## Seminar information archive

Seminar information archive ～07/13｜Today's seminar 07/14 | Future seminars 07/15～

### 2024/07/10

#### Number Theory Seminar

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

On special v-adic gamma values after Gross-Koblitz-Thakur (英語)

**Chieh-Yu Chang**(National Tsing Hua University)On special v-adic gamma values after Gross-Koblitz-Thakur (英語)

[ Abstract ]

In this talk, we will introduce special v-adic arithmetic gamma values in positive characteristic, which play the function field analogue of the special values of Morita’s p-adic gamma function. In the function field case, Thakur established a formula à la Gross-Koblitz, and hence obtained algebraicity of certain special v-adic arithmetic gamma values. In a joint work with Fu-Tsun Wei and Jing Yu, we prove that all algebraic relations among these special v-adic gamma values are coming from the three types of functional equations that the v-adic arithmetic gamma function satisfies, and Thakur’s analogue of Gross-Koblitz’s formula.

In this talk, we will introduce special v-adic arithmetic gamma values in positive characteristic, which play the function field analogue of the special values of Morita’s p-adic gamma function. In the function field case, Thakur established a formula à la Gross-Koblitz, and hence obtained algebraicity of certain special v-adic arithmetic gamma values. In a joint work with Fu-Tsun Wei and Jing Yu, we prove that all algebraic relations among these special v-adic gamma values are coming from the three types of functional equations that the v-adic arithmetic gamma function satisfies, and Thakur’s analogue of Gross-Koblitz’s formula.

### 2024/07/09

#### Operator Algebra Seminars

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

Actions of tensor categories on Kirchberg algebras

[ Reference URL ]

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

**Kan Kitamura**(RIKEN)Actions of tensor categories on Kirchberg algebras

[ Reference URL ]

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

#### Tuesday Seminar of Analysis

16:00-17:30 Room #128 (Graduate School of Math. Sci. Bldg.)

The topological nature of resonance(s) for 2D Schroedinger operators (English)

https://forms.gle/2fypneTA8CjYrLTX9

**Serge Richard**(Nagoya University)The topological nature of resonance(s) for 2D Schroedinger operators (English)

[ Abstract ]

In 1986, Gesztesy et al. revealed the surprising behavior of thresholds resonances for two-dimensional scattering systems: their contributions to Levinson's theorem are either 0 or 1, but not 1/2 as previously known for systems in dimension 1 and 3. During this seminar, we shall review this result, and explain how a C*-algebraic framework leads to a better understanding of this surprise. The main algebraic tool consists of a hexagonal algebra of Cordes, replacing a square algebra sufficient for systems in 1D and 3D. No prior C*-knowledge is expected from the audience. This presentation is based on a joint work with A. Alexander, T.D. Nguyen, and A. Rennie.

[ Reference URL ]In 1986, Gesztesy et al. revealed the surprising behavior of thresholds resonances for two-dimensional scattering systems: their contributions to Levinson's theorem are either 0 or 1, but not 1/2 as previously known for systems in dimension 1 and 3. During this seminar, we shall review this result, and explain how a C*-algebraic framework leads to a better understanding of this surprise. The main algebraic tool consists of a hexagonal algebra of Cordes, replacing a square algebra sufficient for systems in 1D and 3D. No prior C*-knowledge is expected from the audience. This presentation is based on a joint work with A. Alexander, T.D. Nguyen, and A. Rennie.

https://forms.gle/2fypneTA8CjYrLTX9

#### Numerical Analysis Seminar

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

The transformation of stabilizations into spaces for Galerkin methods for PDEs (English)

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

**Bernardo Cockburn**(University of Minnesota)The transformation of stabilizations into spaces for Galerkin methods for PDEs (English)

[ Abstract ]

We describe a novel technique which allows us to transform the terms which render Galerkin methods stable into spaces (JJIAM, 2023). We begin by applying this technique to show that the Continuous and Discontinuous Galerkin (DG) methods for ODEs produce the very same approximation of the time derivative, and use this to obtain superconvergence points of the DG method. We then apply this technique to mixed methods for second-order elliptic equations to show that they can always be recast as hybridizable DG (HDG) methods. We then show that this recating makes the implementation from 10% to 20% better for polynomial degrees ranging from 1 to 20.We end by sketching or ongoing and future work.

[ Reference URL ]We describe a novel technique which allows us to transform the terms which render Galerkin methods stable into spaces (JJIAM, 2023). We begin by applying this technique to show that the Continuous and Discontinuous Galerkin (DG) methods for ODEs produce the very same approximation of the time derivative, and use this to obtain superconvergence points of the DG method. We then apply this technique to mixed methods for second-order elliptic equations to show that they can always be recast as hybridizable DG (HDG) methods. We then show that this recating makes the implementation from 10% to 20% better for polynomial degrees ranging from 1 to 20.We end by sketching or ongoing and future work.

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

#### Tuesday Seminar on Topology

17:00-18:00 Online

Pre-registration required. See our seminar webpage.

Knitted surfaces in the 4-ball and their chart description (JAPANESE)

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

Pre-registration required. See our seminar webpage.

**Inasa Nakamura**(Saga University)Knitted surfaces in the 4-ball and their chart description (JAPANESE)

[ Abstract ]

Knits (or BMW tangles) are tangles in a cylinder generated by generators of the BMW (Birman-Murakami-Wenzl) algebras, consisting of standard generators of the braid group and their inverses, and splices of crossings called pairs of hooks. We give a new construction of surfaces in $D^2 \times B^2$, called knitted surfaces (or BMW surfaces), that are described as the trace of deformations of knits, and we give the notion of charts for knitted surfaces, that are finite graphs in $B^2$. We show that a knitted surface has a chart description. Knitted surfaces and their chart description include 2-dimensional braids and their chart description. This is joint work with Jumpei Yasuda (Osaka University).

[ Reference URL ]Knits (or BMW tangles) are tangles in a cylinder generated by generators of the BMW (Birman-Murakami-Wenzl) algebras, consisting of standard generators of the braid group and their inverses, and splices of crossings called pairs of hooks. We give a new construction of surfaces in $D^2 \times B^2$, called knitted surfaces (or BMW surfaces), that are described as the trace of deformations of knits, and we give the notion of charts for knitted surfaces, that are finite graphs in $B^2$. We show that a knitted surface has a chart description. Knitted surfaces and their chart description include 2-dimensional braids and their chart description. This is joint work with Jumpei Yasuda (Osaka University).

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

### 2024/07/08

#### Seminar on Geometric Complex Analysis

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

. (Japanese)

[ Reference URL ]

https://forms.gle/gTP8qNZwPyQyxjTj8

**Tomoyuki Hisamoto**(Tokyo Metropolitan Univ.). (Japanese)

[ Reference URL ]

https://forms.gle/gTP8qNZwPyQyxjTj8

#### 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.

Maximum of the Gaussian interface model in random external fields (日本語)

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

**Hironobu Sakagawa**(Keio University)Maximum of the Gaussian interface model in random external fields (日本語)

[ Abstract ]

相分離の界面モデルの一つとして格子上のGauss型界面モデル（離散Gauss自由場）を取り上げ，そこにランダムな外場（化学ポテンシャル）を加えた（ランダムな）Gibbs測度の下での最大値について考える．特に，外場の確率変数の末尾確率の挙動に応じて最大値の挙動が変わることを示し，その主要項を特徴付ける．

相分離の界面モデルの一つとして格子上のGauss型界面モデル（離散Gauss自由場）を取り上げ，そこにランダムな外場（化学ポテンシャル）を加えた（ランダムな）Gibbs測度の下での最大値について考える．特に，外場の確率変数の末尾確率の挙動に応じて最大値の挙動が変わることを示し，その主要項を特徴付ける．

### 2024/07/04

#### Algebraic Geometry Seminar

13:00-14:30 Room #ハイブリッド開催/118 (Graduate School of Math. Sci. Bldg.)

On a principle of Ogus: the Hasse invariant's order of vanishing and "Frobenius and the Hodge filtration'' (English)

**Stefan Reppen**(University of Tokyo)On a principle of Ogus: the Hasse invariant's order of vanishing and "Frobenius and the Hodge filtration'' (English)

[ Abstract ]

In joint work with W. Goldring we generalize a result of Ogus that, under certain technical conditions, the vanishing order of the Hasse invariant of a family $Y/X$ of $n$-dimensional Calabi-Yau varieties in characteristic $p$ at a point $x$ of $X$ equals the "conjugate line position" of $H^n_{\dR}(Y/X)$ at $x$, i.e. the largest $i$ such that the line of the conjugate filtration is contained in $\text{Fil}^i$ of the Hodge filtration. For every triple $(G,\mu,r)$ consisting of a connected, reductive $\mathbb{F}_p$-group $G$, a cocharacter $\mu \in X_*(G)$ and an $\mathbb{F}_p$-representation $r$ of $G$, we state a generalized Ogus Principle. If $\zeta:X \to \GZip^{\mu}$ is a smooth morphism, then the group theoretic Ogus Principle implies an Ogus Principle on $X$. We deduce an Ogus Principle for several Hodge and abelian-type Shimura varieties and the moduli space of K3 surfaces. In the talk I will present this work.

In joint work with W. Goldring we generalize a result of Ogus that, under certain technical conditions, the vanishing order of the Hasse invariant of a family $Y/X$ of $n$-dimensional Calabi-Yau varieties in characteristic $p$ at a point $x$ of $X$ equals the "conjugate line position" of $H^n_{\dR}(Y/X)$ at $x$, i.e. the largest $i$ such that the line of the conjugate filtration is contained in $\text{Fil}^i$ of the Hodge filtration. For every triple $(G,\mu,r)$ consisting of a connected, reductive $\mathbb{F}_p$-group $G$, a cocharacter $\mu \in X_*(G)$ and an $\mathbb{F}_p$-representation $r$ of $G$, we state a generalized Ogus Principle. If $\zeta:X \to \GZip^{\mu}$ is a smooth morphism, then the group theoretic Ogus Principle implies an Ogus Principle on $X$. We deduce an Ogus Principle for several Hodge and abelian-type Shimura varieties and the moduli space of K3 surfaces. In the talk I will present this work.

### 2024/07/03

#### Lectures

16:00-17:30 Room #052 (Graduate School of Math. Sci. Bldg.)

Boundary Rigidity and the Geodesic X-ray Transform in Low Regularity (English)

**Kelvin Lam**(Department of Mathematics, University of Washington, U.S.A.)Boundary Rigidity and the Geodesic X-ray Transform in Low Regularity (English)

### 2024/07/02

#### Tuesday Seminar on Topology

17:00-18:30 Room #ハイブリッド開催/117 (Graduate School of Math. Sci. Bldg.)

Pre-registration required. See our seminar webpage.

The second quandle homology group of the knot $n$-quandle (JAPANESE)

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

Pre-registration required. See our seminar webpage.

**Kokoro Tanaka**(Tokyo Gakugei University)The second quandle homology group of the knot $n$-quandle (JAPANESE)

[ Abstract ]

We compute the second quandle homology group of the knot $n$-quandle for each integer $n>1$, where the knot $n$-quandle is a certain quotient of the knot quandle (of an oriented classical knot in the $3$-sphere). Although the second quandle homology group of the knot quandle can only detect the unknot, it turns out that that of its 3-quandle can detect the unknot, the trefoil and the cinqfoil. This is a joint work with Yuta Taniguchi.

[ Reference URL ]We compute the second quandle homology group of the knot $n$-quandle for each integer $n>1$, where the knot $n$-quandle is a certain quotient of the knot quandle (of an oriented classical knot in the $3$-sphere). Although the second quandle homology group of the knot quandle can only detect the unknot, it turns out that that of its 3-quandle can detect the unknot, the trefoil and the cinqfoil. This is a joint work with Yuta Taniguchi.

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

### 2024/06/28

#### Tokyo-Nagoya Algebra Seminar

16:30-18:00 Online

Classifying KE-closed subcategories (Japanese)

[ Reference URL ]

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

**Shunya Saito**(The University of Tokyo)Classifying KE-closed subcategories (Japanese)

[ Reference URL ]

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

#### Seminar on Probability and Statistics

13:00-14:10 Room #128 (Graduate School of Math. Sci. Bldg.)

医学における予測モデルの活用と階層構造を持つ順序回帰の提案 (日本語)

[ Reference URL ]

https://u-tokyo-ac-jp.zoom.us/meeting/register/tZUpd-ispjIqG9NfJk7_kjW2pBcvq_KMXHPW

**原田 和治**(東京医科大学医療データサイエンス分野)医学における予測モデルの活用と階層構造を持つ順序回帰の提案 (日本語)

[ Reference URL ]

https://u-tokyo-ac-jp.zoom.us/meeting/register/tZUpd-ispjIqG9NfJk7_kjW2pBcvq_KMXHPW

#### Algebraic Geometry Seminar

13:30-15:00 Room #ハイブリッド開催/117 (Graduate School of Math. Sci. Bldg.)

Stable rationality of hypersurfaces in mock toric varieties (日本語)

**Taro Yoshino**(The University of Tokyo)Stable rationality of hypersurfaces in mock toric varieties (日本語)

[ Abstract ]

In recent years, there has been a development in approaching rationality problems through motivic methods. This approach requires the explicit construction of degeneration families over curves with favorable properties. However, the specific construction is generally difficult. Nicaise and Ottem combined combinatorial methods to construct degeneration families of hypersurfaces in toric varieties and mentioned the stable rationality of a very general hypersurface in projective spaces. In this talk, we mention the following two points: First, I introduce the notion of mock toric varieties, which are generalizations of toric varieties. Second, I combinatorially construct degeneration families of hypersurfaces in mock toric varieties, and I mention the irrationality of a very general hypersurface in the complex Grassmannian variety Gr(2, n).

In recent years, there has been a development in approaching rationality problems through motivic methods. This approach requires the explicit construction of degeneration families over curves with favorable properties. However, the specific construction is generally difficult. Nicaise and Ottem combined combinatorial methods to construct degeneration families of hypersurfaces in toric varieties and mentioned the stable rationality of a very general hypersurface in projective spaces. In this talk, we mention the following two points: First, I introduce the notion of mock toric varieties, which are generalizations of toric varieties. Second, I combinatorially construct degeneration families of hypersurfaces in mock toric varieties, and I mention the irrationality of a very general hypersurface in the complex Grassmannian variety Gr(2, n).

### 2024/06/27

#### Applied Analysis

16:00-17:30 Room #128 (Graduate School of Math. Sci. Bldg.)

A Control Theory in Mathematical Demography (Japanese)

**Ryo OIZUMI**(National Institute of Population and Social Security Research)A Control Theory in Mathematical Demography (Japanese)

[ Abstract ]

Multistate Age-Structured Population Model is a fundamental mathematical model in mathematical demography that describes population structure and dynamics with state variables that are not uniform with age (e.g., body size, place of residence, genetic characteristics, etc.). The model's eigensystems have been used in various demographic analyses, providing essential indicators for discussing evolutionary theory. In this study, we derive a control equation (HJB equation) that maximizes the spectral radius from the eigensystem of the multistate age-structured population model and discuss the control process that generates an evolutionarily adaptive life history.

Multistate Age-Structured Population Model is a fundamental mathematical model in mathematical demography that describes population structure and dynamics with state variables that are not uniform with age (e.g., body size, place of residence, genetic characteristics, etc.). The model's eigensystems have been used in various demographic analyses, providing essential indicators for discussing evolutionary theory. In this study, we derive a control equation (HJB equation) that maximizes the spectral radius from the eigensystem of the multistate age-structured population model and discuss the control process that generates an evolutionarily adaptive life history.

### 2024/06/25

#### Operator Algebra Seminars

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

Polynomial family of quantum flag manifolds via deformed QEA

[ Reference URL ]

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

**Mao Hoshino**(Univ. Tokyo)Polynomial family of quantum flag manifolds via deformed QEA

[ Reference URL ]

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

#### Tuesday Seminar on Topology

17:00-18:30 Room #056 (Graduate School of Math. Sci. Bldg.)

Joint with RIKEN iTHEMS. Pre-registration required. See our seminar webpage.

Liouville symmetry groups and pseudo-isotopies (ENGLISH)

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

Joint with RIKEN iTHEMS. Pre-registration required. See our seminar webpage.

**Emmy Murphy**(University of Toronto)Liouville symmetry groups and pseudo-isotopies (ENGLISH)

[ Abstract ]

Even though $\mathbb{C}^n$ is the most basic symplectic manifold, when $n>2$ its compactly supported symplectomorphism group remains mysterious. For instance, we do not know if it is connected. To understand it better, one can define various subgroups of the symplectomorphism group, and a number of Serre fibrations between them. This leads us to the Liouville pseudo-isotopy group of a contact manifold, important for relating (for instance) compactly supported symplectomorphisms of $\mathbb{C}^n$, and contactomorphisms of the sphere at infinity. After explaining this background, the talk will focus on a new result: that the pseudo-isotopy group is connected, under a Liouville-vs-Weinstein hypothesis.

[ Reference URL ]Even though $\mathbb{C}^n$ is the most basic symplectic manifold, when $n>2$ its compactly supported symplectomorphism group remains mysterious. For instance, we do not know if it is connected. To understand it better, one can define various subgroups of the symplectomorphism group, and a number of Serre fibrations between them. This leads us to the Liouville pseudo-isotopy group of a contact manifold, important for relating (for instance) compactly supported symplectomorphisms of $\mathbb{C}^n$, and contactomorphisms of the sphere at infinity. After explaining this background, the talk will focus on a new result: that the pseudo-isotopy group is connected, under a Liouville-vs-Weinstein hypothesis.

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

### 2024/06/24

#### Seminar on Geometric Complex Analysis

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

. (Japanese)

[ Reference URL ]

https://forms.gle/gTP8qNZwPyQyxjTj8

**Kazumasa Narita**(Nagoya Univ.). (Japanese)

[ Reference URL ]

https://forms.gle/gTP8qNZwPyQyxjTj8

#### 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.

Temperley - Lieb 演算子の持ち上げとRazumov - Stroganov 予想について (日本語)

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

**Fumihiko Nakano**(Tohoku University)Temperley - Lieb 演算子の持ち上げとRazumov - Stroganov 予想について (日本語)

[ Abstract ]

Razumov - Stroganov 予想とはリンクパターン上の生成する線型空間上のあるハミルトニアンの基底状態に対応するFPLの個数が現れるという予想で、２０１０年に解決されたが、O(1)-loop model, 交代符号行列を介して２次元統計力学の模型や組み合わせ論との様々なつながりがあり、今も注目されている。Temperley - Lieb 演算子の持ち上げを用いたRS予想のより平易な証明について議論する。

Razumov - Stroganov 予想とはリンクパターン上の生成する線型空間上のあるハミルトニアンの基底状態に対応するFPLの個数が現れるという予想で、２０１０年に解決されたが、O(1)-loop model, 交代符号行列を介して２次元統計力学の模型や組み合わせ論との様々なつながりがあり、今も注目されている。Temperley - Lieb 演算子の持ち上げを用いたRS予想のより平易な証明について議論する。

### 2024/06/21

#### Tokyo-Nagoya Algebra Seminar

16:30-18:00 Online

松井スペクトラムを用いた復元定理の再解釈 (Japanese)

[ Reference URL ]

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

**Daigo Ito**(UC Berkeley)松井スペクトラムを用いた復元定理の再解釈 (Japanese)

[ Reference URL ]

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

#### Colloquium

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

In order to contact you in case of an outbreak of infections, we appreciate your regitration by following the link in the [Reference URL] field below.

The minimal exponent of hypersurface singularities (English)

https://docs.google.com/forms/d/e/1FAIpQLSdUrEZYZ4fvi8So3pUVkxF08M2jbVdo7hTew_B1S5l-opFyzg/viewform?usp=sharing

In order to contact you in case of an outbreak of infections, we appreciate your regitration by following the link in the [Reference URL] field below.

**Mircea Mustaţă**(The University of Michigan)The minimal exponent of hypersurface singularities (English)

[ Abstract ]

The log canonical threshold of a hypersurface is an invariant of singularities that plays an important role in birational geometry, but which arises in many other contexts and admits different characterizations. A refinement of this invariant is Saito's minimal exponent, whose definition relies on the theory of b-functions, an important concept in D-module theory. The new information (by comparison with the log canonical threshold) provides a numerical measure of rational singularities. In this talk I will give an introduction to minimal exponents, highlighting recent progress and open questions.

[ Reference URL ]The log canonical threshold of a hypersurface is an invariant of singularities that plays an important role in birational geometry, but which arises in many other contexts and admits different characterizations. A refinement of this invariant is Saito's minimal exponent, whose definition relies on the theory of b-functions, an important concept in D-module theory. The new information (by comparison with the log canonical threshold) provides a numerical measure of rational singularities. In this talk I will give an introduction to minimal exponents, highlighting recent progress and open questions.

https://docs.google.com/forms/d/e/1FAIpQLSdUrEZYZ4fvi8So3pUVkxF08M2jbVdo7hTew_B1S5l-opFyzg/viewform?usp=sharing

#### Algebraic Geometry Seminar

13:30-15:00 Room #ハイブリッド開催/056 (Graduate School of Math. Sci. Bldg.)

ON THE POWER SERIES OF DENEF AND LOESER'S MOTIVIC VANISHING CYCLES OF JET POLYNOMIALS (English)

**Kien Nguyen Huu**(Normandie Université/KU Leuven)ON THE POWER SERIES OF DENEF AND LOESER'S MOTIVIC VANISHING CYCLES OF JET POLYNOMIALS (English)

[ Abstract ]

Let f be a non-constant polynomial in n variables over a field k of characteristic

0. Denef and Loeser introduced the notion of motivic vanishing cycles of f as an element in

the localization Mμˆ of the Grothendieck ring Kμˆ(Var ) of k-varieties with a good action of k0k

μˆ := lim μm by inverting the affne line equipped with the trivial action of μˆ, where μm

is the group scheme over k of mth roots of unity. In particular, if k is the field of complex

numbers then Denef and Loeser showed that their motivic vanishing cycles and the complex

φf [n − 1] has the same Hodge characteristic, where φf is the complex of vanishing cycles

in the usual sense. Motivated by the Igusa conjecture for exponential sums and the strong

monodromy conjecture, we introduce the notion of Poincaré series of Denef-Loeser's van-

ishing cycles of jet polynomials of f, where jet polynomials of f are polynomials appearing

naturally when we compute the jet schemes of f. By using Davison-Meinhardt's conjecture

which was proved by Nicaise and Payne in 2019, we can show that our Poincaré series is a

rational function over a quotient ring of Mμˆ by very natural relations. In particular, we can k

recovery Denef and Loeser's motivic vanishing cycles from our Poincaré series. Moreover, we can show that our Poincaré series owns a universal property in the sense that if k is a number field then the Igusa local zeta functions, the motivic Igusa zeta functions, the Poincaré series of exponential sums modulo pm of f can be obtained from our Poincaré se- ries by suitable specialization maps preserving the rationality. If time permits, I will present some initial consequences that have arisen during the study of our Poincaré series.

Let f be a non-constant polynomial in n variables over a field k of characteristic

0. Denef and Loeser introduced the notion of motivic vanishing cycles of f as an element in

the localization Mμˆ of the Grothendieck ring Kμˆ(Var ) of k-varieties with a good action of k0k

μˆ := lim μm by inverting the affne line equipped with the trivial action of μˆ, where μm

is the group scheme over k of mth roots of unity. In particular, if k is the field of complex

numbers then Denef and Loeser showed that their motivic vanishing cycles and the complex

φf [n − 1] has the same Hodge characteristic, where φf is the complex of vanishing cycles

in the usual sense. Motivated by the Igusa conjecture for exponential sums and the strong

monodromy conjecture, we introduce the notion of Poincaré series of Denef-Loeser's van-

ishing cycles of jet polynomials of f, where jet polynomials of f are polynomials appearing

naturally when we compute the jet schemes of f. By using Davison-Meinhardt's conjecture

which was proved by Nicaise and Payne in 2019, we can show that our Poincaré series is a

rational function over a quotient ring of Mμˆ by very natural relations. In particular, we can k

recovery Denef and Loeser's motivic vanishing cycles from our Poincaré series. Moreover, we can show that our Poincaré series owns a universal property in the sense that if k is a number field then the Igusa local zeta functions, the motivic Igusa zeta functions, the Poincaré series of exponential sums modulo pm of f can be obtained from our Poincaré se- ries by suitable specialization maps preserving the rationality. If time permits, I will present some initial consequences that have arisen during the study of our Poincaré series.

### 2024/06/20

#### Tuesday Seminar on Topology

17:00-18:30 Room #002 (Graduate School of Math. Sci. Bldg.)

Joint with RIKEN iTHEMS. Pre-registration required. See our seminar webpage.

Rigidity and Flexibility of Iosmetric Embeddings (ENGLISH)

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

Joint with RIKEN iTHEMS. Pre-registration required. See our seminar webpage.

**Dominik Inauen**(University of Leipzig)Rigidity and Flexibility of Iosmetric Embeddings (ENGLISH)

[ Abstract ]

The problem of embedding abstract Riemannian manifolds isometrically (i.e. preserving the lengths) into Euclidean space stems from the conceptually fundamental question of whether abstract Riemannian manifolds and submanifolds of Euclidean space are the same. As it turns out, such embeddings have a drastically different behaviour at low regularity (i.e. $C^1$) than at high regularity (i.e. $C^2$). For example, by the famous Nash--Kuiper theorem it is possible to find $C^1$ isometric embeddings of the standard $2$-sphere into arbitrarily small balls in $\mathbb{R}^3$, and yet, in the $C^2$ category there is (up to translation and rotation) just one isometric embedding, namely the standard inclusion. Analoguous to the Onsager conjecture in fluid dynamics, one might ask if there is a sharp regularity threshold in the Hölder scale which distinguishes these flexible and rigid behaviours. In my talk I will review some known results and argue why the Hölder exponent 1/2 can be seen as a critical exponent in the problem.

[ Reference URL ]The problem of embedding abstract Riemannian manifolds isometrically (i.e. preserving the lengths) into Euclidean space stems from the conceptually fundamental question of whether abstract Riemannian manifolds and submanifolds of Euclidean space are the same. As it turns out, such embeddings have a drastically different behaviour at low regularity (i.e. $C^1$) than at high regularity (i.e. $C^2$). For example, by the famous Nash--Kuiper theorem it is possible to find $C^1$ isometric embeddings of the standard $2$-sphere into arbitrarily small balls in $\mathbb{R}^3$, and yet, in the $C^2$ category there is (up to translation and rotation) just one isometric embedding, namely the standard inclusion. Analoguous to the Onsager conjecture in fluid dynamics, one might ask if there is a sharp regularity threshold in the Hölder scale which distinguishes these flexible and rigid behaviours. In my talk I will review some known results and argue why the Hölder exponent 1/2 can be seen as a critical exponent in the problem.

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

### 2024/06/19

#### Number Theory Seminar

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

Prismatic $F$-crystals and Wach modules (English)

**Abhinandan**(University of Tokyo)Prismatic $F$-crystals and Wach modules (English)

[ Abstract ]

For an absolutely unramified extension $K/\mathbb{Q}_p$ with perfect residue field, by the works of Fontaine, Colmez, Wach and Berger, it is well known that the category of Wach modules over a certain integral period ring is equivalent to the category of lattices inside crystalline representations of $G_K$ (the absolute Galois group of $K$). Moreover, by the recent works of Bhatt and Scholze, we also know that lattices inside crystalline representations of $G_K$ are equivalent to the category of prismatic $F$-crystals on the absolute prismatic site of $O_K$, the ring of integers of $K$. The goal of this talk is to present a direct construction of the categorical equivalence between Wach modules and prismatic $F$-crystals over the absolute prismatic site of $O_K$. If time permits, we will also mention a generalisation of these results to the case of a "small" base ring.

For an absolutely unramified extension $K/\mathbb{Q}_p$ with perfect residue field, by the works of Fontaine, Colmez, Wach and Berger, it is well known that the category of Wach modules over a certain integral period ring is equivalent to the category of lattices inside crystalline representations of $G_K$ (the absolute Galois group of $K$). Moreover, by the recent works of Bhatt and Scholze, we also know that lattices inside crystalline representations of $G_K$ are equivalent to the category of prismatic $F$-crystals on the absolute prismatic site of $O_K$, the ring of integers of $K$. The goal of this talk is to present a direct construction of the categorical equivalence between Wach modules and prismatic $F$-crystals over the absolute prismatic site of $O_K$. If time permits, we will also mention a generalisation of these results to the case of a "small" base ring.

### 2024/06/18

#### Operator Algebra Seminars

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

Lie theoretic approach to the unitary groups of $C^*$-algebras

[ Reference URL ]

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

**Hiroshi Ando**(Chiba Univ.)Lie theoretic approach to the unitary groups of $C^*$-algebras

[ Reference URL ]

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

#### Tuesday Seminar of Analysis

16:00-17:30 Room #128 (Graduate School of Math. Sci. Bldg.)

Blocking and propagation in two-dimensional cylinders with spatially undulating boundary (Japanese)

https://forms.gle/TrFmSZQ1ZeqvSjfP7

**MORI Ryunosuke**(Meiji University)Blocking and propagation in two-dimensional cylinders with spatially undulating boundary (Japanese)

[ Abstract ]

We consider blocking and propagation phenomena of mean curvature flow with a driving force in two-dimensional cylinders with spatially undulating boundary. In this problem, Matano, Nakamura and Lou in 2006, 2013 characterize the effect of the shape of the boundary to blocking and propagation of the solutions under some slop condition about the boundary that implies time global existence of the classical solutions. In this talk, we consider the effect of the shape of the boundary to blocking and propagation of this problem under more general situation that the solutions may develop singularities near the boundary.

[ Reference URL ]We consider blocking and propagation phenomena of mean curvature flow with a driving force in two-dimensional cylinders with spatially undulating boundary. In this problem, Matano, Nakamura and Lou in 2006, 2013 characterize the effect of the shape of the boundary to blocking and propagation of the solutions under some slop condition about the boundary that implies time global existence of the classical solutions. In this talk, we consider the effect of the shape of the boundary to blocking and propagation of this problem under more general situation that the solutions may develop singularities near the boundary.

https://forms.gle/TrFmSZQ1ZeqvSjfP7

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