Date/Time : 12 Dec 2024 14:00
Venue : S17 #04-05 (SR2)
Speaker : Yihong Du, University of New England, Australia

A new phenomenon in nonlocal diffusion models is that accelerated propagation may happen, that is, the propagation speed could be infinite, which never occurs in the corresponding local diffusion model with compactly supported initial data. In this talk, we will first briefly review the history of the KPP model used to describe the propagation of biological/chemical species, and then look at some very recent results on the KPP equation with nonlocal diffusion and free boundaries. For several natural classes of kernel functions appearing in the nonlocal diffusion term, we will show how the exact rate of acceleration can be determined. The talk is based on joint works with several collaborators.

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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 : 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 #04-04 (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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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 : 25 Nov 2024 14:00
Venue : S17 #04-06 (SR1)
Speaker : Tomasz Przebinda, University of Oklahoma

We describe an integral kernel operator which should map a Harish-Chandra character of a representation of one member of a dual pair to a distribution on the other member of the dual pair. A well founded hope is that this distribution coincides with the character of the big theta lift. In part this is a joint work with Hung Yean Loke.

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Date/Time : 21 Nov 2024 15:00
Venue : S17 #04-04 (SR3)
Speaker : Gunther Uhlmann, University of Washington

We will consider the question on whether we can determine the structure of space time by making measurements near the worldline of an observer. We will consider both active and passive measurements. For the case of passive measurements one measures the fronts of light sources near the observer. For the case of active measurements we couple Einstein equations with matter or electromagnetic fields and formulate the question of determining the structure of space time as the problem of recovering the metric from observations of waves near the observer, The method applies to several other inverse problems for nonlinear equations, for example, the Westervelt equation arising in nonlinear acoustics.

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