Date/Time : 13 May 2024 14:00
Venue : S17 #04-04 (SR3)
Speaker : Sheng Meng, East China Normal University

A critical point x of a differentiable self-map f of X=R^n or manifolds means the determinant of the Jacobian |J_f(x)|=0. We collect these points and call it the critical set C_f. This set C_f also carries algebraic data, denoted as R_f which is nothing but the zeros (counting multiplicities) of the equation |J_f(x)|=0, called the ramification.

In the algebraic geometric setting, R_f turns out to be an effective divisor, but not necessarily smooth anymore. In this talk, I shall show you the behaviour of the coefficients of R_f (multiplicity in codimension 1), and also the much more complicated, the singularities of R_f (multiplicity in higher codimension).
Some applications will be discussed if time permits.

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Date/Time : 10 May 2024 15:00
Venue : S17 #04-06 (SR1)
Speaker : Erez Lapid, Weissman Institute of Science

Prehomogeneous vector spaces are ubiquitous in the theory of nilpotent orbits of semisimple Lie algebras. Their study was initiated by M. Sato in the 1960s. Zeta functions attached to them are useful for the asymptotics of arithmetic invariants. They are also important in the geometric side of Arthur’s trace formula. I will discuss some recent work on the analysis of these zeta functions.

Joint work with Tobias Finis.

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Date/Time : 08 May 2024 16:30
Venue : S17 #04-05 (SR2)
Speaker : Junfu Yao, Johns Hopkins University

Minimal surfaces are known as critical points to the area functional. In this talk, I’ll focus on the question of finding minimal surfaces with prescribed behavior at the asymptotic infinity. This is called the asymptotic Plateau problem. I’ll briefly review previous results and present a min-max theory for finding solutions to this problem. As a corollary of the mountain-pass theorem, I’ll also discuss a topological uniqueness result.

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Date/Time : 08 May 2024 14:00
Venue : S17 #04-05 (SR2)
Speaker : Xiaofan Yuan, Arizona State University

Resolving a recent problem of Bell, Frieze, and Marbach, we establish the threshold result of Frankston–Kahn–Narayanan–Park in the rainbow setting. This is joint work with Jie Han.

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Date/Time : 08 May 2024 17:00
Venue : S17 #04-04 (SR3)
Speaker : Vittorio Cipriani, Vienna University of Technology

In this talk, we combine computable structure theory and inductive inference to study learning of families of structures. All the structures we consider are relational and countable and all the families of structures we consider are at most countable. The framework we use was defined in a series of papers by Bazhenov, Fokina, Kotzing, and San Mauro. In a nutshell, the framework models the scenario in which, given a family of structures K, a learner receives more and more information about the atomic diagram of a copy of some A ∈ K and, at each stage, is required to output a conjecture about the isomorphism type of such a structure. In this context, a natural criterion to consider is Ex-learning in which we require the learner to stabilize to the correct conjecture after finitely many steps.

Together with Bazhenov and San Mauro we gave a descriptive set-theoretic characterization of Ex-learning. Namely, we showed that a family of structures is Ex-learnable if and only if the corresponding isomorphism problem continuously reduces to E_0, the equivalence relation of eventual agreement on infinite binary sequences. Replacing E0 with other equivalence relations, one obtains a hierarchy to rank such isomorphism problems. That is, a family of structures K is E-learnable, for an equivalence relation E, if there is a continuous reduction from the isomorphism problem associated with K to E.

We aim to obtain model-theoretic characterization of E-learning for different equivalence relations E. Some characterizations are already present in the literature: here we show that a family of structures K such that for any A_i, A_j ∈ K there is a Σ_n^{inf} formula satisfied by A_i but not by A_j is E-learnable if and only if E is the (iteration of the) Friedman-Stanely jump of the identity either on natural numbers or on Cantor space. We also show that other learning criteria coming from the classical setting of inductive inference of formal languages or recursive functions have a nice model-theoretic characterization.

This talk collects joint works with Bazhenov, Jain, Marcone, San Mauro and Stephan.

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Date/Time : 02 May 2024 14:00
Venue : S17 #06-11 (SR6)
Speaker : Xuecheng Wang, Tsinghua University

The 3D relativistic Vlasov-Maxwell system is one of the fundamental models in the plasma physics, which describes the dynamics of electron and ions under the electromagnetic field created by particles themselves. In this talk, we discuss the existence of global solution and the propagation of regularity of the 3D relativistic Vlasov-Maxwell system for both the small initial data case and the large initial data case.

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Date/Time : 02 May 2024 15:00
Venue : S17 #06-11 (SR6)
Speaker : Pengyu Le, BIMSA

In this talk, we shall introduce the Lorentz polarisation in the Minkowski spacetime, which is the analogy of the polarisation in the Euclidean space. Similar to the application of the Euclidean polarisation to the classical isoperimetric inequality, we shall apply the Lorentz polarisation to study an analogous isoperimetric inequality for the volume of domain of dependence in Minkowski spacetime, and some other geometric variational problems.

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Date/Time : 24 Apr 2024 14:00
Venue : S17 #06-11 (SR6)
Speaker : Arick Shao, Queen Mary University of London

We consider heat operators on a bounded domain, with a critically singular potential diverging as the inverse square of the distance to the boundary of the domain. We establish the first boundary control results for such operators in all spatial dimensions.

First, we prove null controllability for a range of parameters representing the strength of the singular potential, and with control imposed on the entire boundary. Next, we prove a weaker approximate control result, but for the full range of parameters with a viable well-posedness theory, and with the control localised to any open subset of the boundary.

This is joint work with Alberto Enciso (ICMAT) and Bruno Vergara (Brown).

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Date/Time : 23 Apr 2024 16:00
Venue : S8 #03-14
Speaker : Philipp Kunde, Jagiellonian University

One of the oldest and most influential problems in ergodic theory is the classification of invertible measure-preserving transformations (MPT’s) up to isomorphism or Kakutani equivalence. Starting from the late 1990s, so-called anti-classification results use concepts from descriptive set theory to show precisely that classification for ergodic MPT’s is impossible in terms of computable invariants, even with a very inclusive understanding of “computability”. In this talk, we present more details about our techniques to prove such anti-classification results.

In the search for classification or anti-classification results, we also place increasingly more stringent conditions on the family of transformations being considered for classification. Bernoulli shifts, which can be viewed as the most random type of transformations, have the strongest type of classification, a complete Borel invariant (namely measure-theoretic entropy). But what about classes of transformations that are more random than ergodic transformations, but less so than Bernoulli shifts, such as weakly mixing transformations, mixing transformations, and K-automorphisms?

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