Veranstaltungen

Conical symplectic resolutions are a rather loosely defined class of hyperkähler varieties arising from canonical constructions in representation theory. Important examples include hypertoric varieties, Nakajima quiver varieties and Hitchin spaces. I will talk about Fukaya categories of conical symplectic resolutions. These are very rich objects, and they also turn out to be related to geometric representation theory. I will try to give an overview of this circle of ideas (much of which is not new in the symplectic literature). Any new content discussed in this talk is joint work with (subsets of) Benjamin Gammage, Justin Hilburn, Christopher Kuo, David Nadler and Vivek Shende.

In recent joint work with M. Dindos and L. Li, we show that the Regularity problem for a second order divergence form parabolic operator is solvable when the data belongs to an appropriate L^p space. The parabolic operator has coefficients that vary in both space and time, and satisfy a certain minimal smoothness assumption defined by a Carleson measure condition - one that has been well-studied for the elliptic analog of these equations. The long term goal, of which this is a step, is to bring the parabolic theory to the level of understanding that has now been achieved for elliptic boundary value problems and beyond, to free boundary problems.  

In this talk, I will present my research on the application of mathematical techniques to the analysis of data from STIX and HXI, two indirect imagers that measure X-rays emitted by the Sun during solar flares. The presentation will introduce the underlying physical problem, followed by a mathematical formulation that leads to an ill-posed inverse problem. I will discuss the stability of the problem and explore suitable algorithms for its solution. Finally, I will showcase preliminary reconstructions obtained by applying these methods to real observational data.

The Markoff triples are the natural solutions of the Markoff relation $x^2+y^2+z^2=3xyz$. They can be viewed as points in the orbit of the mapping class group acting on the character variety of the punctured torus. In this case the action is transitive. However, when considering surfaces with a higher genus or number of holes, this is no longer the case. We will discuss the possible orbits closures of the general case; the algebraic relations that are preserved under the action and the exceptional cases. This is a joint work with Alireza Salehi-Golsefidy.

I will describe two new constructions of finitely presented infinite simple groups. First, I will present a construction of finitely presented (and type F_∞) simple groups that act by orientation preserving homeomorphisms on the real line. These are the first such examples. Next, I will present a construction of a family of finitely presented infinite uniformly simple groups, where the Ulam width can get arbitrarily large. Among the class of finitely generated (but not finitely presentable) groups, the existence of such examples was demonstrated in the work of Ivanov from 1989. Our construction provides the first such family of examples in the class of infinite finitely presented groups. This is joint work with James Hyde.

There is a mystery at the heart of operator learning: how can one recover a non-self-adjoint operator from data without probing the adjoint? Current practical approaches suggest that one can accurately recover an operator while only using data generated by the forward action of the operator without access to the adjoint. However, naively, it seems essential to sample the action of the adjoint for learning time-dependent PDEs. In this talk, we will first explore connections with low-rank matrix recovery problems in numerical linear algebra. Then, we will show that one can approximate a family of non-self-adjoint infinite-dimensional compact operators via projection onto a Fourier basis without querying the adjoint.

Since the pioneering work of Aldous in the 1990s, it has been well established that large random trees converge towards a universal object: the Brownian tree. This object, which has become a pillar of modern probability, is a real random compact tree of fractal dimension 2. In this presentation, we will focus on different tree growth algorithms, such as the Rémy algorithm and the Luczak-Winkler algorithm. We will see how, by passing them to the limit, they give rise to diffusions taking values in the space of real trees, of which the Brownian tree constitutes the invariant law.

Since numerically solving high-dimensional semidefinite programs is challenging, solvers should exploit the structure of the problem at hand, when it allows for speedups. In this talk, we will consider the setting where we have the a priori information that the solution is of low rank (for instance, rank 1). A standard way to exploit this information is through the Burer-Monteiro factorization: we represent the unknown matrix as a product of "thin" matrices (i.e. with few columns); then, we optimize the factors instead of the whole square matrix. This technique reduces the dimensionality of the problem, allowing for important computational savings. However, it makes the problem non-convex, thereby possibly introducing non-optimal critical points which can cause the solver to fail. With Faniriana Rakoto Endor, we have considered the specific category of so-called "MaxCut-type" semidefinite problems. We have exhibited a simple property which guarantees that the resulting non-convex problem has no non-optimal critical point, and hence ensures the success of the algorithm.

Let S be a closed surface of genus g > 1. A classical theorem due to Fricke asserts that the mapping class group of S acts properly discontinuously on its Teichmüller space. This group action can be viewed from an algebraic perspective by examining the natural action by precomposition of Out(π₁(S)) on the set of conjugacy classes of discrete and faithful representations of π₁(S) into PSL₂(ℝ). In this talk, we will introduce these fundamental objects, clarify the aforementioned identifications, and prove a generalization of Fricke's theorem using exclusively Gromov-hyperbolic geometry.

I will discuss a series of recent works written jointly with M. Ifrim, B. Pineau and D. Tataru where we develop a new \emph{fully Eulerian} approach to free boundary fluid dynamics, obtaining the first sharp well-posedness theorems for the free boundary Euler and MHD equations. Our well-posedness theories include (i) Local well-posedness in the Hadamard sense, i.e., local existence, uniqueness, and the first proof of continuous dependence on the data, all at optimal Sobolev regularity; (ii) Enhanced uniqueness: Our uniqueness result holds at the level of scaling, requiring merely integrability in time of the Lipschitz norm of \((v,B)\) and the \(C^{1,\frac{1}{2}}\) norm of the free surface; (iii) Stability bounds: We construct a nonlinear functional which measures, in a suitable sense, the distance between two solutions (even when defined on different domains) and we show that this distance is propagated by the flow; (iv) Energy estimates: We prove refined, essentially scale invariant energy estimates for solutions, relying on a newly constructed family of elliptic estimates; (v) Continuation criterion: We give the first proof of a sharp continuation criterion in the physically relevant pointwise norms, at the level of scaling. In particular, for the Euler equations we show that solutions can be continued as long as the velocity is in \(L_T^1W^{1,\infty}\), the free surface is in \(L_T^1C^{1,\frac{1}{2}}\) and the surface does not self-intersect; (vi) A novel construction of regular solutions, which was not known at any regularity for MHD.

We investigate Markovian lifts of stochastic Volterra equations(SVEs) with completely monotone kernels and general coefficients within the framework of weighted Sobolev spaces. Our primary focus is developing a comprehensive solution theory for a class of non-local stochastic evolution equations (SEEs) encompassing these Markovian lifts. This enables us to extend known results for the lifted equation such as existence of solutions and of invariant measures. Additionally our Framework allows us to obtain an Itô-type formula for the stochastic Volterra equations.