Date/Time : 13 Nov 2025 15:00
Venue : S16 #05-17
Speaker : Myoungjean Bae , Korea Advanced Institute of Science & Technology (KAIST)

In the study of inviscid compressible flows, the Mach number, defined by

M =local speed of a flow/sound speed,

plays a significant role in that not only it determines a physical feature of a flow, but also it determines a mathematical feature of the governing equations. Especially when $M=1$, it is directly related to the degeneracy of a component in the compressible Euler system (or the Euler-Poisson system). The degeneracy brings a great challenge to establish the well-posedness of a boundary value problem for a transonic flow containing a sonic interface.

In this talk, I present two examples of sonic interfaces:
1. a sonic arc occurring in a self-similar flow of the compressible Euler system with a pseudo-transonic transition;
2. a sonic interface occurring in an accelerated transonic flow of compressible Euler-Poisson system.
The most interesting aspect is that the first result shows weak discontinuity across which all the flow variable are continuous but their derivatives are not, while the second result shows regular interface across which all the flow variable are $C^1$. And, it leads us to the following question: What is a general criterion to determine the type of a sonic interface?

This talk is based on collaborations with G.-Q. Chen, B. Duan, M. Feldman, and C. Xie.

Add to calendar :

Date/Time : 12 Nov 2025 15:00
Venue : S17 #04-06 (Seminar Room 1)
Speaker : Hong Wang, Courant Institute of Mathematical Sciences

A Kakeya set is a compact subset of R^n that contains a unit line segment pointing in every direction. Kakeya set conjecture asserts that every Kakeya set has Minkowski and Hausdorff dimension n. We prove this conjecture in R^3 as a consequence of a more general statement about union of tubes.
This is joint work with Josh Zahl.

Add to calendar :

Date/Time : 12 Nov 2025 17:00
Venue : S17 #05-11 (Seminar Room 5)
Speaker : Daniel Mourad, Nanjing University

We characterize the continuous-Weihrauch-least problem that is discontinuous on a countable set of Cantor-Bendixson (CB) rank alpha. Consider the following conceptualization of CB rank: Suppose that X is a countable subset of Baire space and that x is an element of X which you want to write down. Call x your target. Suppose that while writing down x via increasing initial segments you decide to switch your target to a different element of X that is compatible with what you have already written. The Cantor-Bendixson rank of x in X is an ordinal alpha keeping track of how many times you may switch your target after committing to arbitrarily long initial segments in between each target switch. Applying this conceptualization to Weihrauch problems leads to the intuition that a multifunction P being discontinuous on a set of Cantor-Bendixson rank alpha tells us that, when showing P is discontinuous, we can diagonalizing against the continuous functions in steps counting down from alpha. We formalize this notion and prove the following result.

The problem of all-or-co-unique choice on the natural numbers (ACC_N) can be thought of as the problem of diagonalizing a single time. Building on work of Pauly and Soldà, we show that a specific notion of solving ACC_N within alpha many attempts is the continuous-Weihrauch-least problem discontinuous on a countable subset X of its domain such that all points in X have CB rank at most alpha.

Add to calendar :

Date/Time : 11 Nov 2025 13:00
Venue : S17 #06-11 (Seminar Room 6)
Speaker : Amir Mohammadi, University of California, Berkeley

There is a profound connection between homogeneous dynamics and number theory, particularly in the study of Diophantine approximation. A landmark example is Margulis’s resolution of the Oppenheim conjecture in the mid-1980s using tools from dynamics, followed by further strengthenings by Eskin, Margulis, and Mozes based on Ratner’s seminal work. In this talk, we will present effective results in this direction, with an emphasis on obtaining polynomial rates. This is based on joint works with Elon Lindenstrauss, Zhiren Wang, and Lei Yang.

Add to calendar :

Date/Time : 10 Nov 2025 15:00
Venue : S17 #04-05 (Seminar Room 2)
Speaker : Jhan-Cyuan Syu, National University of Singapore

Let G be a split connected reductive group over a non-archimedean local field. The categorical local Langlands conjecture (CLLC) predicts an equivalence between lisse-étale sheaves on Bun_G, the moduli stack of G-bundles on relative Fargues-Fontaine curves, and ind-coherent sheaves on Par_G, the moduli stack of l-adically continuous L-parameters. One question is to find t-structures on the automorphic side and the spectral side such that the categorical equivalence is t-exact. On the automorphic side, there is a perverse t-structure. However, the story is still mysterious on the spectral side. After localizing at certain types of L-parameters, we can say more about t-structures. In this talk, we will localize at the trivial L-parameter and compute the multiplicity of certain perverse sheaves with the help of CLLC.

Add to calendar :

Date/Time : 05 Nov 2025 15:00
Venue : S17 #06-11 (Seminar Room 6)
Speaker : Ander Lamaison, Institute for Basic Science, South Korea

We study Turán problems for hypergraphs with an additional uniformity condition on the edge distribution. This kind of Turán problems was introduced by Erdős and Sós in the 1980s but it took more than 30 years until the first non-trivial exact results were obtained. Central to the study of the uniform Turán density of hypergraphs are palette constructions, which were implicitly introduced by Rödl in the 1980s. We prove that palette constructions always yield tight lower bounds, unconditionally confirming present empirical evidence. This results in new and simpler approaches to determining uniform turán densities, which completely bypass the use of the hypergraph regularity method.

Add to calendar :

Date/Time : 05 Nov 2025 17:00
Venue : S17 #05-11 (Seminar Room 5)
Speaker : Linus Richter, National University of Singapore

I prove two theorems which classify the complexity of certain sets of Borel codes. These are especially useful in the recursive construction of sets of reals in which Borel codes take on the role of “requirements”; our theorems show that certain classical constructions of sets of reals cannot be straightforwardly effectivised.

Add to calendar :

Date/Time : 03 Nov 2025 15:00
Venue : S17 #04-05 (Seminar Room 2)
Speaker : Joao Lourenco, Universite Sorbonne Paris Nord

Given a reductive group $G$ over a $p$-adic field $F$ with parahoric group model $\mathcal{G}$ over the ring of integers $O$, we can consider its associated flag variety using Witt vectors on perfect $k$-algebras: this is representable by a perfect ind-scheme by a deep theorem of Bhatt-Scholze. This time, we want to study its $\mathbb{F}_p$-étale cohomology, but its behavior is linked with coherent cohomology due to the Artin-Schreier sequence. We are going to explain how the Cohen-Macaulayness of a perfect scheme relates to its perverse $\mathbb{F}_p$-sheaves. Then, we will explain a new and uniform proof of this property for Schubert varieties in the minuscule range based on inversion of adjunction. This is joint work with Robert Cass.

Add to calendar :

Date/Time : 31 Oct 2025 15:00
Venue : S16 #04-35
Speaker : Vincenzo Vespri, University of Florence

We first consider Harnack-type inequalities for solutions to equations involving the doubly nonlinear operator, a natural bridge between the porous medium and the p-Laplacian operators. This class of degenerate and singular equations exhibits a rich interplay between diffusion and nonlinearity, leading to delicate regularity properties. The development of intrinsic scaling techniques has been essential in establishing local bounds and continuity estimates. After discussing the isotropic case, we turn to anisotropic operators, where diffusion acts differently along distinct spatial directions. This emerging field is still at the beginning of systematic investigation, posing numerous open questions concerning the geometry of intrinsic metrics, Harnack inequalities for general operators, and the role of directional degeneracies in regularity theory.

Add to calendar :