[ Abstract ]
Colding-Minicozzi entropy is a natural quantity associated to mean curvature flow which measures complexity of submanifolds of Euclidean space. We discuss some (nearly) optimal relationships between entropy and areas of (minimal) submanifolds of the sphere.

[ Abstract ]
Parabolic partial differential equations tend to improve regularity. Generally one can control strong norms (e.g. $C^k$) of a solution at time $t$ principally in terms of $t$ and a weak norm of the initial data. Curve shortening flow is a geometric flow for which the story is more weird and wonderful. I will explain some recent works with Arjun Sobnack from 2026 and before where this is manifest. I expect to be able to make the talk accessible to a relatively broad audience.

[ Abstract ]
Algom-Kfir and Bestvina introduced the mapping class groups of locally finite, infinite graphs in 2021. Since Out(Fn) can be realized as the mapping group of a finite graph, their construction may be viewed as a "big" version of Out(Fn). In this talk, we survey the algebraic and coarse geometric properties of these groups and discuss a relationship with mapping class groups of infinite-type surfaces ("big mapping class groups"). This talk is based on joint work with Ryan Dickmann, George Domat, and Hannah Hoganson, in various collaborations.

[ Abstract ]
We show that the cohomology of the structure sheaf of smooth and proper varieties over a complete non-archimedean field K with the ring R of integers of characteristic zero, can be refined to a birational cohomology theory of smooth (not necessarily proper) varieties over K with values in R-lattices, and the same holds for K of positive characteristic in dimensions at most 3.
As one application, we obtain that the automorphism group of the function field of a proper smooth variety X of dimension at most 3 over any field of positive characteristic acts quasi-unipotently on the cohomology of the canonical sheaf of X. The proof relies on some results from rigid analytic geometry on the cohomology of twisted integral rigid structure sheaves due to Bartenwerfer and van der Put.
This is a joint work with A. Merici and Kay Ruelling.

[ Abstract ]
Polylogarithms are special functions with many remarkable properties, notably their functional identities. The most interesting identities of this kind involve several variables and are known only in low weights. In weights 1 and 2, there is essentially one fundamental identity in each case: Cauchy’s equation for the logarithm and Abel’s five-term identity for the dilogarithm.
After introducing the subject, I will present natural generalizations, up to weight 6, of Cauchy’s and Abel’s identities. The new identities are no longer merely polylogarithmic, but hyperlogarithmic, and they arise naturally from the geometry of del Pezzo surfaces.
In the second part of the talk, I will discuss a generalization of an approach due to Gelfand and MacPherson in the weight 2 case, leading to a more canonical viewpoint on these hyperlogarithmic functional equations.
The first part of the talk is based on joint work with Ana-Maria Castravet.

[ Abstract ]
In this talk, I will explain the main results of my recent paper (arXiv:2602.17435).

Khovanov homology is a categorification of the Jones polynomial, introduced by M. Khovanov. A persistent theme in the subject is that adding extra structures on Khovanov homology strengthens the invariant, and often detects phenomena invisible at the level of polynomials or bigraded vector spaces.

Motivated by the y-ification of HOMFLY--PT homology by Gorsky and Hogancamp, and the sl₂-action constructed by Gorsky, Hogancamp and Mellit, we construct a y-ification of Khovanov homology and define an action of the element e in sl₂ on these y-ifications. Our construction is compatible with the previous ones via Rasmussen's spectral sequence from HOMFLY--PT homology to Khovanov homology; yet our approach is more elementary and suited to diagrammatic and algorithmic computations. As an application, we show that the additional structure can distinguish knots with identical Khovanov homology and identical HOMFLY--PT homology, in particular the Conway knot and the Kinoshita--Terasaka knot.

[ 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).