Date/Time : 28 Jul 2026 15:00
Venue : S17 #04-05 (Seminar Room 2)
Speaker : Zhongkai Tao, Institut des Hautes Études Scientifiques

The initial data of the vacuum Einstein equations on a spacelike hypersurface satisfy the Einstein constraint equations. It is an underdetermined PDE system. We exploit the underdeterminedness to find nontrivial solutions of the Einstein constraint equations with interesting properties. A key ingredient is the construction of Bogovskii-type operators for underdetermined PDEs satisfying a robust symbol condition. The Bogovskii-type operators have favorable support and regularizing properties, allowing us to manipulate the solutions flexibly. This is based on joint work (partly in progress) with Philip Isett (Caltech), Yuchen Mao (UC Berkeley), and Sung-Jin Oh (UC Berkeley)

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Date/Time : 28 Jul 2026 16:00
Venue : S17 #04-05 (Seminar Room 2)
Speaker : Chengyang Shao, Institut des Hautes Études Scientifiques

In this talk, I will present a mathematical proof for the existence of time-periodic oscillating capillary droplet. The major difficulty is that the non-flat underlying geometry makes toric Fourier techniques unavailable, while the quasilinear nature still requires delicate spectral analysis. The analysis starts with paraproduct on compact Lie groups based on invariant Littlewood-Paley decomposition, followed by a toolset of explicit formulas for paradifferential operators. In case of zonal symmetry on the 3-sphere, an effectively computable symbolic calculus is available. It will then be shown how this paradifferential toolset is used to construct time-periodic solution to the spherical capillary water waves equation, giving mathematical justification of Rayleigh’s formula for oscillating capillary droplet. The paradifferential method not only incorporates the underlying geometry and analysis, but also effectively balances the loss of regularity caused by small denominators, allowing a “fixed point” style proof instead of Nash-Moser/KAM iteration. The talk is based on a working paper co-authored with Haocheng Yang.

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Date/Time : 28 Jul 2026 16:00
Venue : S17 #05-11 (Seminar Room 5)
Speaker : Tianqi Wang, Yale University

We introduce notions of ending currents for properly discontinuous isometric actions of hyperbolic groups on Gromov hyperbolic spaces, generalizing Thurston’s ending laminations from the setting of Kleinian surface groups. In contrast with the 3-dimensional picture, we construct an example of geometrically finite representations of a closed surface group into $SO^+(4,1)$ whose parabolicity appears only at a non-simple closed geodesic, and an example of hyperbolic group extensions where different notions of ending currents, which coincide in the 3-dimensional setting, become distinct. This is joint work with Yair Minsky.

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Date/Time : 27 Jul 2026 14:00
Venue : S17 #04-04 (Seminar Room 3)
Speaker : Pavel Turek, Okinawa Institute of Science and Technology

A central open problem of the modular representation theory of symmetric groups (and Hecke algebras) is the computation of decomposition numbers, which describe irreducible constituents of Specht modules in prime characteristic. These numbers are organised in a block matrix, called the decomposition matrix, whose structure remains a mystery. In this talk, we present new whole columns of this matrix labelled by a novel family of partitions called d-balanced partitions; this seems to be the first description of whole columns in each block to date! Depending on the number d, we either get an algebraic interpretation (for d=2, this is joint with David Hemmer) or characteristic-free columns applicable to Hecke algebras (for d>2, this is joint with Bim Gustavsson, Stacey Law, Lorenzo Putignano and Liron Speyer).

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Date/Time : 24 Jul 2026 15:00
Venue : S17 #04-05 (Seminar Room 2)
Speaker : Tsuyoshi Kato, Kyoto University

We associate to each proper discrete metric space X a Lie algebra that acts by locally generated derivations on an infinite tensor product of matrix algebras indexed by the points of X. We compute the homology of this Lie algebra with trivial scalar coefficients when X is the integer lattice in n-dimensional Euclidean space.

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Date/Time : 23 Jul 2026 15:00
Venue : S17 #04-05 (Seminar Room 2)
Speaker : Xuantao Chen, Sorbonne Universite

We revisit the problem of solving the Einstein constraint equations in vacuum by a new method, which allows us to prescribe four scalar quantities, representing the full dynamical degrees of freedom of the constraint system. We show that once appropriate gauge conditions have been chosen and four scalars freely specified (modulo $\ell\leq 1$ modes), we can rewrite the constraint equations as a well-posed system of coupled transport and elliptic equations on two-spheres. Our method can be applied to construct various types of initial data, including data with arbitrarily fast decay, borderline decay, and short-pulse type data. This is joint work with Sergiu Klainerman.

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Date/Time : 15 Jul 2026 14:00
Venue : S17 #04-04 (Seminar Room 3)
Speaker : Luca Makowiec, Leipzig University

In this talk, we consider a special instance of a random spanning tree in random environment on the complete graph. Each edge is given weight infinity with probability 1/|V| and weight one otherwise. We will discuss a result about the approximate order of the diameter of this tree. In particular, we will explain how a critical Erdős–Rényi graph formed by the infinite-weight edges influences the geometry of the spanning tree. Along the way, we will encounter interesting results and proof techniques associated with the birthday problem and their relation to (weighted) uniform spanning trees (USTs).

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Date/Time : 15 Jul 2026 14:00
Venue : S17 #05-11 (Seminar Room 5)
Speaker : Wang Hao, Sichuan University

Solid-state phase transitions typically follow a Minimum Energy Path (MEP). Classical methods such as G-SSNEB compute the MEP in the full high-dimensional configuration space, which becomes prohibitive when paths must be scanned over pressures, defect types, and system sizes. We propose the Preset Structure Strain Path (PSSP) method, which uses feature extraction and transfer learning to compute MEPs efficiently across these conditions. PSSP is based on the fact that in defective systems, lattice strain sets the direction of the MEP, while the local atomistic environment changes little but governs the energy. PSSP fully exploits this separation by presetting the displacement field as a transferable feature and optimizing only the lattice evolution in a low-dimensional space. By doing this, the strain paths learned in one setting can be reused under different pressures, defects and system sizes. The advantage of PSSP is verified that it reduces the energy and force evaluations to less than 2% of G-SSNEB but preserving the accuracy of the energy barrier in a satisfactory level.

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Date/Time : 08 Jul 2026 15:00
Venue : S17 #04-04 (Seminar Room 3)
Speaker : Jean-Bernard Lasserre, LAAS-CNRS

Anomaly detection plays a critical role in identifying unusual patterns across domains such as fraud detection, network intrusion, and system fault diagnosis. Relatively recently, Christoffel Function (CF) based methods have emerged as powerful and easy-to-use tools for support inference and outlier (or anomaly) detection in data mining.  However in their original form, their suffer from a scalability issue even for moderate data dimension. In this talk we describe two ways of addressing this challenge.

In a first direction we briefly describe how to construct a Christoffel-like rational function when $\mu satisfies a correlative sparsity pattern associated with a graphical model. Its computational complexity is parametrized by the tree-width of the associated graph. In a second direction we consider the general cas with no assumption on the underlying measure $\mu$ associated with the data cloud. One introduces the mapping $y\mapsto g_x(y):=norm(x-y)^2$ and the CF associated the univariate pushforward $\nu_x$ of $\mu$ by $g_x$. The score function is now the CF of $\nu_x$ evaluated at $z=0$ (and so must be recomputed when $x$ changes). We describe its remarkable properties for support inference and density approximation, and Its efficiency on a large set of benchmarks.

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