Date/Time : 04 Dec 2024 15:00
Venue : S17 #04-06 (SR1)
Speaker : Thomas Karam, University of Oxford

This talk shall be essentially composed of four parts, each consisting of a short introduction to a research area followed by some contributions. We shall begin by discussing a central conjecture in Ramsey theory unifying several generalisations of the van der Waerden theorem. Thereafter, our focus shall move to the extension of basic properties of the rank of matrices to the different notions of rank of tensors, followed by the extension of basic properties of polynomials over finite fields to restrictions of these polynomials with Boolean variables. Finally, we shall study the Brunn-Minkowski inequality in more arbitrary ambient spaces than R^n, in particular in the torus (R/Z)^n for subsets satisfying various levels of constraints.

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Date/Time : 04 Dec 2024 14:00
Venue : S17 #04-06 (Seminar Room 1)
Speaker : Federico Rodriguez Hertz, The Pennsylvania State University

In the 80’s two type of related rigidity results emerged, one of geometric flavor and the other dynamical, but both have a dynamical background. Otal and Croke showed that for negatively curved surface the marked length spectrum determines the isometry type of the surface. On the other hand, Shub ans Sullivan for expanding maps and later de la LLave, Marco and Moriyon for Anosov diffeomorphisms in dimension 2 showed that the marked Lyapunov spectrum determines the smooth isomorphism type of the system. In this talk I will discuss new developments along this line of problems, discussing a more general framework where these theory can be developed. This project is joint with A. Gogolev and M. Leguil.

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Date/Time : 03 Dec 2024 14:00
Venue : S17 #04-06 (SR1)
Speaker : An Xinliang, National University of Singapore

In this talk, I will give an overview of my recent research. We prove a series of mathematical theorems that rigorously describe the entire process of black hole formation. With scale-critical short-pulse initial data, we solve Einstein’s equations and show that: “in the course of gravitational collapse, a whole trapped region and its boundary (apparent horizon) emerge from a singular spacetime point. The apparent horizon then expands as the spacetime evolves. Passing through a transition period, the spacetime later enters a regime, close to a slowly-rotating Kerr black hole solution. Nonlinear Kerr stability then comes into play and the spacetime metric eventually settles into the Kerr solution with small angular momentum.”

Meanwhile, I also introduce a new framework for proving the high co-dimensional nonlinear instability of naked-singularities. Specifically, I study the censorship of Christodoulou’s naked singularity and show that even a tiny anisotropic perturbation can trigger the formation of an anisotropic apparent horizon, thereby covering and censoring the naked singularity.

If time permits, sharp low-regularity illposedness and wellposedness results for nonlinear wave systems will also be presented.

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Date/Time : 03 Dec 2024 15:30
Venue : S17 #04-04 (Seminar Room 3)
Speaker : Federico Rodriguez Hertz, The Pennsylvania State University

In a joint work in progress with Pablo Carrasco we show that to any quasi-morphism for a hyperbolic group can be associated a function with the Bowen property for some hyperbolic dynamics. In this talk I plan to describe the construction and how to prove the CLT for a quasi-morphism w.r.t. some length function.

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Date/Time : 02 Dec 2024 16:00
Venue : S17 #04-04 (SR3)
Speaker : Changguang Dong, Nankai University

We will discuss a dynamically defined zeta function for partially hyperbolic systems. This is a generalization of the classical Ruelle zeta function for Anosov (more generally Axiom A) diffeomorphisms/flows, which may reflect dynamical invariants such as entropy and pressure, as well as the geometry/topology of the manifolds. Based on an ongoing project with Qiujie Qiao.

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Date/Time : 28 Nov 2024 13:30
Venue : S17 #04-05 (SR2)
Speaker : Peter Buhlmann, ETH Zurich

Nonparametric function estimation and prediction with moderate or large dimension of the covariates are particularly susceptible to extrapolation, because data points are typically far apart from each other in such moderate or higher dimension. Thus, there is a need to have machine learning methods that are extrapolation-aware, i.e. that automatically perform well (in a sense) when extrapolation occurs. Without such extrapolation-aware techniques, inference from standard machine learning and nonparametric procedures may be poor or invalid. We introduce a novel conceptual framework and introduce *Xtrapolation* which allows for extrapolation-aware inference with any ML algorithm.

This is joint work with Niklas Pfister (University of Copenhagen)

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Date/Time : 27 Nov 2024 14:00
Venue : S17 #04-06 (SR1)
Speaker : Patrick Lutz, University of California, Berkeley

The field of computability theory studies the complexity of uncomputable problems (where a “problem” is usually formally defined as either a set of natural numbers or a set of binary strings). In this study, a special role is played by the Halting Problem—i.e. the problem of determining whether a given program stops after a finite number of steps or runs forever. Not only is it the first problem proved to be uncomputable, it also seems to be the simplest “natural” uncomputable problem.

Martin’s Conjecture is a long-standing open question in computability theory which partially explains why the Halting Problem plays such a special role. A key idea behind Martin’s Conjecture is to view the Halting Problem not just as an individual problem, but as an operator on problems, which takes any problem to a strictly harder one. Martin’s Conjecture consists of a classification of such operators, which says, in part, that the Halting Problem is the minimal non-trivial operator. I will discuss the background and motivation for Martin’s Conjecture, as well as recent progress by Benjamin Siskind and myself which essentially completes a proof of the conjecture for a special class of operators called “order-preserving” (i.e. those which preserve the relative complexity of problems).

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Date/Time : 27 Nov 2024 14:00
Venue : S17 #04-04 (SR3)
Speaker : Marco Vergamini, Scuola Normale Superiore

Let f be a complex Hénon map and µ its unique measure of maximal entropy. Recently, Bianchi-Dinh proved that µ is exponentially mixing of all orders for all Hölder observables, and that all such observables satisfy the central limit theorem with respect to µ. De Thélin-Vigny generalized these results for a certain class of bounded plurisubharmonic observables. We prove that these properties hold for all, not necessarily bounded, plurisubharmonic observables. This is a joint work with Hao Wu.

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Date/Time : 27 Nov 2024 15:00
Venue : S17 #04-06 (SR1)
Speaker : Ruiyi Yang, Princeton University

Gaussian processes (GPs) are important random function models with desirable analytic properties that have found wide applications in inverse problems, spatial statistics, and machine learning. In this talk, we shall investigate a generalization of the popular Matérn GP to the manifold setting. In the first part, we formalize its definition and introduce a graph-based approximation that is computable given only a point cloud of samples from the manifold. The resulting graph Matérn GP enjoys a sparsity structure that facilitates computation and is widely applicable in many aspects of Bayesian methodologies. In the second part, we discuss in detail its application in a manifold optimization problem where gradients are intractable to obtain. Exploiting tools from Bayesian optimization, we propose an efficient algorithm with provable guarantees, whose effectiveness is further demonstrated by numerical examples.

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