Date/Time : 10 May 2023 17:00
Venue : S17 #04-05 (SR2)
Speaker : Jan Baars,

For a Tychonov space X we define Cp(X) to be the linear subspace of R^X consisting of all real-valued continuous functions on X. Let βX be the ˇCech-Stone compactifaction of X and let X^* = βX \ X be the remainder of X. For u ∈ X^*, we denote Xu = X ∪ {u} ⊆ βX.

For the countable discrete space ω, elements of ω^* can be identified with free ultrafilters on ω. In 1990 Gul’ko proved in [3] for u, v ∈ ω^* that Cp(ωu) and Cp(ωu) are linearly home- omorphic if and only if ωu and ωv are homeomorphic. In [1] this result was generalized for finite sums of spaces ωu as follows: For n, m ≥ 1 and {u1,…,un, v1,…,vm} ⊆ ω∗, let X and Y be the direct sums of all ω_ui and ω_vi, respectively.

Then the spaces Cp(X) and Cp(Y) are linearly homeomorphic if and only X and Y are homeomorphic, in particular n = m. Gul’ko’s result does not hold for all spaces X. For example in [2] it was shown that for each ordinal space α, where ω < α < ω1 is a limit ordinal, there are u, v ∈ α^* such that Cp(αu) and Cp(αv) are linearly homeomorphic but αu and αv are not homeomorphic.

Here the references are as follows:

[1] J. Baars, On the lp-equivalence of ultrafilters, Bull. Pol. Acad. Sci. Math. Vol 70 No 1. (2022), 63–82.

[2] J. Baars and J. van Mill, Function spaces and points in ˇCech-Stone remainders, To appear in Topology Appl.

[3] S.P. Gul’ko, Spaces of continuous functions on ordinals and ultrafilters, Math. Notes, Vol. 47,4 (1990), 329–334.

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Date/Time : 04 May 2023 16:00
Venue : S17 #04-06 (SR1)
Speaker : Zhang Zhen, Southern University of Science and Technology

We develop energy stable and bound preserving schemes for phase-field surfactant model with moving contact lines. The desired properties of the schemes are rigorously proved. We numerically validate the accuracies of the schemes and apply them in simulating droplet impact problems. Qualitative agreements with experiments are obtained. Moreover, surfactants are observed to have effects on enhancing droplet deformation and reducing dissipations.

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Date/Time : 03 May 2023 15:00
Venue : S17 #04-04 (SR3)
Speaker : Marco Abate, University of Pisa

In this talk we shall describe the asymptotic behaviour of sequences of functions obtained by iteratively left-composing or right-composing holomorphic self-maps of a hyperbolic Riemann surface $X$. We shall consider in detail two cases: when the holomorphic self-maps to be composed have values in a Bloch subdomain of~$X$; and when the holomorphic self-maps to be composed are sufficiently close to a given self-map. This is a joint work with Argyrios Christodoulou and Ian Short.

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Date/Time : 27 Apr 2023 16:00
Venue : S17 #04-06 (SR1)
Speaker : Franck Sueur, University of Bordeaux

In this talk, I will give an overview of some recent results on the so-called transport-Stokes system which describes sedimentation of inertialess particles in a viscous flow. This system couples a transport equation and the steady Stokes equations in the full 3D space. I will address the Cauchy problem, the regularity of the particle trajectories and the controllability issue. We will see that the system has some similarity to the incompressible Euler system. Some of the exposed results have been obtained in collaboration with Amina Mecherbet.

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Date/Time : 19 Apr 2023 15:00
Venue : S17 #04-04 (SR3)
Speaker : Lucas Kaufmann, University of Orleans

Consider a (singular) holomorphic foliation F on a complex manifold. In the 1970s, Baum-Bott constructed residue classes associated with each singular component of F. These classes satisfy an index theorem, which computes characteristic classes of F.

A natural question since their work is how to compute or find natural representatives of such cohomology classes. Apart from the case where the foliation is of rank one and has isolated singularities, no general result is known. In this talk I will show that each Baum-Bott class can be naturally represented by a residue current supported by the singular component of F. This is a joint work with Richard Lärkäng and Elizabeth Wulcan.

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Date/Time : 18 Apr 2023 11:00
Venue : S17 #04-05 (SR2)
Speaker : Simon Gabriel, University of Warwick

We study the Allen-Cahn equation, commonly used to describe convection phenomena in physics. It is natural to start the system from chaos, corresponding to an initial condition described by spatial white noise. In two spatial dimensions, this leads to a critical initial datum, for which the equation can not be made sense of a priori.

In this talk we give a gentle introduction into the methods used to study this equation using an infinite expansion of stochastic integrals. Both the convergence and truthfulness of such an approximation are non-trivial, since the criticality has the effect that each term in the infinite expansion contributes towards the solution. Thus, it is crucial to determine each term’s statistic exactly, for which we exploit a tree-like notation. This allows us to manipulate stochastic integrals using solely their underlying tree structure. After performing such manipulations, we can conclude convergence of the expansion to a solution with the help of a series expansion of an explicit ODE.

The talk is based on joint work with Tommaso Rosati and Nikos Zygouras.

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Date/Time : 13 Apr 2023 11:00
Venue : S17 #05-12 (SR4)
Speaker : Lucas Mason-Brown, University of Oxford

Let G and H be real reductive groups. To any L-homomorphism e: H^L \to G^L one can associate a map e_* from virtual representations of H to virtual representations of G. This map was predicted by Langlands and defined (in the real case) by Adams, Barbasch, and Vogan. Without further restrictions on e, this map can be very poorly behaved. A special case in which e_* exhibits especially nice behavior is the case when H is an endoscopic group. In this talk, I will introduce a more general class of L-homomorphisms which exhibit similar behavior to the endoscopic case. I will explain how this more general notion of endoscopic lifting relates to the theory of cohomological induction. I will also explain how this generalized notion of endoscopic lifting can be used to prove the unitarity of many Arthur packets. This is based on joint work with Jeffrey Adams and David Vogan.

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Date/Time : 13 Apr 2023 10:00
Venue : via Zoom
Speaker : Daniel Sanz-Alonso, University of Chicago

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Semi-supervised learning refers to the problem of recovering an input-output map using many unlabeled examples and a few labeled ones. In this talk I will survey several mathematical questions arising from the Bayesian formulation of graph-based semi-supervised learning. These questions include the modeling of prior distributions for functions on graphs, the derivation of continuum limits for the posterior, the design of scalable posterior sampling algorithms, and the contraction of the posterior in the large data limit.

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Date/Time : 13 Apr 2023 16:00
Venue : S17 #04-06 (SR1)
Speaker : Xavier Fernández-Real, Swiss Federal Institute of Technology Lausanne (EPFL)

In this talk we present an introduction to integro-differential operators and the obstacle problem, and how they interact with each other in what is known as the fractional obstacle problem.

In $\R^n$, the fractional obstacle problem with a given obstacle $\varphi\in C_c^\infty(\R^n)$ can be written as

\[\min\{(-\Delta)^s u, u – \varphi\} = 0\quad\text{in}\quad \R^n.\]

The set $\{u = \varphi\}$ is called the contact set, and its boundary $\partial\{u = \varphi\}$ is called the free boundary, which are unknowns of the problem. The free boundary can be divided into two sets: regular points (where it is regular) and degenerate points. In general, degenerate points can exist in any dimension.

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