[ Abstract ]
This talk starts from a simple question: how can one define a geometric mean for sparse positive definite matrices without destroying their zero pattern?
For the arrowhead pattern, this leads naturally to the five-dimensional Vinberg cone, a basic non-symmetric homogeneous cone.
I will present two intrinsic means on this cone: a Cholesky-Vinberg mean built from triangular Cholesky factors, and a logarithmic Vinberg mean built from global clan (Vinberg algebra) coordinates.
The first is tied to a flat affine geometry with torsion, while the second belongs to a torsion-free flat geometry.
I will also explain why these means differ from the classical Riemannian midpoint and why this difference is a genuinely non-symmetric phenomenon.
If time permits, I will give an application to quantum information theory.

[ Abstract ]
This talk starts from a simple question: how can one define a geometric mean for sparse positive definite matrices without destroying their zero pattern?
For the arrowhead pattern, this leads naturally to the five-dimensional Vinberg cone, a basic non-symmetric homogeneous cone.
I will present two intrinsic means on this cone: a Cholesky-Vinberg mean built from triangular Cholesky factors, and a logarithmic Vinberg mean built from global clan (Vinberg algebra) coordinates.
The first is tied to a flat affine geometry with torsion, while the second belongs to a torsion-free flat geometry.
I will also explain why these means differ from the classical Riemannian midpoint and why this difference is a genuinely non-symmetric phenomenon.
If time permits, I will give an application to quantum information theory.

[ Abstract ]
The vortex equation is a second-order PDE on a Riemann surface, defined in terms of a triple consisting of a holomorphic line bundle, a section, and a Hermitian metric. Its solutions are closely related to Hermitian–Einstein metrics and to geometric structures such as metrics with conical singularities. In https://arxiv.org/abs/1612.06710, Manton introduced several generalizations of the vortex equation, leading to five distinct types of vortex equations, which we refer to as Manton’s exotic vortex equations. In this talk, I will introduce these equations and discuss the existence of their solutions. I will also explain how these solutions can be obtained via dimensional reduction of a solution of Hermitian–Einstein equation.

[ Abstract ]
Directed polymers can be described as a tilting of the simple random walk, where some local random noise can attract or repel the trajectory of the walk. In the subcritical regime of the two-dimensional model, the partition function is known to be asymptotically approximated by a Gaussian log-correlated field. In a work in collaboration with Shuta Nakajima and Ofer Zeitouni, we could refine this result by proving that the maximum of the partition function field converges to that of a branching Brownian motion, which is the source of the log-correlation. In this talk, I will introduce the model as well as the objects related to it and present our result.

[ Abstract ]
We establish asymptotically Gaussian fluctuations for functionals of a large class of spin models and strongly correlated random point fields, achieving near-optimal rates.
For spin models, we demonstrate Gaussian asymptotics for the magnetization (i.e., the total spin) for a wide class of ferromagnetic spin systems on Euclidean lattices, in particular those with continuous spins. Specific applications include, in particular, the celebrated XY and Heisenberg models under ferromagnetic conditions, and more broadly, systems with very general rotationally invariant spins in arbitrary dimensions. We address both the setting of free boundary conditions and a large class of ferromagnetic boundary conditions, and our CLTs are endowed with near-optimal rate of O(log |Λ| · |Λ|−1/2) in the Kolmogorov-Smirnov distance, where the system size is |Λ|. Our approach leverages the classical Lee-Yang theory for the zeros of partition functions, and subsumes as a special case results of Lebowitz, Ruelle, Pittel and Speer on CLTs in discrete statistical mechanical models for which we obtain sharper convergence rates.
In a different direction, we obtain CLTs for linear statistics of a wide class of point processes known as α-determinantal point processes which interpolate between negatively and positively associated random point fields (including the usual determinantal, permanental and Poisson point processes).
We contribute a unified approach to CLTs in such models (agnostic to the parameter α that modulates the nature of association). Our methods are able to address a broad class of kernels including in particular those with slow spatial decay (such as the Bessel kernel in general dimensions). Significantly, our approach is able to analyse such processes in dimensions ≥ 3, where structural alternatives such as connections to random matrix theory are not available, and obtain explicit rates for fast convergence in a wide spectrum of models.
A key ingredient of our approach is a broad, quantitative extension of the classical Marcinkiewicz Theorem that holds under the limited condition that the characteristic function is non-vanishing only on a bounded disk. This technique complements classical work of Ostrovskii, Linnik, Zimogljad and others, as well as recent work of Michelen and Sahasrabudhe, and Eremenko and Fryntov. In spite of the general applicability of the results, including to heavy-tailed setups, our rates for the CLT match the classic Berry- Esseen bounds for independent sums up to a log factor.
Based on joint work with T.C. Dinh, H.S. Tran and M.H. Tran. Under revision at Annals of Applied Probability.

[ Abstract ]
The aim of this talk is to explore a possible weakening of the Birch–Swinnerton-Dyer conjecture, framed as a dichotomy, in which neither the equality of the analytic and Mordell–Weil ranks nor the finiteness of the Tate–Shafarevich group is assumed to hold individually, but rather that if one fails, then so does the other – and in a very specific way. We will explain our motivations coming from (1) the analogy with Iwasawa theory, (2) connections with known results onelliptic curves, and (3) comparison with the function field case.

This talk is based on joint work with Don Zagier (MPIM Bonn).

[ Abstract ]
Wall's stabilization principle suggests that exotic phenomena in dimension four in the orientable category disappear after taking connected sums with sufficiently many S²xS². Since most known exotic pairs of closed 4-manifolds become diffeomorphic after one stabilization, a natural question was: is a single S²xS² enough? Recently, Jianfeng Lin constructed an exotic diffeomorphism on a closed 4-manifold-a diffeomorphism topologically isotopic to the identity but not smoothly isotopic-that survives one stabilization. In this talk, we provide a relative exotic diffeomorphism on a compact contractible 4-manifold that survives two stabilizations. This gives the first exotic phenomenon in the orientable category that survives two stabilizations. This is joint work with Sungkyung Kang and Junghwan Park.

[ Abstract ]
Let $M$ be a compact complex manifold, $L$ be a positive holomorphic line bundle over $M$, and $E$ be a holomorphic vector bundle over $M$. It is known that the cohomology groups $H^i(M, L^k \otimes E)$ vanish for $i > 0$ when $k$ is sufficiently large. This vanishing theorem is typically proved by solving the $\overline{\partial}$-equation using H\"ormander’s $L^2$-estimates in the complex geometry. In this talk, we solve the $\overline{\partial}$-equation not by H\"ormander’s method, but by means of weighted integral formulas. In particular, we apply the weighted integral formula of Andersson--Berndtsson (1982) in a semi-classical setting and obtain $C^{\ell}$-norm estimates for solutions of the $\overline{\partial}$-equation.

[ Abstract ]
We will discuss the flat flow solution for the surface diffusion equation via a discrete minimizing movements scheme proposed in 1994 in a celebrated paper by J.W. Cahn and J.E. Taylor. We will show that in dimension three the scheme converges to the unique smooth solution of the equation, provided the initial set is sufficiently regular. Joint paper with Marco Cicalese, Vesa Julin and Andrea Kubin.

[ Abstract ]
The unit cotangent bundle of the Euclidean space R³ has a canonical contact structure. In this talk, we discuss loops of Legendrian tori in this 5-dimensional contact manifold. In particular, we focus on loops arising as families of the unit conormal bundles of knots in R³, and I will explain a topological method to compute the monodromy on the Legendrian contact homology in degree 0 induced by those loops. As an application, we get examples of non-contractible loops of Legendrian tori which are contractible in the space of smoothly embedded tori. This is joint work with Marián Poppr.

[ Abstract ]
よく知られているように、4次元ブラウン運動の軌跡は単純曲線になる。一方で4次元単純ランダムウォークの軌跡は、ループ除去ランダムウォークの長さに非自明な対数項が現れる程度のループを持つ。本講演では、4次元単純ランダムウォークの軌跡の交叉とある長距離パーコレーションモデルとの関係性について解説する。

[ Abstract ]
The quasi-projective groups are the fundamental groups of smooth quasi-projective complex varieties. Aguilar and Campana provided the problem about the torsion-free nilpotent quasi-projective groups. The problem asks whether such groups are 2-step nilpotent or abelian groups(arXiv:2301.11232,Question26). In this talk, I introduce some my result related to the torsion-free nilpotent quasi-projective groups and the above question. In particular, the latest result suggests the existence of torsion-free nilpotent quasi-projective groups with three or more steps.

[ Abstract ]
Our ultimate goal is to discretize Seiberg-Witten theory.
Considering PL = DIFF in dimension four, we would like to construct something like PL Seiberg-Witten theory.
As a first step towards this goal, we study the discretization of the analytic index of Dirac operators.
However, the analytic index of Fredholm operators is an essentially infinite-dimensional phenomenon, while the index theory of finite-dimensional self-adjoint operators is trivial.
Thus, a naive discretization of Dirac operators does not work.
In this talk, I will explain how the “Wilson-Dirac operator” from lattice gauge theory provides a correct discretization, at least from the viewpoint of the analytic index.
This talk is based on a joint work with Shoto Aoki, Hidenori Fukaya, Mikio Furuta, Tetsuya Oonogi, and Satoshi Yamaguchi.
https://arxiv.org/abs/2602.12576
https://arxiv.org/abs/2407.17708

[ Abstract ]
A cut-and-paste operation along an embedded real projective plane in a 4-manifold is called a Price twist. A Price twist on the 4-sphere produces, up to diffeomorphism, at most three 4-manifolds: the 4-sphere itself, a homotopy 4-sphere, and a non-simply connected closed 4-manifold. In general, the classification of diffeomorphism types of non-simply connected closed 4-manifolds is still far from being well understood. In this talk, we focus on Price twists on the 4-sphere associated with embeddings of the real projective plane of Kinoshita type that yield non-simply connected 4-manifolds. We present several properties of these manifolds and results on the classification of their diffeomorphism types. We also explain pochette surgery, introduced by Zjuñici Iwase and Yukio Matsumoto, which is closely related to the results of this work. This talk is based on joint work with Tsukasa Isoshima (Keio University).

[ Abstract ]
My studies in probability and statistics have been guided by numerous coincidences, even though my direction has at times been seemingly random. Aware of the bias inherent in my selection of topics and retrospective evaluation of events, I reflect on the trajectory that has brought me to the present.

[ Abstract ]
The advent of artificial intelligence raises an important question: can AI assist mathematicians in solving open problems in mathematics? This talk explores this question from multiple perspectives. We will explore how different types of AI models can be trained to provide valuable insights into mathematical questions from different areas of mathematics and applied mathematics. We will also present recent works on AI models specifically designed for automated theorem proving.

[ Abstract ]
Control theory consists in asking: if we can act on a system, what can we make it do? One of the main problems is the stabilization problem: how can we act on a system to guarantee the long-term behavior of its solutions? In this presentation, we will examine this problem from several angles. First, we will look at the problem of stabilizing PDEs from an abstract perspective and present a recent approach called F-equivalence (or sometimes Fredholm backstepping). The principle is simple: instead of trying to find a control that makes the system stable, we look at another problem: we try find a control that renders the PDE system dynamically equivalent to a simpler system for which stability is already known. Besides being interesting in itself, this approach has also resulted in new results in control theory, and we will review the progress that has been made in the last three years. In a second part, we will focus on a more concrete problem: the stabilization of hyperbolic equations modeling road traffic. We will show how abstract mathematical concepts, such as entropic solutions, can have tangible impacts in real-world scenarios, and we will discuss their application to traffic regulation and the reduction of traffic jams.

[ Abstract ]
Subject of my presentation is a novel approach for optimizing both the macroscopic shape and the porous mesoscopic structure of components. In the first part of my presentation I will introduce the concept of phasefield based topology optimization. The second part of my presentation is devoted to two-scale topology optimization. The key feature here is the introduction of an additional local volume control (LVC), which allows to adjust the desired spatial scales. The main novelty is that the radius of the LVC may depend both on space and a local stress measure. This allows for creating optimal topologies with heterogeneous mesostructures enforcing any desired spatial grading and accommodating stress concentrations by stress dependent pore size. I will present some analytical results for the resulting optimal control problem and conclude with numerical results showing the versatility of our approach for creating optimal macroscopic designs with tailored mesostructures.

[ Abstract ]
The presentation consists of two parts. In the first part, we review -- from a novice point of view -- the categorical local Langlands correspondence due to Fargues and Scholze. Topics include: the structure of Bun_G and LocSys_{\hat G}, spectral action via Hecke operators, geometric Satake transform, and some conjectural consequences proposed by Fargues. (Apologies: the p-adic geometry underlying the relative Fargues-Fontaine curve is not included.)
In the second part, I will present a conjectural construction of Arthur packets in Fargues-Scholze's framework. This construction is based on the vanishing cycle functor introduced by Cunningham-Fiori-Moussaoui-Mracek-Xu, which is in turn inspired by Adams-Barbasch-Vogan for real groups. (A confession for curious audiences: this presentation offers essentially no new results. My goal is to illustrate how the legacy of James Arthur may influence other theories.)