Date/Time : 20 Aug 2025 13:30
Venue : S17 #04-05 (Seminar Room 2)
Speaker : Nguyen Tat Thang, Institute of Mathematics, Vietnam Academy of Science and Technology

Let $f:\R^n\to \R^m$ be a real polynomial mapping, $\Sing(f)$ be the set of singular points of $f$ in $\R^n$, and $K_0(f)=f(\Sing(f))$. A {\it bifurcation set} of $f$ is the smallest set of values in $\R^m$ outside which $f$ is a locally trivial fibration. A regular value $\lambda\in f(\R^n)\setminus K_0(f)$ is called a {\it typical value at $\infty$} of $f$ if there is an open neighborhood over which $f$ is a trivial fibration. Otherwise, $\lambda$ is called an {\it atypical value at $\infty$} of $f$. In this talk, we characterize atypical values at infinity of a real polynomial function of three variables by a certain sum of indices of the gradient vector field of the function restricted to a sphere with a sufficiently large radius. This is an analogy of a result of Coste and de la Puente for real polynomial functions with two variables. We also give a characterization of atypical values at infinity of a real polynomial map whose regular fibers are 2-dimensional surfaces. This is a joint work with Masaharu Ishikawa.

Add to calendar :

Date/Time : 20 Aug 2025 15:00
Venue : S17 #04-06 (Seminar Room 1)
Speaker : Andreas Wieser, Institute for Advanced Study

A classical question in number theory asks: which integers are represented by a given quadratic form? More generally, one can ask when an integral quadratic form q in m variables can be represented by another integral quadratic form Q in n > m variables—that is, whether there exists an integer matrix T such that Q(Tx) = q(x).

Naturally, a necessary condition is that such a representation exists locally, meaning over the real numbers and modulo N for every positive integer N. In the absence of local obstructions, does a (global) representation of q by Q necessarily exist?

A breakthrough result of Ellenberg and Venkatesh in 2008 used deep methods from homogeneous dynamics to prove such a local-global principle when the codimension n – m is at least 5. In this talk, we present joint work with Manfred Einsiedler, Elon Lindenstrauss, and Amir Mohammadi in which we establish a local-global principle in codimension n – m at least 3. The key new ingredient is an effective equidistribution result for periodic orbits of semisimple groups on compact adelic homogeneous spaces.

Add to calendar :

Date/Time : 20 Aug 2025 16:00
Venue : S17 #04-04 (Seminar Room 3)
Speaker : Jia Jia, Tsinghua University

We study the moduli space of higher rank marginally stable pairs (E,s), where E is a torsion free coherent sheaf of rank r on a smooth projective surface and s = (s_1, …, s_r) is a collection of r sections of E. Fixing the Chern character of E, the moduli space is realised as a subscheme of an appropriate Quot–scheme that parametrises quotient sheaves with the corresponding Hilbert polynomial.

We establish a precise link between these moduli spaces and the stable minimal models determined by E and its sections, together with the (analytic) minimal log resolutions of the base surface. Using the birational geometry of such minimal models, we analyse in detail the components of the Hilbert–Chow morphism from the moduli space to the Hilbert scheme of Cartier divisors on the surface. This is a work in progress.

Add to calendar :

Date/Time : 19 Aug 2025 16:00
Venue : S17 #05-11 (Seminar Room 5)
Speaker : York Ye, National University of Singapore

Configuration spaces of flags can be used to describe many algebraic varieties appearing in representation theory, such as double Bruhat cells and surface character varieties. In this talk, I will discuss the relationship between configuration spaces and cluster algebras. Then I will introduce my joint work with Huanchen Bao on the upper cluster structure of Kac-Moody Richardson varieties and explain some of the new techniques.

Add to calendar :

Date/Time : 13 Aug 2025 17:00
Venue : S17 #05-11 (Seminar Room 5)
Speaker : Kenny Gill, La Salle University

A countable structure S is indivisible if it satisfies a pigeonhole principle: for any bounded coloring of any presentation of S, there is a monochromatic induced substructure isomorphic to S. This naturally gives rise to an instance-solution problem Ind(S) which outputs such a substructure given a presentation and coloring. We seek to understand both (1) the computational difficulty of solving Ind(S) for various specific S; and (2) how properties of S can translate to properties of Ind(S) and vice versa. This talk will survey what is known about both (1) and (2), highlighting some of the many open problems in the area. In particular, we propose several notions of uniform embeddability of one structure in another which allow one to obtain desirable properties of Ind(S) for large classes of S. This is in part ongoing joint work with Damir Dzhafarov and Reed Solomon.

Add to calendar :

Date/Time : 13 Aug 2025 16:30
Venue : S17 #04-04 (SR3)
Speaker : Liu Wenjie, Harbin Institute of Technology

In this talk, we introduce a new theoretical framework built upon fractional spaces for optimal error estimates of Chebyshev/Legendre polynomial approximations to functions with limited regularity. It naturally arises from exact representations of orthogonal polynomial expansion coefficients. Here, the essential pieces of the puzzle for the error analysis include (i) fractional integration by parts (under the weakest possible conditions); (ii) generalised Gegenbauer functions of fractional degree (GGF-Fs): a new family of special functions with notable fractional calculus properties, and (iii) asymptotic formulas or uniform upper bounds of GGF-Fs. Finally we introduce the hypothesis of Babuska and Hakula. The talk is based on joint work with Li-Lian Wang, Huiyuan Li, Boying Wu and Ruiyi Xie.

Add to calendar :

Date/Time : 12 Aug 2025 14:00
Venue : S17 #04-05 (Seminar Room 2)
Speaker : Matias Delgadino, University of Texas at Austin

We prove sharp estimates for the mean-field limit of weakly interacting diffusions with repulsive logarithmic interaction in arbitrary dimension. More precisely, we show that the associated partition function is uniformly bounded in the number of particles N for an arbitrary bounded base measure. Combined with the modulated free energy method, this amounts to a logarithmic improvement in N of the current best available closeness estimates in the literature. Our arguments are inspired by and borrow ideas from Nelson’s classical construction of the \Phi^4_2 Euclidean quantum field theory.

Add to calendar :

Date/Time : 08 Aug 2025 10:00
Venue : S17 #04-04 (Seminar Room 3)
Speaker : Timo Sprekeler, Texas A&M University

We begin by discussing the finite element approximation of stationary Fokker–Planck–Kolmogorov (FPK) equations subject to periodic boundary conditions in two settings: one with weakly differentiable coefficients, and one with merely essentially bounded measurable coefficients under a Cordes-type condition. These problems arise as governing equations for the invariant measure in the homogenization of nondivergence-form equations with large drifts. We then suggest and rigorously analyze an approximation scheme for the effective diffusion matrix in both settings, based on the finite element scheme for stationary FPK problems developed in the first part. This talk includes joint work with Endre Süli (University of Oxford) and Zhiwen Zhang (The University of Hong Kong).

Add to calendar :

Date/Time : 30 Jul 2025 14:00
Venue : S17 #05-12 (Seminar Room 4)
Speaker : Sang-hyun Kim, Korea Institute for Advanced Study

Two groups are elementarily equivalent if they have the same sets of true first order group theoretic sentences. We prove that for two real numbers r,s ≥ 1, and for two smooth closed manifolds M and N, the C^r diffeomorphism group of M is elementarily equivalent to the C^s diffeomorphism group of N if and only if r=s and M is diffeomorphic to N. In the case of r=s=0, we can even weaken the hypothesis so that the manifolds are only assumed to be compact, dropping the smoothability or closedness hypotheses. This strengthens (1) Whittaker’s theorem (1963) on the C^0 case, and (2) Takens–Filipkiewicz theorems (1982) on C^p case with an integer p. Joint work with Thomas Koberda (UVa) and Javier de la Nuez-Gonzalez (KIAS)

Add to calendar :