## Seminar information archive

Seminar information archive ～09/18｜Today's seminar 09/19 | Future seminars 09/20～

#### Colloquium

15:30-16:30 Room #大講義室(auditorium) (Graduate School of Math. Sci. Bldg.)

Riemannian manifolds and their limit spaces (JAPANESE)

**Shouhei Honda**(Graduate School of Mathematical Sciences, University of Tokyo)Riemannian manifolds and their limit spaces (JAPANESE)

[ Abstract ]

The Gromov-Hausdorff (GH) distance defines a distance on the set A of all isometry classes of Riemannian manifolds. Gromov established a precompactness result with respect to the GH distance, under assuming a lower bound on Ricci curvature. In particular we are able to discuss limit nonsmooth spaces of Riemannian manifolds with Ricci curvature bounded below. In this talk, we explain recent developments about this topic.

The Gromov-Hausdorff (GH) distance defines a distance on the set A of all isometry classes of Riemannian manifolds. Gromov established a precompactness result with respect to the GH distance, under assuming a lower bound on Ricci curvature. In particular we are able to discuss limit nonsmooth spaces of Riemannian manifolds with Ricci curvature bounded below. In this talk, we explain recent developments about this topic.

### 2024/04/24

#### Numerical Analysis Seminar

16:30-18:00 Room #002 (Graduate School of Math. Sci. Bldg.)

Generalization analysis of neural networks based on Koopman operators

(Japanese)

[ Reference URL ]

https://sites.google.com/g.ecc.u-tokyo.ac.jp/utnas-bulletin-board/

**Yuka Hashimoto**(NTT Network Service Systems Laboratories)Generalization analysis of neural networks based on Koopman operators

(Japanese)

[ Reference URL ]

https://sites.google.com/g.ecc.u-tokyo.ac.jp/utnas-bulletin-board/

#### FJ-LMI Seminar

15:00-16:00 Room #056 (Graduate School of Math. Sci. Bldg.)

Some aspects of Schrödinger models (英語)

https://fj-lmi.cnrs.fr/seminars/

**Laurent Di Menza**(Université de Reims Champagne-Ardenne, CNRS)Some aspects of Schrödinger models (英語)

[ Abstract ]

In this talk, we focus on basic facts about the Schrödinger equation that arises in various physical contexts, from quantum mechanics to gravita-tional systems. This kind of equation has been intensively studied in the literature and many properties are known, either from a qualitative and quantitative point of view. The goal of this presentation is to give basic properties of solutions in different regimes. A particular effort will be paid for the numerical computation of solitons that consist in solutions that propagate with shape invariance.

[ Reference URL ]In this talk, we focus on basic facts about the Schrödinger equation that arises in various physical contexts, from quantum mechanics to gravita-tional systems. This kind of equation has been intensively studied in the literature and many properties are known, either from a qualitative and quantitative point of view. The goal of this presentation is to give basic properties of solutions in different regimes. A particular effort will be paid for the numerical computation of solitons that consist in solutions that propagate with shape invariance.

https://fj-lmi.cnrs.fr/seminars/

#### Discrete mathematical modelling seminar

13:30-15:00 Room #126 (Graduate School of Math. Sci. Bldg.)

Dynamical degrees of birational maps from indices of polynomials with respect to blow-ups (English)

**Jaume Alonso**(Technische Universität Berlin)Dynamical degrees of birational maps from indices of polynomials with respect to blow-ups (English)

[ Abstract ]

In this talk we propose a new method for the exact computation of the degree $\deg (f^n)$ of the iterates of a birational map $f:\mathbb{P}^n \dashrightarrow \mathbb{P}^n$. The method is based on two main ingredients. Firstly, the factorisation of a polynomial under the pull-back by $f$, based on local indices of a polynomial associated to blow-ups used to resolve the singularity. Secondly, the propagation of these indices along the orbits of $f$. We will illustrate the method in different examples, showing its flexibility, since it does not require the construction of an algebraically stable lift of $f$, unlike other methods based on the Picard group.

This is a joint work with Yuri Suris and Kangning Wei.

In this talk we propose a new method for the exact computation of the degree $\deg (f^n)$ of the iterates of a birational map $f:\mathbb{P}^n \dashrightarrow \mathbb{P}^n$. The method is based on two main ingredients. Firstly, the factorisation of a polynomial under the pull-back by $f$, based on local indices of a polynomial associated to blow-ups used to resolve the singularity. Secondly, the propagation of these indices along the orbits of $f$. We will illustrate the method in different examples, showing its flexibility, since it does not require the construction of an algebraically stable lift of $f$, unlike other methods based on the Picard group.

This is a joint work with Yuri Suris and Kangning Wei.

### 2024/04/23

#### Tuesday Seminar on Topology

17:00-18:30 Room #ハイブリッド開催/056 (Graduate School of Math. Sci. Bldg.)

Pre-registration required. See our seminar webpage.

Pochette surgery on 4-manifolds and the Ozsváth--Szabó $d$-invariants of Brieskorn homology 3-spheres (JAPANESE)

https://park.itc.u-tokyo.ac.jp/MSF/topology/TuesdaySeminar/index_e.html

Pre-registration required. See our seminar webpage.

**Tatsumasa Suzuki**(Meiji University)Pochette surgery on 4-manifolds and the Ozsváth--Szabó $d$-invariants of Brieskorn homology 3-spheres (JAPANESE)

[ Abstract ]

This talk consists of the following two research contents:

I. The boundary sum of $S^1 \times D^3$ and $D^2 \times S^2$ is called a pochette. The pochette surgery, which is a generalization of Gluck surgery and a special case of torus surgery, was discovered by Zjuñici Iwase and Yukio Matsumoto in 2004. For a pochette $P$ embedded in a 4-manifold $X$, a pochette surgery on $X$ is the operation of removing the interior of $P$ and gluing $P$ by a diffeomorphism of the boundary of $P$. In this talk, we focus on the fact that pochette surgery is a surgery with a cord and the 2-sphere $S^2$, and attempt to classify the diffeomorphism type of pochette surgery on the 4-sphere $S^4$.

II. In 2003, Peter Ozsváth and Zoltán Szabó introduced a homology cobordism invariant for homology 3-spheres called a $d$-invariant. In this talk, we present new computable examples by refining the Karakurt--Şavk formula for any Brieskorn homology 3-sphere $\Sigma(p,q,r)$ with $p$ is odd and $pq+pr-qr=1$. Furthermore, by refining the Can--Karakurt formula for the $d$-invariant of any $\Sigma(p,q,r)$, we also introduce the relationship with the $d$-invariant of $\Sigma(p,q,r)$ and those of lens spaces.

This talk includes contents of joint work with Motoo Tange (University of Tsukuba).

[ Reference URL ]This talk consists of the following two research contents:

I. The boundary sum of $S^1 \times D^3$ and $D^2 \times S^2$ is called a pochette. The pochette surgery, which is a generalization of Gluck surgery and a special case of torus surgery, was discovered by Zjuñici Iwase and Yukio Matsumoto in 2004. For a pochette $P$ embedded in a 4-manifold $X$, a pochette surgery on $X$ is the operation of removing the interior of $P$ and gluing $P$ by a diffeomorphism of the boundary of $P$. In this talk, we focus on the fact that pochette surgery is a surgery with a cord and the 2-sphere $S^2$, and attempt to classify the diffeomorphism type of pochette surgery on the 4-sphere $S^4$.

II. In 2003, Peter Ozsváth and Zoltán Szabó introduced a homology cobordism invariant for homology 3-spheres called a $d$-invariant. In this talk, we present new computable examples by refining the Karakurt--Şavk formula for any Brieskorn homology 3-sphere $\Sigma(p,q,r)$ with $p$ is odd and $pq+pr-qr=1$. Furthermore, by refining the Can--Karakurt formula for the $d$-invariant of any $\Sigma(p,q,r)$, we also introduce the relationship with the $d$-invariant of $\Sigma(p,q,r)$ and those of lens spaces.

This talk includes contents of joint work with Motoo Tange (University of Tsukuba).

https://park.itc.u-tokyo.ac.jp/MSF/topology/TuesdaySeminar/index_e.html

### 2024/04/22

#### Seminar on Geometric Complex Analysis

10:30-12:00 Room #128 (Graduate School of Math. Sci. Bldg.)

Neighborhood of a compact curve whose intersection matrix has a positive eigenvalue (Japanese)

[ Reference URL ]

https://forms.gle/gTP8qNZwPyQyxjTj8

**Takayuki Koike**(Osaka Metropolitan Univ.)Neighborhood of a compact curve whose intersection matrix has a positive eigenvalue (Japanese)

[ Reference URL ]

https://forms.gle/gTP8qNZwPyQyxjTj8

### 2024/04/17

#### Number Theory Seminar

17:00-18:00 Room #117 (Graduate School of Math. Sci. Bldg.)

Functoriality of the p-adic Simpson correspondence by proper push forward (English)

**Ahmed Abbes**(Institut des Hautes Études Scientifiques, University of Tokyo)Functoriality of the p-adic Simpson correspondence by proper push forward (English)

[ Abstract ]

Faltings initiated in 2005 a p-adic analogue of the (complex) Simpson correspondence whose construction has been taken up by various authors, according to several approaches. After recalling the one initiated by myself with Michel Gros, I will present our initial result on the functoriality of the p-adic Simpson correspondence by proper push forward, leading to a generalization of the relative Hodge-Tate spectral sequence. If time permits, I will give a brief overview of an ongoing project with Michel Gros and Takeshi Tsuji, aimed at establishing a more robust framework for achieving broader functoriality results of the p-adic Simpson correspondence, by both proper push forward and pullback.

Faltings initiated in 2005 a p-adic analogue of the (complex) Simpson correspondence whose construction has been taken up by various authors, according to several approaches. After recalling the one initiated by myself with Michel Gros, I will present our initial result on the functoriality of the p-adic Simpson correspondence by proper push forward, leading to a generalization of the relative Hodge-Tate spectral sequence. If time permits, I will give a brief overview of an ongoing project with Michel Gros and Takeshi Tsuji, aimed at establishing a more robust framework for achieving broader functoriality results of the p-adic Simpson correspondence, by both proper push forward and pullback.

#### Discrete mathematical modelling seminar

13:30-15:00 Room #126 (Graduate School of Math. Sci. Bldg.)

Discrete Painlevé equations and pencils of quadrics in 3D (English)

**Jaume Alonso**(Technische Universität Berlin)Discrete Painlevé equations and pencils of quadrics in 3D (English)

[ Abstract ]

In this talk we propose a new geometric interpretation of discrete Painlevé equations. From this point of view, the equations are birational transformations of $\mathbb{P}^3$ that preserve a pencil of quadrics and map each quadric of the pencil to a different one, according to a Möbius transformation of the pencil parameter. This allows for a classification of discrete Painlevé equations based on the classification of pencils of quadrics in $\mathbb{P}^3$. In this scheme, discrete Painlevé equations are obtained as deformations of the 3D QRT maps introduced in the previous talk, which consist of the composition of two involutions along the generators of the quadrics of a pencil of quadrics until they meet a second pencil. The deformation is then a birational (often linear) transformation in $\mathbb{P}^3$ under which the pencil remains invariant, but the individual quadrics do not.

This is a joint work with Yuri Suris and Kangning Wei.

In this talk we propose a new geometric interpretation of discrete Painlevé equations. From this point of view, the equations are birational transformations of $\mathbb{P}^3$ that preserve a pencil of quadrics and map each quadric of the pencil to a different one, according to a Möbius transformation of the pencil parameter. This allows for a classification of discrete Painlevé equations based on the classification of pencils of quadrics in $\mathbb{P}^3$. In this scheme, discrete Painlevé equations are obtained as deformations of the 3D QRT maps introduced in the previous talk, which consist of the composition of two involutions along the generators of the quadrics of a pencil of quadrics until they meet a second pencil. The deformation is then a birational (often linear) transformation in $\mathbb{P}^3$ under which the pencil remains invariant, but the individual quadrics do not.

This is a joint work with Yuri Suris and Kangning Wei.

### 2024/04/16

#### Tuesday Seminar on Topology

17:00-18:00 Online

Pre-registration required. See our seminar webpage.

Skein algebras and quantum tori in view of pants decompositions (JAPANESE)

https://park.itc.u-tokyo.ac.jp/MSF/topology/TuesdaySeminar/index_e.html

Pre-registration required. See our seminar webpage.

**Hiroaki Karuo**(Gakushuin University)Skein algebras and quantum tori in view of pants decompositions (JAPANESE)

[ Abstract ]

To understand the algebraic structures of skein algebras and their generalizations, we usually try to embed these algebras into quantum tori using ideal triangulations of a surface and the splitting map. However, such a construction does not work for the skein algebras of closed surfaces and the Roger--Yang skein algebras of punctured surfaces.

In the talk, we define filtrations on these algebras using pants decompositions and embed the associated graded algebras into quantum tori. As a consequence, Roger--Yang skein algebras are quantizations of decorated Teichmuller spaces. This talk is based on a joint work with Wade Bloomquist (Morningside University) and Thang Le (Georgia Institute of Technology).

[ Reference URL ]To understand the algebraic structures of skein algebras and their generalizations, we usually try to embed these algebras into quantum tori using ideal triangulations of a surface and the splitting map. However, such a construction does not work for the skein algebras of closed surfaces and the Roger--Yang skein algebras of punctured surfaces.

In the talk, we define filtrations on these algebras using pants decompositions and embed the associated graded algebras into quantum tori. As a consequence, Roger--Yang skein algebras are quantizations of decorated Teichmuller spaces. This talk is based on a joint work with Wade Bloomquist (Morningside University) and Thang Le (Georgia Institute of Technology).

https://park.itc.u-tokyo.ac.jp/MSF/topology/TuesdaySeminar/index_e.html

#### Operator Algebra Seminars

16:45-18:15 Room #126 (Graduate School of Math. Sci. Bldg.)

Perturbations of von Neumann algebras

[ Reference URL ]

https://www.ms.u-tokyo.ac.jp/~yasuyuki/tokyo-seminar.htm

**Jean Roydor**(Sorbonne Université)Perturbations of von Neumann algebras

[ Reference URL ]

https://www.ms.u-tokyo.ac.jp/~yasuyuki/tokyo-seminar.htm

### 2024/04/15

#### Tokyo Probability Seminar

16:00-17:30 Room #126 (Graduate School of Math. Sci. Bldg.)

We are having teatime from 15:15 in the common room on the second floor. Please join us.

Quantitative stochastic homogenization of elliptic equations with unbounded coefficients (日本語)

We are having teatime from 15:15 in the common room on the second floor. Please join us.

**Tomohiro Aya**(Kyoto University)Quantitative stochastic homogenization of elliptic equations with unbounded coefficients (日本語)

#### Seminar on Geometric Complex Analysis

10:30-12:00 Room #128 (Graduate School of Math. Sci. Bldg.)

Kohn-Rossi cohomology of spherical CR manifolds (Japanese)

[ Reference URL ]

https://forms.gle/gTP8qNZwPyQyxjTj8

**Yuya Takeuchi**(Tsukuba Univ.)Kohn-Rossi cohomology of spherical CR manifolds (Japanese)

[ Reference URL ]

https://forms.gle/gTP8qNZwPyQyxjTj8

### 2024/04/11

#### Applied Analysis

16:00-17:30 Room #126 (Graduate School of Math. Sci. Bldg.)

Non-Markovian models of collective motion (English)

https://forms.gle/5cZ4WzqBjhsXrxgU6

**Jan Haskovec**(KAUST, Saudi Arabia)Non-Markovian models of collective motion (English)

[ Abstract ]

I will give an overview of recent results for models of collective behavior governed by functional differential equations with non-Markovian structure. The talk will focus on models of interacting agents with applications in biology (flocking, swarming), social sciences (opinion formation) and engineering (swarm robotics), where latency (delay) plays a significant role. I will characterize two main sources of delay - inter-agent communications ("transmission delay") and information processing ("reaction delay") - and discuss their impacts on the group dynamics. I will give an overview of analytical methods for studying the asymptotic behavior of the models in question and their mean-field limits. In particular, I will show that the transmission vs. reaction delay leads to fundamentally different mathematical structures and requires appropriate choice of analytical tools. Finally, motivated by situations where finite speed of information propagation is significant, I will introduce an interesting class of problems where the delay depends nontrivially and nonlinearly on the state of the system, and discuss the available analytical results and open problems here.

[ Reference URL ]I will give an overview of recent results for models of collective behavior governed by functional differential equations with non-Markovian structure. The talk will focus on models of interacting agents with applications in biology (flocking, swarming), social sciences (opinion formation) and engineering (swarm robotics), where latency (delay) plays a significant role. I will characterize two main sources of delay - inter-agent communications ("transmission delay") and information processing ("reaction delay") - and discuss their impacts on the group dynamics. I will give an overview of analytical methods for studying the asymptotic behavior of the models in question and their mean-field limits. In particular, I will show that the transmission vs. reaction delay leads to fundamentally different mathematical structures and requires appropriate choice of analytical tools. Finally, motivated by situations where finite speed of information propagation is significant, I will introduce an interesting class of problems where the delay depends nontrivially and nonlinearly on the state of the system, and discuss the available analytical results and open problems here.

https://forms.gle/5cZ4WzqBjhsXrxgU6

### 2024/04/10

#### FJ-LMI Seminar

16:00-17:00 Room #056 (Graduate School of Math. Sci. Bldg.)

Galois outer representation and the problem of Oda

(英語)

https://fj-lmi.cnrs.fr/seminars/

**Séverin PHILIP**(京都大学 数理解析研究所, RIMS, Kyoto University)Galois outer representation and the problem of Oda

(英語)

[ Abstract ]

Oda’s problem stems from considering the pro-l outer Galois actions on the moduli spaces of hyperbolic curves. These actions come from a generalization by Oda of the standard étale homotopy exact sequence for algebraic varieties over the rationals. We will introduce these geometric Galois actions and present some of the mathematics that they have stimulated over the past 30 years along with the classical problem of Oda. In the second and last part of this talk, we will see how a cyclic special loci version of this problem can be formulated and resolved in the case of simple cyclic groups using the maximal degeneration method of Ihara and Nakamura adapted to this setting.

[ Reference URL ]Oda’s problem stems from considering the pro-l outer Galois actions on the moduli spaces of hyperbolic curves. These actions come from a generalization by Oda of the standard étale homotopy exact sequence for algebraic varieties over the rationals. We will introduce these geometric Galois actions and present some of the mathematics that they have stimulated over the past 30 years along with the classical problem of Oda. In the second and last part of this talk, we will see how a cyclic special loci version of this problem can be formulated and resolved in the case of simple cyclic groups using the maximal degeneration method of Ihara and Nakamura adapted to this setting.

https://fj-lmi.cnrs.fr/seminars/

#### Seminar on Probability and Statistics

13:30-14:40 Room #126 (Graduate School of Math. Sci. Bldg.)

Limit theorems for additive functionals of stationary Gaussian fields (English)

https://forms.gle/uMKm3gVquLpYaVdc6

**Ivan Nourdin**(University of Luxembourg)Limit theorems for additive functionals of stationary Gaussian fields (English)

[ Abstract ]

In this talk, we will investigate central and non-central limit theorems for additive functionals of stationary Gaussian fields. Our main tool will be the Malliavin-Stein approach. Based on joint works with Nikolai Leonenko, Leonardo Maini and Francesca Pistolato.

[ Reference URL ]In this talk, we will investigate central and non-central limit theorems for additive functionals of stationary Gaussian fields. Our main tool will be the Malliavin-Stein approach. Based on joint works with Nikolai Leonenko, Leonardo Maini and Francesca Pistolato.

https://forms.gle/uMKm3gVquLpYaVdc6

#### Discrete mathematical modelling seminar

13:30-15:00 Room #056 (Graduate School of Math. Sci. Bldg.)

Room change: Room 470 → Room 056

Integrable birational maps and a generalisation of QRT to 3D (English)

Room change: Room 470 → Room 056

**Jaume Alonso**(Technische Universität Berlin)Integrable birational maps and a generalisation of QRT to 3D (English)

[ Abstract ]

When completely integrable Hamiltonian systems are discretised, the resulting discrete-time systems are often no longer integrable themselves. This is the so-called problem of integrable discretisation. Two known exceptions to this situation in 3D are the Kahan-Hirota-Kimura discretisations of the Euler top and the Zhukovski-Volterra gyrostat with one non-zero linear parameter β, both birational maps of degree 3. The integrals of these systems define pencils of quadrics. By analysing the geometry of these pencils, we develop a framework that generalises QRT maps and QRT roots to 3D, which allows us to create new integrable maps as a composition of two involutions. We show that under certain geometric conditions, the new maps become of degree 3. We use these results to create new families of discrete integrable maps and we solve the problem of integrability of the Zhukovski-Volterra gyrostat with two β’s.

This is a joint work with Yuri Suris and Kangning Wei.

When completely integrable Hamiltonian systems are discretised, the resulting discrete-time systems are often no longer integrable themselves. This is the so-called problem of integrable discretisation. Two known exceptions to this situation in 3D are the Kahan-Hirota-Kimura discretisations of the Euler top and the Zhukovski-Volterra gyrostat with one non-zero linear parameter β, both birational maps of degree 3. The integrals of these systems define pencils of quadrics. By analysing the geometry of these pencils, we develop a framework that generalises QRT maps and QRT roots to 3D, which allows us to create new integrable maps as a composition of two involutions. We show that under certain geometric conditions, the new maps become of degree 3. We use these results to create new families of discrete integrable maps and we solve the problem of integrability of the Zhukovski-Volterra gyrostat with two β’s.

This is a joint work with Yuri Suris and Kangning Wei.

### 2024/04/09

#### Tuesday Seminar on Topology

17:00-18:30 Room #ハイブリッド開催/056 (Graduate School of Math. Sci. Bldg.)

Pre-registration required. See our seminar webpage.

Topological stability theorem and Gromov-Hausdorff convergence (JAPANESE)

https://park.itc.u-tokyo.ac.jp/MSF/topology/TuesdaySeminar/index_e.html

Pre-registration required. See our seminar webpage.

**Shouhei Honda**(The University of Tokyo)Topological stability theorem and Gromov-Hausdorff convergence (JAPANESE)

[ Abstract ]

Gromov-Hausdorff distance defines a distance on the set of all isometry classes of compact metric spaces. It is natural to ask about topological relationships between two compact metric spaces whose Gromov-Hausdorff distance is small. Cheeger-Colding provided a striking result about this question, under a (lower) curvature bound on Ricci curvature. In this talk we will improve this result sharply. This is a joint work with Yuanlin Peng (Tohoku University). If time permits, along this direction, we will also discuss a recent work about a topological stability result to flat tori via harmonic maps, where this is a joint work with Christian Ketterer (University of Freiburg), Ilaria Mondello (Université de Paris Est Créteil), Chiara Rigoni (University of Vienna) and Raquel Perales (CIMAT).

[ Reference URL ]Gromov-Hausdorff distance defines a distance on the set of all isometry classes of compact metric spaces. It is natural to ask about topological relationships between two compact metric spaces whose Gromov-Hausdorff distance is small. Cheeger-Colding provided a striking result about this question, under a (lower) curvature bound on Ricci curvature. In this talk we will improve this result sharply. This is a joint work with Yuanlin Peng (Tohoku University). If time permits, along this direction, we will also discuss a recent work about a topological stability result to flat tori via harmonic maps, where this is a joint work with Christian Ketterer (University of Freiburg), Ilaria Mondello (Université de Paris Est Créteil), Chiara Rigoni (University of Vienna) and Raquel Perales (CIMAT).

https://park.itc.u-tokyo.ac.jp/MSF/topology/TuesdaySeminar/index_e.html

#### Operator Algebra Seminars

16:45-18:15 Room #126 (Graduate School of Math. Sci. Bldg.)

Towards lattice construction of quantum field theories

[ Reference URL ]

https://www.ms.u-tokyo.ac.jp/~yasuyuki/tokyo-seminar.htm

**Yoh Tanimoto**(Univ Rome, Tor Vergata)Towards lattice construction of quantum field theories

[ Reference URL ]

https://www.ms.u-tokyo.ac.jp/~yasuyuki/tokyo-seminar.htm

### 2024/03/21

#### Applied Analysis

16:00-17:30 Room #126 (Graduate School of Math. Sci. Bldg.)

Symmetry Results for Nonlinear PDEs (English)

**Mostafa Fazly**(University of Texas at San Antonio)Symmetry Results for Nonlinear PDEs (English)

[ Abstract ]

The study of qualitative behavior of solutions of Partial Differential Equations (PDEs) started roughly in mid-18th century. Since then scientists and mathematicians from different fields have put in a great effort to expand the theory of nonlinear PDEs. PDEs can be divided into two kinds: (a) the linear ones, which are relatively easy to analyze and can often be solved completely, and (b) the nonlinear ones, which are much harder to analyze and can almost never be solved completely.

We begin this talk by an introduction on foundational ideas behind the De Giorgi’s conjecture (1978) for the Allen-Cahn equation that is inspired by the Bernstein’s problem (1910). This conjecture brings together three groups of mathematicians: (a) a group specializing in nonlinear partial differential equations, (b) a group in differential geometry, and more specially on minimal surfaces and constant mean curvature surfaces, and (c) a group in mathematical physics on phase transitions. We then present natural generalizations and counterparts of the problem. These generalizations lead us to introduce certain novel concepts, and we illustrate why these novel concepts seem to be the right concepts in the context and how they can be used to study particular systems and models arising in Sciences. We give a survey of recent results.

The study of qualitative behavior of solutions of Partial Differential Equations (PDEs) started roughly in mid-18th century. Since then scientists and mathematicians from different fields have put in a great effort to expand the theory of nonlinear PDEs. PDEs can be divided into two kinds: (a) the linear ones, which are relatively easy to analyze and can often be solved completely, and (b) the nonlinear ones, which are much harder to analyze and can almost never be solved completely.

We begin this talk by an introduction on foundational ideas behind the De Giorgi’s conjecture (1978) for the Allen-Cahn equation that is inspired by the Bernstein’s problem (1910). This conjecture brings together three groups of mathematicians: (a) a group specializing in nonlinear partial differential equations, (b) a group in differential geometry, and more specially on minimal surfaces and constant mean curvature surfaces, and (c) a group in mathematical physics on phase transitions. We then present natural generalizations and counterparts of the problem. These generalizations lead us to introduce certain novel concepts, and we illustrate why these novel concepts seem to be the right concepts in the context and how they can be used to study particular systems and models arising in Sciences. We give a survey of recent results.

### 2024/03/14

#### Colloquium

14:30-17:00 Room #大講義室(auditorium) (Graduate School of Math. Sci. Bldg.)

If you do not belong to Graduate School of Mathematical Sciences, the University of Tokyo, please apply from the form at [Reference URL].

Many years from now (JAPANESE)

https://forms.gle/m38f1KRi67ECuA7MA

Mathematics, which I eventually found that I like: from the viewpoint of some marginal areas (JAPANESE)

https://forms.gle/m38f1KRi67ECuA7MA

If you do not belong to Graduate School of Mathematical Sciences, the University of Tokyo, please apply from the form at [Reference URL].

**Toshiyasu Arai**(Graduate School of Mathematical Sciences, The University of Tokyo) 14:30-15:30Many years from now (JAPANESE)

[ Abstract ]

I have been studying proof theory since the 1980's. In this talk I will talk about what happened to me in these 40 years, and let me report the latest result on ordinal analysis.

[ Reference URL ]I have been studying proof theory since the 1980's. In this talk I will talk about what happened to me in these 40 years, and let me report the latest result on ordinal analysis.

https://forms.gle/m38f1KRi67ECuA7MA

**Masahiro Yamamoto**(Graduate School of Mathematical Sciences, The University of Tokyo) 16:00-17:00Mathematics, which I eventually found that I like: from the viewpoint of some marginal areas (JAPANESE)

[ Abstract ]

Looking back on my experiences over 40 years, I have been convinced that I have been loving my own mathematics among others.

After all, I can sum up as that all my mathematics are concerned with the three topics: control theories, inverse problems and time-fractional partial differential equations. Some of these research fields has already developed to major topics, while others keep still minor interests.

When I started studies on inverse problems in 1980's, there were very few population of mathematicians as specialists in Japan. In particular, inverse problems did not call great attention of mathematicians and were understood as marginal mathematical topics in spite of practical significance and demands On the other hand, possibly available methodologies and ideas have been exploited and integrated gradually. As consequence, main research partners have been outside Japan.

I have been enjoying not only the research contents, but also such wider collaboration.

Aiming at non-meaningless reference for the youngers, and trying not to be too retrospective, I will describe how I have done in mathematics as well as my research contents.

[ Reference URL ]Looking back on my experiences over 40 years, I have been convinced that I have been loving my own mathematics among others.

After all, I can sum up as that all my mathematics are concerned with the three topics: control theories, inverse problems and time-fractional partial differential equations. Some of these research fields has already developed to major topics, while others keep still minor interests.

When I started studies on inverse problems in 1980's, there were very few population of mathematicians as specialists in Japan. In particular, inverse problems did not call great attention of mathematicians and were understood as marginal mathematical topics in spite of practical significance and demands On the other hand, possibly available methodologies and ideas have been exploited and integrated gradually. As consequence, main research partners have been outside Japan.

I have been enjoying not only the research contents, but also such wider collaboration.

Aiming at non-meaningless reference for the youngers, and trying not to be too retrospective, I will describe how I have done in mathematics as well as my research contents.

https://forms.gle/m38f1KRi67ECuA7MA

#### Tokyo-Nagoya Algebra Seminar

10:30-12:00 Online

Lattices of torsion classes in representation theory of finite groups (Japanese)

[ Reference URL ]

http://www.math.nagoya-u.ac.jp/~aaron.chan/TNAseminar.html

**Arashi Sakai**(Nagoya University)Lattices of torsion classes in representation theory of finite groups (Japanese)

[ Reference URL ]

http://www.math.nagoya-u.ac.jp/~aaron.chan/TNAseminar.html

### 2024/03/13

#### Numerical Analysis Seminar

16:30-17:30 Online

Approximating Langevin Monte Carlo with ResNet-like neural network architectures (English)

https://sites.google.com/g.ecc.u-tokyo.ac.jp/utnas-bulletin-board/

**David Sommer**(Weierstrass Institute for Applied Analysis and Stochastics)Approximating Langevin Monte Carlo with ResNet-like neural network architectures (English)

[ Abstract ]

We analyse a method to sample from a given target distribution by constructing a neural network which maps samples from a simple reference distribution, e.g. the standard normal, to samples from the target distribution. For this, we propose using a neural network architecture inspired by the Langevin Monte Carlo (LMC) algorithm. Based on LMC perturbation results, approximation rates of the proposed architecture for smooth, log-concave target distributions measured in the Wasserstein-2 distance are shown. The analysis heavily relies on the notion of sub-Gaussianity of the intermediate measures of the perturbed LMC process. In particular, we derive bounds on the growth of the intermediate variance proxies under different assumptions on the perturbations. Moreover, we propose an architecture similar to deep residual neural networks (ResNets) and derive expressivity results for approximating the sample to target distribution map.

[ Reference URL ]We analyse a method to sample from a given target distribution by constructing a neural network which maps samples from a simple reference distribution, e.g. the standard normal, to samples from the target distribution. For this, we propose using a neural network architecture inspired by the Langevin Monte Carlo (LMC) algorithm. Based on LMC perturbation results, approximation rates of the proposed architecture for smooth, log-concave target distributions measured in the Wasserstein-2 distance are shown. The analysis heavily relies on the notion of sub-Gaussianity of the intermediate measures of the perturbed LMC process. In particular, we derive bounds on the growth of the intermediate variance proxies under different assumptions on the perturbations. Moreover, we propose an architecture similar to deep residual neural networks (ResNets) and derive expressivity results for approximating the sample to target distribution map.

https://sites.google.com/g.ecc.u-tokyo.ac.jp/utnas-bulletin-board/

#### Numerical Analysis Seminar

17:30-18:30 Online

Analysis of the Scattering Matrix Algorithm (RCWA) for Diffraction by Periodic Surface Structures (English)

https://sites.google.com/g.ecc.u-tokyo.ac.jp/utnas-bulletin-board/

**Andreas Rathsfeld**(Weierstrass Institute for Applied Analysis and Stochastics)Analysis of the Scattering Matrix Algorithm (RCWA) for Diffraction by Periodic Surface Structures (English)

[ Abstract ]

The scattering matrix algorithm is a popular numerical method for the diffraction of optical waves by periodic surfaces. The computational domain is divided into horizontal slices and, by a clever recursion, an approximated operator, mapping incoming into outgoing waves, is obtained. Combining this with numerical schemes inside the slices, methods like RCWA and FMM have been designed.

The key for the analysis is the scattering problem with special radiation conditions for inhomogeneous cover materials. If the numerical scheme inside the slices is the FEM, then the scattering matrix algorithm is nothing else than a clever version of a domain decomposition method.

[ Reference URL ]The scattering matrix algorithm is a popular numerical method for the diffraction of optical waves by periodic surfaces. The computational domain is divided into horizontal slices and, by a clever recursion, an approximated operator, mapping incoming into outgoing waves, is obtained. Combining this with numerical schemes inside the slices, methods like RCWA and FMM have been designed.

The key for the analysis is the scattering problem with special radiation conditions for inhomogeneous cover materials. If the numerical scheme inside the slices is the FEM, then the scattering matrix algorithm is nothing else than a clever version of a domain decomposition method.

https://sites.google.com/g.ecc.u-tokyo.ac.jp/utnas-bulletin-board/

### 2024/03/12

#### Tuesday Seminar of Analysis

16:00-17:30 Room #123 (Graduate School of Math. Sci. Bldg.)

On the generic regularity of min-max CMC hypersurfaces (English)

https://forms.gle/7mqzgLqhtBuAovKB8

**Kobe Marshall-Stevens**(University College London)On the generic regularity of min-max CMC hypersurfaces (English)

[ Abstract ]

Smooth constant mean curvature (CMC) hypersurfaces serve as effective tools to study the geometry and topology of Riemannian manifolds. In high dimensions however, one in general must account for their singular behaviour. I will discuss how such hypersurfaces are constructed via min-max techniques and some recent progress on their generic regularity, allowing for certain isolated singularities to be perturbed away.

[ Reference URL ]Smooth constant mean curvature (CMC) hypersurfaces serve as effective tools to study the geometry and topology of Riemannian manifolds. In high dimensions however, one in general must account for their singular behaviour. I will discuss how such hypersurfaces are constructed via min-max techniques and some recent progress on their generic regularity, allowing for certain isolated singularities to be perturbed away.

https://forms.gle/7mqzgLqhtBuAovKB8

### 2024/03/11

#### FJ-LMI Seminar

13:30-14:30 Room #117 (Graduate School of Math. Sci. Bldg.)

Fractional Nonlinear Diffusion Equation: Numerical Analysis and Large-time Behavior. (英語)

https://fj-lmi.cnrs.fr/seminars/

**Florian SALIN**(Université de Lyon - 東北大学)Fractional Nonlinear Diffusion Equation: Numerical Analysis and Large-time Behavior. (英語)

[ Abstract ]

This talk will discuss a fractional nonlinear diffusion equation on bounded domains. This equation arises by combining fractional (in space) diffusion, with a nonlinearity of porous medium or fast diffusion type. It is known that, in the porous medium case, the energy of the solutions to this equation decays algebraically, and in the fast diffusion case, solutions extinct in finite time. Based on these estimates, we will study the fine large-time asymptotic behavior of the solutions. In particular, we will show that the solutions approach separate variable solutions as the time converges to infinity in the porous medium case, or as it converges to the extinction time in the fast diffusion case. However, the extinction time is not known analytically, and to compute it, we will introduce a numerical scheme that satisfies the same decay estimates as the continuous equation.

[ Reference URL ]This talk will discuss a fractional nonlinear diffusion equation on bounded domains. This equation arises by combining fractional (in space) diffusion, with a nonlinearity of porous medium or fast diffusion type. It is known that, in the porous medium case, the energy of the solutions to this equation decays algebraically, and in the fast diffusion case, solutions extinct in finite time. Based on these estimates, we will study the fine large-time asymptotic behavior of the solutions. In particular, we will show that the solutions approach separate variable solutions as the time converges to infinity in the porous medium case, or as it converges to the extinction time in the fast diffusion case. However, the extinction time is not known analytically, and to compute it, we will introduce a numerical scheme that satisfies the same decay estimates as the continuous equation.

https://fj-lmi.cnrs.fr/seminars/

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