## Seminar information archive

Seminar information archive ～10/10｜Today's seminar 10/11 | Future seminars 10/12～

### 2023/11/21

#### Tuesday Seminar on Topology

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

Pre-registration required. See our seminar webpage.

Shadows, divides and hyperbolic volumes (JAPANESE)

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

Pre-registration required. See our seminar webpage.

**Yuya Koda**(Keio University)Shadows, divides and hyperbolic volumes (JAPANESE)

[ Abstract ]

In 2008, Costantino and D.Thurston revealed that the combinatorial structure of the Stein factorizations of stable maps from 3-manifolds into the real plane can be used to describe the hyperbolic structures of the complement of the set of definite fold points, which is a link. The key was that the Stein factorizations can naturally be embedded into 4-manifolds, and nice ideal polyhedral decompositions become visible on their boundaries. In this talk, we consider divides, which are the images of a proper and generic immersions of compact 1-manifolds into the 2-disk. Due to A'Campo's theory, each divide is associated with a link in the 3-sphere. By embedding a polyhedron induced from a given divide into the 4-ball as was done to Stein factorization, we can read off the ideal polyhedral decompositions on the boundary. We then show that the complement of the link of the divide can be obtained by Dehn filling a hyperbolic 3-manifold that admits a decomposition into several ideal regular hyperbolic polyhedra, where the number of each polyhedron is determined by types of the double points of the divide. This immediately gives an upper bound of the hyperbolic volume of the links of divides, which is shown to be asymptotically sharp. As in the case of Stein factorizations, an idea from the theory of Turaev's shadows plays an important role here. This talk is based on joint work with Ryoga Furutani (Hiroshima University).

[ Reference URL ]In 2008, Costantino and D.Thurston revealed that the combinatorial structure of the Stein factorizations of stable maps from 3-manifolds into the real plane can be used to describe the hyperbolic structures of the complement of the set of definite fold points, which is a link. The key was that the Stein factorizations can naturally be embedded into 4-manifolds, and nice ideal polyhedral decompositions become visible on their boundaries. In this talk, we consider divides, which are the images of a proper and generic immersions of compact 1-manifolds into the 2-disk. Due to A'Campo's theory, each divide is associated with a link in the 3-sphere. By embedding a polyhedron induced from a given divide into the 4-ball as was done to Stein factorization, we can read off the ideal polyhedral decompositions on the boundary. We then show that the complement of the link of the divide can be obtained by Dehn filling a hyperbolic 3-manifold that admits a decomposition into several ideal regular hyperbolic polyhedra, where the number of each polyhedron is determined by types of the double points of the divide. This immediately gives an upper bound of the hyperbolic volume of the links of divides, which is shown to be asymptotically sharp. As in the case of Stein factorizations, an idea from the theory of Turaev's shadows plays an important role here. This talk is based on joint work with Ryoga Furutani (Hiroshima University).

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

#### Operator Algebra Seminars

16:45-18:15 Room #128 (Graduate School of Math. Sci. Bldg.)

Finite index quantum subgroups of DQGs

[ Reference URL ]

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

**Mao Hoshino**(Univ. Tokyo)Finite index quantum subgroups of DQGs

[ Reference URL ]

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

### 2023/11/20

#### Tokyo Probability Seminar

17:00-18:30 Room #126 (Graduate School of Math. Sci. Bldg.)

Yet another construction of “Sobolev” spaces on metric spaces (日本語)

**Jun Kigami**(Kyoto University)Yet another construction of “Sobolev” spaces on metric spaces (日本語)

### 2023/11/16

#### Information Mathematics Seminar

16:50-18:35 Room #128 (Graduate School of Math. Sci. Bldg.)

Isogeny graphs on superspecial abelian varieties: Eigenvalues, Connection to Bruhat-Tits buildings, and Property (T) (Japanese)

**Takuya Yamauchi**(Tohoku Univ.)Isogeny graphs on superspecial abelian varieties: Eigenvalues, Connection to Bruhat-Tits buildings, and Property (T) (Japanese)

[ Abstract ]

For each fixed integer $g\ge 2$, a prime $p$, and all primes $\ell$ with $\ell\neq p$, we can consider finite regular directed graphs associated with the set of equivalence classes of $\ell$-marked principally polarized superspecial abelian varieties of dimension $g$ in characteristic $p$. In this talk, I will explain that we can study such graphs in terms of the corresponding Bruhat-Tits buildings. I also discuss the eigenvalues values of the random walk matrices in view of the theory of automorphic representations when $g=2$. This is a joint work with Y. Aikawa (Tokyo university) and R. Tanaka (Kyoto university).

For each fixed integer $g\ge 2$, a prime $p$, and all primes $\ell$ with $\ell\neq p$, we can consider finite regular directed graphs associated with the set of equivalence classes of $\ell$-marked principally polarized superspecial abelian varieties of dimension $g$ in characteristic $p$. In this talk, I will explain that we can study such graphs in terms of the corresponding Bruhat-Tits buildings. I also discuss the eigenvalues values of the random walk matrices in view of the theory of automorphic representations when $g=2$. This is a joint work with Y. Aikawa (Tokyo university) and R. Tanaka (Kyoto university).

### 2023/11/14

#### 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.

Dependency of the positive and negative long-time behaviors of flows on surfaces (JAPANESE)

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

Pre-registration required. See our seminar webpage.

**Tomoo Yokoyama**(Saitama University)Dependency of the positive and negative long-time behaviors of flows on surfaces (JAPANESE)

[ Abstract ]

We discuss the dependence of a flow's positive and negative limit behaviors on a surface. In particular, I introduce the list of possible pairs of positive and negative limit behaviors that can and cannot occur. The idea of the dependence mechanism is illustrated using the dependence of the limit behavior of a toy model, a circle homeomorphism. We overview with as few prior knowledge assumptions as possible.

[ Reference URL ]We discuss the dependence of a flow's positive and negative limit behaviors on a surface. In particular, I introduce the list of possible pairs of positive and negative limit behaviors that can and cannot occur. The idea of the dependence mechanism is illustrated using the dependence of the limit behavior of a toy model, a circle homeomorphism. We overview with as few prior knowledge assumptions as possible.

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

#### Numerical Analysis Seminar

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

On some dynamical systems and their prediction using data assimilation (Japanese)

[ Reference URL ]

ハイブリッド開催です。参加の詳細は参考URLをご覧ください。

**Ken Furukawa**(RIKEN)On some dynamical systems and their prediction using data assimilation (Japanese)

[ Reference URL ]

ハイブリッド開催です。参加の詳細は参考URLをご覧ください。

#### Tuesday Seminar of Analysis

16:15-17:15 Room #126 (Graduate School of Math. Sci. Bldg.)

Resolvent expansions for magnetic Schrödinger operators (English)

https://forms.gle/qyEUeo4kVuPL1s289

**Arne Jensen**(Aalborg University)Resolvent expansions for magnetic Schrödinger operators (English)

[ Abstract ]

I will present some new results resolvent expansions around threshold zero for magnetic Schrödinger operators in dimension three. The magnetic field and the electric potential are assumed to decay sufficiently fast. Analogous results for Pauli operators will also be presented.

Joint work with H. Kovarik, Brescia, Italy.

[ Reference URL ]I will present some new results resolvent expansions around threshold zero for magnetic Schrödinger operators in dimension three. The magnetic field and the electric potential are assumed to decay sufficiently fast. Analogous results for Pauli operators will also be presented.

Joint work with H. Kovarik, Brescia, Italy.

https://forms.gle/qyEUeo4kVuPL1s289

#### Operator Algebra Seminars

16:45-18:15 Room #128 (Graduate School of Math. Sci. Bldg.)

On the Scottish Book Problem 155 by Mazur and Sternbach

[ Reference URL ]

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

**Michiya Mori**(Univ. Tokyo)On the Scottish Book Problem 155 by Mazur and Sternbach

[ Reference URL ]

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

### 2023/11/09

#### Information Mathematics Seminar

16:50-18:35 Room #128 (Graduate School of Math. Sci. Bldg.)

Mathematical Aspects of Isogeny-Based Cryptography (Japanese)

**Katsuyuki Takashima**(Waseda Univ.)Mathematical Aspects of Isogeny-Based Cryptography (Japanese)

[ Abstract ]

I will explain mathematical aspects of isogeny-based cryptography.

I will explain mathematical aspects of isogeny-based cryptography.

### 2023/11/07

#### Operator Algebra Seminars

16:45-18:15 Room #128 (Graduate School of Math. Sci. Bldg.)

A quantum analogue of the special linear group and its proper cocycle

[ Reference URL ]

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

**Masato Tanaka**(Nagoya Univ.)A quantum analogue of the special linear group and its proper cocycle

[ Reference URL ]

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

#### 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.

Hodge numbers of moduli spaces of principal bundles on curves (ENGLISH)

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

Pre-registration required. See our seminar webpage.

**Florent Schaffhauser**(Heidelberg University)Hodge numbers of moduli spaces of principal bundles on curves (ENGLISH)

[ Abstract ]

The Poincaré series of moduli stacks of semistable G-bundles on curves has been computed by Laumon and Rapoport. In this joint work with Melissa Liu, we show that the Hodge-Poincaré series of these moduli stacks can be computed in a similar way. As an application, we obtain a new proof of a joint result of the speaker with Erwan Brugallé, on the maximality on moduli spaces of vector bundles over real algebraic curves.

[ Reference URL ]The Poincaré series of moduli stacks of semistable G-bundles on curves has been computed by Laumon and Rapoport. In this joint work with Melissa Liu, we show that the Hodge-Poincaré series of these moduli stacks can be computed in a similar way. As an application, we obtain a new proof of a joint result of the speaker with Erwan Brugallé, on the maximality on moduli spaces of vector bundles over real algebraic curves.

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

### 2023/11/02

#### Information Mathematics Seminar

16:50-18:35 Room #128 (Graduate School of Math. Sci. Bldg.)

Algorithms for solving lattice problems and their applications (Japanese)

**Masaya Yasuda**(Rikkyo Univ.)Algorithms for solving lattice problems and their applications (Japanese)

[ Abstract ]

In this talk, I introduce lattice algorithms such as LLL and BKZ reduction algorithms,

which are mandatory for solving lattice problems. I also describe how to solve LWE and

NTRU problems using lattice algorithms. In addition, I describe an application of lattice

algorithms for solving the integer factorization problem.

In this talk, I introduce lattice algorithms such as LLL and BKZ reduction algorithms,

which are mandatory for solving lattice problems. I also describe how to solve LWE and

NTRU problems using lattice algorithms. In addition, I describe an application of lattice

algorithms for solving the integer factorization problem.

### 2023/11/01

#### Number Theory Seminar

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

Prismatic realization functor for Shimura varieties of abelian type (English)

**Alex Youcis**(University of Tokyo)Prismatic realization functor for Shimura varieties of abelian type (English)

[ Abstract ]

Shimura varieties are certain classes of schemes which play an important role in various studies of number theory. The Langlands program is one of such examples. While far from known in general, it is expected that Shimura varieties are moduli spaces of certain motives with extra structure. In this talk I discuss joint work with Naoki Imai and Hiroki Kato, which constructs prismatic objects on the integral canonical models of Shimura varieties of abelian type at hyperspecial level. These may be thought of as the prismatic realization of such a hypothetical universal motive. I will also discuss how one can use this object to characterize these integral models, even at finite level.

Shimura varieties are certain classes of schemes which play an important role in various studies of number theory. The Langlands program is one of such examples. While far from known in general, it is expected that Shimura varieties are moduli spaces of certain motives with extra structure. In this talk I discuss joint work with Naoki Imai and Hiroki Kato, which constructs prismatic objects on the integral canonical models of Shimura varieties of abelian type at hyperspecial level. These may be thought of as the prismatic realization of such a hypothetical universal motive. I will also discuss how one can use this object to characterize these integral models, even at finite level.

### 2023/10/31

#### Tuesday Seminar on Topology

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

Pre-registration required. See our seminar webpage.

On Harada Conjecture II (JAPANESE)

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

Pre-registration required. See our seminar webpage.

**Naoki Chigira**(Kumamoto University)On Harada Conjecture II (JAPANESE)

[ Abstract ]

The Character table of finite group has a lot of information about the group. In this talk, we discuss about a conjecture of Koichiro Harada (so called Harada conjecture II) which is related to the product of all irreducible characters and the product of all conjugacy class sizes.

[ Reference URL ]The Character table of finite group has a lot of information about the group. In this talk, we discuss about a conjecture of Koichiro Harada (so called Harada conjecture II) which is related to the product of all irreducible characters and the product of all conjugacy class sizes.

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

#### Classical Analysis

10:30-14:30 Room #126 (Graduate School of Math. Sci. Bldg.)

The Wild Riemann-Hilbert Correspondence via Groupoid Representations (ENGLISH)

The Wild Riemann-Hilbert Correspondence via Groupoid Representations (ENGLISH)

**Benedetta Facciotti**(University of Birmingham) 10:30-11:30The Wild Riemann-Hilbert Correspondence via Groupoid Representations (ENGLISH)

[ Abstract ]

In this talk, through simple examples, I will explain the basic idea behind the Riemann-Hilbert correspondence. It is a correspondence between two different moduli spaces: the de Rham moduli space parametrizing meromorphic differential equations, and the Betti moduli space describing local systems of solutions and the representations of the fundamental group defined by them. We will see why such a correspondence breaks down for higher order poles.

In this talk, through simple examples, I will explain the basic idea behind the Riemann-Hilbert correspondence. It is a correspondence between two different moduli spaces: the de Rham moduli space parametrizing meromorphic differential equations, and the Betti moduli space describing local systems of solutions and the representations of the fundamental group defined by them. We will see why such a correspondence breaks down for higher order poles.

**Nikita Nikolaev**(University of Birmingham) 13:30-14:30The Wild Riemann-Hilbert Correspondence via Groupoid Representations (ENGLISH)

[ Abstract ]

I will explain an approach to extending the Riemann-Hilbert correspondence to the setting of equations with higher-order poles using the representation theory of holomorphic Lie groupoids. Each Riemann-Hilbert problem is associated with a suitable Lie algebroid that is integrable to a holomorphic Lie groupoid that can be explicitly constructed as a blowup of the fundamental groupoid. Then the Riemann-Hilbert correspondence can be formulated in rather familiar Lie theoretic terms as the correspondence between representations of algebroids and groupoids. An advantage of this approach is that groupoid representations can be investigated geometrically. Based on joint work with Benedetta Facciotti (Birmingham) and Marta Mazzocco (Birmingham), as well as joint work with Francis Bischoff (Regina) and Marco Gualtieri (Toronto).

I will explain an approach to extending the Riemann-Hilbert correspondence to the setting of equations with higher-order poles using the representation theory of holomorphic Lie groupoids. Each Riemann-Hilbert problem is associated with a suitable Lie algebroid that is integrable to a holomorphic Lie groupoid that can be explicitly constructed as a blowup of the fundamental groupoid. Then the Riemann-Hilbert correspondence can be formulated in rather familiar Lie theoretic terms as the correspondence between representations of algebroids and groupoids. An advantage of this approach is that groupoid representations can be investigated geometrically. Based on joint work with Benedetta Facciotti (Birmingham) and Marta Mazzocco (Birmingham), as well as joint work with Francis Bischoff (Regina) and Marco Gualtieri (Toronto).

### 2023/10/30

#### Tokyo Probability Seminar

16:00-18:50 Room #126 (Graduate School of Math. Sci. Bldg.)

Quantitative homogenization of interacting particle systems (English)

https://chenlin-gu.github.io/index.html

Wasserstein geometry and Ricci curvature bounds for Poisson spaces (English)

https://lzdsmath.github.io

Curvature Bound of the Dyson Brownian Motion (English)

https://www.durham.ac.uk/staff/kohei-suzuki/

**Chenlin Gu**(Tsinghua University) 16:00-16:50Quantitative homogenization of interacting particle systems (English)

[ Abstract ]

This talk presents that, for a class of interacting particle systems in continuous space, the finite-volume approximations of the bulk diffusion matrix converge at an algebraic rate. The models we consider are reversible with respect to the Poisson measures with constant density, and are of non-gradient type. This approach is inspired by recent progress in the quantitative homogenization of elliptic equations. Along the way, a modified Caccioppoli inequality and a multiscale Poincare inequality are developed, which are of independent interest. The talk is based on a joint work with Arianna Giunti and Jean-Christophe Mourrat.

[ Reference URL ]This talk presents that, for a class of interacting particle systems in continuous space, the finite-volume approximations of the bulk diffusion matrix converge at an algebraic rate. The models we consider are reversible with respect to the Poisson measures with constant density, and are of non-gradient type. This approach is inspired by recent progress in the quantitative homogenization of elliptic equations. Along the way, a modified Caccioppoli inequality and a multiscale Poincare inequality are developed, which are of independent interest. The talk is based on a joint work with Arianna Giunti and Jean-Christophe Mourrat.

https://chenlin-gu.github.io/index.html

**Lorenzo Dello-Schiavio**(Institute of Science and Technology Austria (ISTA)) 17:00-17:50Wasserstein geometry and Ricci curvature bounds for Poisson spaces (English)

[ Abstract ]

Let Υ be the configuration space over a complete and separable metric base space, endowed with the Poisson measure π. We study the geometry of Υ from the point of view of optimal transport and Ricci-lower bounds. To do so, we define a formal Riemannian structure on P_1(Y), the space of probability measures over Υ with finite first moment, and we construct an extended distance W on P_1(Y). The distance W corresponds, in our setting, to the Benamou–Brenier variational formulation of the Wasserstein distance. Our main technical tool is a non-local continuity equation defined via the difference operator on the Poisson space. We show that the closure of the domain of the relative entropy is a complete geodesic space, when endowed with W. We establish non-local infinite-dimensional analogues of results regarding the geometry of the Wasserstein space over a metric measure space with synthetic Ricci curvature bounded below. In particular, we obtain that: (a) the Ornstein–Uhlenbeck semi-group is the gradient flow of the relative entropy; (b) the Poisson space has Ricci curvature bounded below by 1 in the entropic sense; (c) the distance W satisfies an HWI inequality.

Base on joint work arXiv:2303.00398 with Ronan Herry (Rennes 1) and Kohei Suzuki (Durham)

[ Reference URL ]Let Υ be the configuration space over a complete and separable metric base space, endowed with the Poisson measure π. We study the geometry of Υ from the point of view of optimal transport and Ricci-lower bounds. To do so, we define a formal Riemannian structure on P_1(Y), the space of probability measures over Υ with finite first moment, and we construct an extended distance W on P_1(Y). The distance W corresponds, in our setting, to the Benamou–Brenier variational formulation of the Wasserstein distance. Our main technical tool is a non-local continuity equation defined via the difference operator on the Poisson space. We show that the closure of the domain of the relative entropy is a complete geodesic space, when endowed with W. We establish non-local infinite-dimensional analogues of results regarding the geometry of the Wasserstein space over a metric measure space with synthetic Ricci curvature bounded below. In particular, we obtain that: (a) the Ornstein–Uhlenbeck semi-group is the gradient flow of the relative entropy; (b) the Poisson space has Ricci curvature bounded below by 1 in the entropic sense; (c) the distance W satisfies an HWI inequality.

Base on joint work arXiv:2303.00398 with Ronan Herry (Rennes 1) and Kohei Suzuki (Durham)

https://lzdsmath.github.io

**Kohei Suzuki**(Durham University) 18:00-18:50Curvature Bound of the Dyson Brownian Motion (English)

[ Abstract ]

The Dyson Brownian Motion (DBM) is an eigenvalue process of a particular Hermitian matrix-valued Brownian motion introduced by Freeman Dyson in 1962, which has been one of the central subjects in the random matrix theory. In this talk, we study the DBM from a geometric perspective. We show that the infinite particle DBM possesses a lower bound of the Ricci curvature à la Bakry-Émery. As a consequence, we obtain various quantitative estimates of the transition probability of the DBM (e.g., the local spectral gap, the local log-Sobolev, and the dimension-free Harnack inequalities) as well as the characterisation of the DBM as the gradient flow of the Boltzmann entropy in a particular Wasserstein-type space, the latter of which provides a new viewpoint of the Dyson Brownian motion.

[ Reference URL ]The Dyson Brownian Motion (DBM) is an eigenvalue process of a particular Hermitian matrix-valued Brownian motion introduced by Freeman Dyson in 1962, which has been one of the central subjects in the random matrix theory. In this talk, we study the DBM from a geometric perspective. We show that the infinite particle DBM possesses a lower bound of the Ricci curvature à la Bakry-Émery. As a consequence, we obtain various quantitative estimates of the transition probability of the DBM (e.g., the local spectral gap, the local log-Sobolev, and the dimension-free Harnack inequalities) as well as the characterisation of the DBM as the gradient flow of the Boltzmann entropy in a particular Wasserstein-type space, the latter of which provides a new viewpoint of the Dyson Brownian motion.

https://www.durham.ac.uk/staff/kohei-suzuki/

#### Seminar on Geometric Complex Analysis

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

The Nonvanishing problem for varieties with nef anticanonical bundle

https://u-tokyo-ac-jp.zoom.us/meeting/register/tZEqceqsrTIjEtRxenSMdPogvCxlWzAogj5A

**Shin-Ichi Matsumura**(Tohoku Univeristy)The Nonvanishing problem for varieties with nef anticanonical bundle

[ Abstract ]

In the framework of the minimal model program for generalized pairs, the abundance conjecture does not hold. However, interestingly, the generalized nonvanishing conjecture is expected to hold. This conjecture asks whether the canonical divisor of generalized pairs can be represented by an effective divisor in its numerical class. In this talk, we discuss the nonvanishing conjecture for generalized LC pairs in three dimensions and prove that the conjecture is true for the nef anti-canonical divisors.

This talk is based on joint work with V. Lazic, Th. Peternell, N. Tsakanikas, and Z. Xie.

[ Reference URL ]In the framework of the minimal model program for generalized pairs, the abundance conjecture does not hold. However, interestingly, the generalized nonvanishing conjecture is expected to hold. This conjecture asks whether the canonical divisor of generalized pairs can be represented by an effective divisor in its numerical class. In this talk, we discuss the nonvanishing conjecture for generalized LC pairs in three dimensions and prove that the conjecture is true for the nef anti-canonical divisors.

This talk is based on joint work with V. Lazic, Th. Peternell, N. Tsakanikas, and Z. Xie.

https://u-tokyo-ac-jp.zoom.us/meeting/register/tZEqceqsrTIjEtRxenSMdPogvCxlWzAogj5A

### 2023/10/27

#### Colloquium

15:30-16:30 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].

Increasing stability and decreasing instability estimates for an inverse boundary value problem (English)

https://forms.gle/9xDcHfHXFFHPfsKW6

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

**Jenn-Nan Wang**(National Taiwan University)Increasing stability and decreasing instability estimates for an inverse boundary value problem (English)

[ Abstract ]

According to Hadamard’s definition, a well-posed problem satisfies three criteria: existence, uniqueness, and continuous dependence on the data. Most of forward problems (e.g., the boundary value problem or Calderón’s problem) can be proved to be well-posed. However, many inverse problems are known to be ill-posed, for example, the inverse boundary value problem in which one would like to determine unknown parameters from the boundary measurements. The failure of the continuous dependence on the data in Hadamard’s sense makes the feasible determination of unknown parameters rather difficult in practice. However, if one restricts the unknown parameters in a suitable subspace, one can restore the continuous dependence or stability. Nonetheless, the ill-posedness nature of the inverse problem may give rise a logarithmic type modulus of continuity. For Calderón’s problem, such logarithmic stability estimate was derived by Alessandrini and Mandache showed that this estimate is optimal by proving an instability estimate of exponential type. When we consider the time-harmonic equation, it was first proved by Isakov that the stability increases as the frequency increases. In this talk, I would like to discuss a refinement of Mandache’s idea aiming to derive explicitly the dependence of the instability estimate on the frequency. If time allows, I also want to discuss the increasing stability phenomenon from the statistical viewpoint based on the Bayes approach. The aim is to show that the posterior distribution contracts around the true parameter at a rate closely related to the decreasing instability estimate derived above.

[ Reference URL ]According to Hadamard’s definition, a well-posed problem satisfies three criteria: existence, uniqueness, and continuous dependence on the data. Most of forward problems (e.g., the boundary value problem or Calderón’s problem) can be proved to be well-posed. However, many inverse problems are known to be ill-posed, for example, the inverse boundary value problem in which one would like to determine unknown parameters from the boundary measurements. The failure of the continuous dependence on the data in Hadamard’s sense makes the feasible determination of unknown parameters rather difficult in practice. However, if one restricts the unknown parameters in a suitable subspace, one can restore the continuous dependence or stability. Nonetheless, the ill-posedness nature of the inverse problem may give rise a logarithmic type modulus of continuity. For Calderón’s problem, such logarithmic stability estimate was derived by Alessandrini and Mandache showed that this estimate is optimal by proving an instability estimate of exponential type. When we consider the time-harmonic equation, it was first proved by Isakov that the stability increases as the frequency increases. In this talk, I would like to discuss a refinement of Mandache’s idea aiming to derive explicitly the dependence of the instability estimate on the frequency. If time allows, I also want to discuss the increasing stability phenomenon from the statistical viewpoint based on the Bayes approach. The aim is to show that the posterior distribution contracts around the true parameter at a rate closely related to the decreasing instability estimate derived above.

https://forms.gle/9xDcHfHXFFHPfsKW6

### 2023/10/25

#### Number Theory Seminar

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

Geometric Eisenstein Series over the Fargues-Fontaine curve (English)

**Linus Hamann**(Stanford University)Geometric Eisenstein Series over the Fargues-Fontaine curve (English)

[ Abstract ]

Geometric Eisenstein series were first studied extensively by Braverman-Gaitsgory, Laumon, and Drinfeld, in the context of function field geometric Langlands. For a Levi subgroup M inside a connected reductive group G, they are functors which send Hecke eigensheaves on the moduli stack of M-bundles to Hecke eigensheaves on the moduli stack of G-bundles via certain relative compactifications of the moduli stack of P-bundles. We will discuss what this theory has to offer in the context of the recent Fargues-Scholze geometric Langlands program. Namely, motivated by the results in the function field setting, we will explicate what the analogous results tell us in this setting of the Fargues-Scholze program, as well as discuss various consequences for the cohmology of local and global Shimura varieties, via the relation between local Shimura varieties and the p-adic shtukas appearing in the Fargues-Scholze program.

Geometric Eisenstein series were first studied extensively by Braverman-Gaitsgory, Laumon, and Drinfeld, in the context of function field geometric Langlands. For a Levi subgroup M inside a connected reductive group G, they are functors which send Hecke eigensheaves on the moduli stack of M-bundles to Hecke eigensheaves on the moduli stack of G-bundles via certain relative compactifications of the moduli stack of P-bundles. We will discuss what this theory has to offer in the context of the recent Fargues-Scholze geometric Langlands program. Namely, motivated by the results in the function field setting, we will explicate what the analogous results tell us in this setting of the Fargues-Scholze program, as well as discuss various consequences for the cohmology of local and global Shimura varieties, via the relation between local Shimura varieties and the p-adic shtukas appearing in the Fargues-Scholze program.

### 2023/10/24

#### Numerical Analysis Seminar

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

Neural Network-based Enclosure of Solutions to Differential Equations and Reconsideration of the Sub- and Super-solution Method (Japanese)

[ Reference URL ]

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

**Kazuaki Tanaka**(Waseda University)Neural Network-based Enclosure of Solutions to Differential Equations and Reconsideration of the Sub- and Super-solution Method (Japanese)

[ Reference URL ]

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.

Index theory for quarter-plane Toeplitz operators via extended symbols (JAPANESE)

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

Pre-registration required. See our seminar webpage.

**Shin Hayashi**(Aoyama Gakuin University)Index theory for quarter-plane Toeplitz operators via extended symbols (JAPANESE)

[ Abstract ]

We consider index theory for some Toeplitz operators on a discrete quarter-plane. Index theory for such operators has been investigated by Simonenko, Douglas-Howe, Park and index formulas are obtained by Coburn-Douglas-Singer, Duducava. In this talk, we revisit Duducava’s idea and discuss an index formula for quarter-plane Toeplitz operators of two-variable rational matrix function symbols from a geometric viewpoint. By using Gohberg-Krein theory for matrix factorizations and analytic continuation, we see that the symbols of Fredholm quarter-plane Toeplitz operators defined originally on a two-dimensional torus can canonically be extended to some three-sphere, and show that their Fredholm indices coincides with the three-dimensional winding number of extended symbols. If time permits, we briefly mention a contact with a topic in condensed matter physics, called (higher-order) topological insulators.

[ Reference URL ]We consider index theory for some Toeplitz operators on a discrete quarter-plane. Index theory for such operators has been investigated by Simonenko, Douglas-Howe, Park and index formulas are obtained by Coburn-Douglas-Singer, Duducava. In this talk, we revisit Duducava’s idea and discuss an index formula for quarter-plane Toeplitz operators of two-variable rational matrix function symbols from a geometric viewpoint. By using Gohberg-Krein theory for matrix factorizations and analytic continuation, we see that the symbols of Fredholm quarter-plane Toeplitz operators defined originally on a two-dimensional torus can canonically be extended to some three-sphere, and show that their Fredholm indices coincides with the three-dimensional winding number of extended symbols. If time permits, we briefly mention a contact with a topic in condensed matter physics, called (higher-order) topological insulators.

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

### 2023/10/19

#### Information Mathematics Seminar

16:50-18:35 Room #128 (Graduate School of Math. Sci. Bldg.)

Mathematical Aspects of Lattice-Based Cryptography (Japanese)

**Katsuyuki Takashima**(Waseda Univ.)Mathematical Aspects of Lattice-Based Cryptography (Japanese)

[ Abstract ]

I will explain mathematical aspects of lattice-based cryptography.

I will explain mathematical aspects of lattice-based cryptography.

### 2023/10/18

#### Number Theory Seminar

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

On Igusa varieties (English)

**Wansu Kim**(KAIST/University of Tokyo)On Igusa varieties (English)

[ Abstract ]

In this talk, we construct Igusa varieties and study some basic properties in the setting of abelian-type Shimura varieties, as well as in the analogous setting for function fields (over shtuka spaces). The is joint work with Paul Hamacher.

In this talk, we construct Igusa varieties and study some basic properties in the setting of abelian-type Shimura varieties, as well as in the analogous setting for function fields (over shtuka spaces). The is joint work with Paul Hamacher.

### 2023/10/17

#### Numerical Analysis Seminar

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

Structure-preserving schemes for the Cahn-Hilliard equation with dynamic boundary conditions in two spatial dimensions (Japanese)

[ Reference URL ]

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

**Makoto Okumura**(Konan University)Structure-preserving schemes for the Cahn-Hilliard equation with dynamic boundary conditions in two spatial dimensions (Japanese)

[ Reference URL ]

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.

Train track combinatorics and cluster algebras (JAPANESE)

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

Pre-registration required. See our seminar webpage.

**Shunsuke Kano**(MathCCS, Tohoku University)Train track combinatorics and cluster algebras (JAPANESE)

[ Abstract ]

The concepts of train track was introduced by W. P. Thurston to study the measured foliations/laminations and the pseudo-Anosov mapping classes on a surface. In this talk, we translate some concepts of train tracks into the language of cluster algebras using the tropicalization of Goncharov--Shen's potential function. Using this, we translate a combinatorial property of a train track associated with a pseudo-Anosov mapping class into the combinatorial property in cluster algebras, called the sign stability which was introduced by Tsukasa Ishibashi and the speaker.

[ Reference URL ]The concepts of train track was introduced by W. P. Thurston to study the measured foliations/laminations and the pseudo-Anosov mapping classes on a surface. In this talk, we translate some concepts of train tracks into the language of cluster algebras using the tropicalization of Goncharov--Shen's potential function. Using this, we translate a combinatorial property of a train track associated with a pseudo-Anosov mapping class into the combinatorial property in cluster algebras, called the sign stability which was introduced by Tsukasa Ishibashi and the speaker.

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

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