectures of Darmon-Rotger.