Date/Time : 19 Nov 2025 16:00
Venue : S17 #05-11 (Seminar Room 5)
Speaker : Srecko Durasinovic, Nanyang Technological University

Mixture models, such as Gaussian mixture models (GMMs), are widely used in machine learning to represent complex data distributions. A key challenge, especially in high-dimensional settings, is to determine the mixture order and estimate the mixture parameters. We study the problem of approximating a target measure, available only through finitely many of its moments, by a mixture of distributions from a parametric family (e.g., Gaussian, exponential, Poisson), with approximation quality measured by the 2-Wasserstein ($\was$) or the total variation ($\tv$) distance. Unlike many existing approaches, the parameter set is not assumed to be finite; it is modeled as a compact basic semi-algebraic set. We introduce a hierarchy of semidefinite relaxations with asymptotic convergence to the desired optimal value. In addition, when a certain rank condition is satisfied, the convergence is even finite and recovery of an optimal mixing measure is obtained. We also present an application to clustering, where our framework serves either as a stand-alone method or as a preprocessing step that yields both the number of clusters and strong initial parameter estimates, thereby accelerating convergence of standard (local) clustering algorithms.

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Date/Time : 19 Nov 2025 16:00
Venue : S17 #04-04 (Seminar Room 3)
Speaker : Arpan Kabiraj, Indian Institute of Technology Palakkad

In 80’s, Goldman introduced a Lie bracket on the linear span of the free homotopy classes of directed closed curves on a surface called the Goldman Lie bracket. In this talk we will discuss some properties of Goldman Lie bracket using hyperbolic geometry. We will use these properties to give characterizations of simple closed curves. This is a joint work with Moira Chas.

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Date/Time : 18 Nov 2025 15:00
Venue : S17 #04-05 (Seminar Room 2)
Speaker : Fernando Camacho Cadena, Strasbourg University

A hyperbolic structure on a surface S can be deformed by cutting along a curve c without self—intersections, twisting, and re-gluing. Such a deformation can also be seen as the Hamiltonian flow of the length function of c on the Teichmüller space of S, equipped with the Weil—Petersson symplectic form. In this talk, I will discuss Hamiltonian flows associated to length functions of self—intersecting curves, and their generalizations to higher rank Teichmüller spaces. This is partially joint work with J. Farre and A. Wienhard.

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Date/Time : 17 Nov 2025 15:00
Venue : S17 #04-04 (Seminar Room 3)
Speaker : He Xuhua, University of Hong Kong

Lusztig’s theory of canonical bases reveals a remarkably rigid and positive algebraic structure on quantum groups and their modules. A fundamental result in symmetric types establishes that the structure constants for multiplication in the negative part $U^-$, as well as for the action of Chevalley generators $E_i$ and $F_i$ on a single simple module, all belong to $\mathbb{N}[v, v^{-1}]$.

Lusztig conjectured that this strong positivity holds for the multiplication within the modified quantum group and the action on the tensor product of modules. In this talk, I will present recent joint work with my postdoc, Jiepeng Fang. Our main result establishes that the structure constants for both the multiplication with arbitrary canonical basis elements in $\dot{\mathbf{U}}$ and the action on the canonical basis elements of arbitrary tensor products of simple lowest and highest weight modules by canonical basis elements arising from spherical parabolic subalgebras belong to $\mathbb{N}[v,v^{-1}]$.

A key innovation in our proof is the “thickening philosophy”, an algebraic technique inspired by geometric ideas from total positivity, building on my earlier work with Huanchen Bao. This method embeds a suitable approximation of the tensor product into the negative part $\tilde{U}^-$ of a larger quantum group, constructed via a framed
quiver. This allows us to inherit the desired positivity directly from the well-established positivity of the canonical basis of $\tilde{U}^-$. This approach demonstrates how wild Kac-Moody groups can provide a powerful framework for elucidating the representation theory of quantum groups beyond finite and affine types.

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Date/Time : 17 Nov 2025 14:00
Venue : S17 #04-05 (Seminar Room 2)
Speaker : Seonwoo Kim, Yonsei University

The main purpose of this lecture course is to introduce the recently-developed resolvent approach [Landim-Marcondes-Seo, JEMS ’25] to metastability and beyond. First, I will review the historical developments in the treatment of the metastability phenomenon and summarize the pros/cons of the previous methods. Next, I will state and prove the resolvent approach in a simple setting, focusing especially on its strength, along with some real-model applications. Finally, I will summarize my recent work with Jungkyoung Lee (KIAS) to extend and generalize this approach to non-compact metastable transitions.

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Date/Time : 17 Nov 2025 16:00
Venue : S17 #04-04 (Seminar Room 3)
Speaker : Wang Weiqiang, University of Virginia

As a degeneration of affine quantum groups, Yangians quantize current algebras. It is known that truncated shifted Yangians quantize affine Grassmannian slices of type ADE; the latter are Coulomb branches of framed quiver gauge theories, and in type A, they can be identified with nilpotent Slodowy slices. As a degeneration of affine iquantum groups, twisted Yangians quantize twisted current algebras. We explain the Poisson geometry of [truncated shifted] twisted Yangians in terms of fixed point loci of affine Grassmannian slices by Poisson involutions; in split type A, the latter are identified will nipotent Slodowy slices of type BCD. This is joint work with Kang Lu and Alex Weekes.

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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.

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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.

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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.

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