## Seminar information archive

Seminar information archive ～02/20｜Today's seminar 02/21 | Future seminars 02/22～

### 2019/07/09

#### Tuesday Seminar on Topology

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

Mod 2 cohomology of moduli stacks of real vector bundles (ENGLISH)

**Florent Schaffhauser**(Université de Strasbourg)Mod 2 cohomology of moduli stacks of real vector bundles (ENGLISH)

[ Abstract ]

The rational cohomology ring of the moduli stack of holomorphic vector bundles of fixed rank and degree over a compact Riemann surface was studied by Atiyah and Bott using tools of differential geometry and algebraic topology: they found generators of that ring and computed its Poincaré series. In joint work with Chiu-Chu Melissa Liu, we study in a similar way the mod 2 cohomology ring of the moduli stack of real vector bundles of fixed topological type over a compact Riemann surface with real structure. The goal of the talk is to explain the principle of that computation, emphasizing the analogies and differences between the real and complex cases, and discuss applications of the method. In particular, we provide explicit generators of mod 2 cohomology rings of moduli stacks of vector bundles over a real algebraic curve.

The rational cohomology ring of the moduli stack of holomorphic vector bundles of fixed rank and degree over a compact Riemann surface was studied by Atiyah and Bott using tools of differential geometry and algebraic topology: they found generators of that ring and computed its Poincaré series. In joint work with Chiu-Chu Melissa Liu, we study in a similar way the mod 2 cohomology ring of the moduli stack of real vector bundles of fixed topological type over a compact Riemann surface with real structure. The goal of the talk is to explain the principle of that computation, emphasizing the analogies and differences between the real and complex cases, and discuss applications of the method. In particular, we provide explicit generators of mod 2 cohomology rings of moduli stacks of vector bundles over a real algebraic curve.

#### Algebraic Geometry Seminar

13:00-14:30 Room #122 (Graduate School of Math. Sci. Bldg.)

Construction of non-Kähler Calabi-Yau 3-folds by smoothing normal crossing varieties (TBA)

**Taro Sano**(Kobe university)Construction of non-Kähler Calabi-Yau 3-folds by smoothing normal crossing varieties (TBA)

[ Abstract ]

It is an open problem whether there are only finitely many diffeomorphism types of projective Calabi-Yau 3-folds. Kawamata--Namikawa developed log deformation theory of normal crossing Calabi-Yau varieties. As an application of their result, one can construct examples of Calabi-Yau manifolds by smoothing SNC varieties. In this talk, I will explain how to construct examples of non-Kähler Calabi-Yau 3-folds with arbitrarily large 2nd Betti numbers. If time permits, I will also explain an example of involutions on a family of K3 surfaces which do not lift biregularly to the total space. This is based on joint work with Kenji Hashimoto.

It is an open problem whether there are only finitely many diffeomorphism types of projective Calabi-Yau 3-folds. Kawamata--Namikawa developed log deformation theory of normal crossing Calabi-Yau varieties. As an application of their result, one can construct examples of Calabi-Yau manifolds by smoothing SNC varieties. In this talk, I will explain how to construct examples of non-Kähler Calabi-Yau 3-folds with arbitrarily large 2nd Betti numbers. If time permits, I will also explain an example of involutions on a family of K3 surfaces which do not lift biregularly to the total space. This is based on joint work with Kenji Hashimoto.

### 2019/07/08

#### Numerical Analysis Seminar

16:50-18:20 Room #056 (Graduate School of Math. Sci. Bldg.)

Parameter estimation and discretization errors for ordinary differential models (Japanese)

**Takeru Matsuda**(University of Tokyo)Parameter estimation and discretization errors for ordinary differential models (Japanese)

#### Seminar on Geometric Complex Analysis

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

A Riemann-Roch theorem on a weighted infinite graph (Japanese)

**Hiroshi Kaneko**(Tokyo University of Science)A Riemann-Roch theorem on a weighted infinite graph (Japanese)

[ Abstract ]

A Riemann-Roch theorem on a connected finite graph was initiated by M. Baker and S. Norine, where connected graph with finite vertices was investigated and unit weight was given on each edge and vertex of the graph. Since a counterpart of the lowest exponents of the complex variable in the Laurent series was proposed as divisor for the Riemann-Roch theorem on graph, its relationships with tropical geometry were highlighted earlier than other complex analytical observations on graphs. On the other hand, M. Baker and F. Shokrieh revealed tight relationships between chip-firing games and potential theory on graphs, by characterizing reduced divisors on graphs as the solution to an energy minimization problem. The objective of this talk is to establish a Riemann-Roch theorem on an edge-weighted infinite graph. We introduce vertex weight assigned by the given weights of adjacent edges other than the units for expression of divisors and assume finiteness of total mass of graph. This is a joint work with A. Atsuji.

A Riemann-Roch theorem on a connected finite graph was initiated by M. Baker and S. Norine, where connected graph with finite vertices was investigated and unit weight was given on each edge and vertex of the graph. Since a counterpart of the lowest exponents of the complex variable in the Laurent series was proposed as divisor for the Riemann-Roch theorem on graph, its relationships with tropical geometry were highlighted earlier than other complex analytical observations on graphs. On the other hand, M. Baker and F. Shokrieh revealed tight relationships between chip-firing games and potential theory on graphs, by characterizing reduced divisors on graphs as the solution to an energy minimization problem. The objective of this talk is to establish a Riemann-Roch theorem on an edge-weighted infinite graph. We introduce vertex weight assigned by the given weights of adjacent edges other than the units for expression of divisors and assume finiteness of total mass of graph. This is a joint work with A. Atsuji.

### 2019/07/05

#### Algebraic Geometry Seminar

10:30-12:00 Room #123 (Graduate School of Math. Sci. Bldg.)

Durfee-type inequality for complete intersection surface singularities

**Makoto Enokizono**(Tokyo university of science)Durfee-type inequality for complete intersection surface singularities

[ Abstract ]

Durfee's negativity conjecture says that the signature of the Milnor fiber of a 2-dimensional isolated complete intersection singularity is always negative. In this talk, I will explain that this conjecture is true (more precisely, the signature is bounded above by the negative number determined by the geometric genus, the embedding dimension and the number of irreducible components of the exceptional set of the minimal resolution) by using the theory of invariants of fibered surfaces. If time permits, I will explain the higher dimensional analogue of Durfee's conjecture for isolated complete intersection singularities.

Durfee's negativity conjecture says that the signature of the Milnor fiber of a 2-dimensional isolated complete intersection singularity is always negative. In this talk, I will explain that this conjecture is true (more precisely, the signature is bounded above by the negative number determined by the geometric genus, the embedding dimension and the number of irreducible components of the exceptional set of the minimal resolution) by using the theory of invariants of fibered surfaces. If time permits, I will explain the higher dimensional analogue of Durfee's conjecture for isolated complete intersection singularities.

### 2019/07/04

#### Information Mathematics Seminar

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

Isogeny-Based Cryptography (Japanese)

**Katsuyuki Takashima**(Mitsubishi Electric Co./Kyushu Univ.)Isogeny-Based Cryptography (Japanese)

[ Abstract ]

Explanation of the isogeny-based cryptography

Explanation of the isogeny-based cryptography

### 2019/07/03

#### Operator Algebra Seminars

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

Tensor product decompositions and rigidity of full factors

**Amine Marrakchi**(RIMS, Kyoto University)Tensor product decompositions and rigidity of full factors

#### Number Theory Seminar

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

Explicit calculation of values of the regulator maps on a certain type of Kummer surfaces (Japanese)

**Ken Sato**(University of Tokyo)Explicit calculation of values of the regulator maps on a certain type of Kummer surfaces (Japanese)

### 2019/07/02

#### Tuesday Seminar on Topology

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

Brane coproducts and their applications (JAPANESE)

**Shun Wakatsuki**(The University of Tokyo)Brane coproducts and their applications (JAPANESE)

[ Abstract ]

The loop coproduct is a coproduct on the homology of the free loop space of a Poincaré duality space (or more generally a Gorenstein space). In this talk, I will introduce two kinds of brane coproducts which are generalizations of the loop coproduct to the homology of a sphere space (i.e. the mapping space from a sphere). Their constructions are based on the finiteness of the dimensions of mapping spaces in some sense. As an application, I will show the vanishing of some cup products on sphere spaces by comparing these two brane coproducts. This gives a generalization of a result of Menichi for the case of free loop spaces.

The loop coproduct is a coproduct on the homology of the free loop space of a Poincaré duality space (or more generally a Gorenstein space). In this talk, I will introduce two kinds of brane coproducts which are generalizations of the loop coproduct to the homology of a sphere space (i.e. the mapping space from a sphere). Their constructions are based on the finiteness of the dimensions of mapping spaces in some sense. As an application, I will show the vanishing of some cup products on sphere spaces by comparing these two brane coproducts. This gives a generalization of a result of Menichi for the case of free loop spaces.

### 2019/07/01

#### Seminar on Geometric Complex Analysis

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

BCOV invariant and birational equivalence (English)

**Yeping Zhang**(Kyoto Univ.)BCOV invariant and birational equivalence (English)

[ Abstract ]

Bershadsky, Cecotti, Ooguri and Vafa constructed a real valued invariant for Calabi-Yau manifolds, which is now called BCOV invariant. Now we consider a pair (X,Y), where X is a Kaehler manifold and $Y ¥subseteq X$ is a canonical divisor. In this talk, we extend the BCOV invariant to such pairs. The extended BCOV invariant is well-behaved under birational equivalence. We expect that these considerations may eventually lead to a positive answer to Yoshikawa's conjecture that the BCOV invariant for Calabi-Yau threefold is a birational invariant.

Bershadsky, Cecotti, Ooguri and Vafa constructed a real valued invariant for Calabi-Yau manifolds, which is now called BCOV invariant. Now we consider a pair (X,Y), where X is a Kaehler manifold and $Y ¥subseteq X$ is a canonical divisor. In this talk, we extend the BCOV invariant to such pairs. The extended BCOV invariant is well-behaved under birational equivalence. We expect that these considerations may eventually lead to a positive answer to Yoshikawa's conjecture that the BCOV invariant for Calabi-Yau threefold is a birational invariant.

#### Colloquium of mathematical sciences and society

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

#### Numerical Analysis Seminar

16:50-18:20 Room #117 (Graduate School of Math. Sci. Bldg.)

The effect of preventing scale formation by ceramic balls and its effect on the human body (Japanese)

**Hideo Kawarada**(AMSOK)The effect of preventing scale formation by ceramic balls and its effect on the human body (Japanese)

#### Mathematical Biology Seminar

11:00-12:00 Room #123 (Graduate School of Math. Sci. Bldg.)

Taylor's Law of Fluctuation Scaling

https://www.rockefeller.edu/our-scientists/heads-of-laboratories/940-joel-e-cohen/

**Joel E. Cohen**(The Rockefeller University and Columbia University)Taylor's Law of Fluctuation Scaling

[ Abstract ]

A family of nonnegative random variables is said to obey Taylor's law when the variance is proportional to some power b of the mean. For example, in the family of exponential distributions, if the mean is m, then the variance is m^2, so the family of exponential distributions obeys Taylor's law exactly with b=2. Many stochastic processes and the prime numbers obey Taylor's law (exactly or asymptotically). Thousands of empirical illustrations of Taylor's law have been published in many different fields including ecology, demography, finance (stock and currency trading), cancer biology, genetics, fisheries, forestry, meteorology, agriculture, physics, cell biology, computer network engineering, and number theory. This survey talk will review some empirical and theoretical results and open problems about Taylor's law, including recently proved versions of Taylor's law for nonnegative stable laws with infinite mean.

[ Reference URL ]A family of nonnegative random variables is said to obey Taylor's law when the variance is proportional to some power b of the mean. For example, in the family of exponential distributions, if the mean is m, then the variance is m^2, so the family of exponential distributions obeys Taylor's law exactly with b=2. Many stochastic processes and the prime numbers obey Taylor's law (exactly or asymptotically). Thousands of empirical illustrations of Taylor's law have been published in many different fields including ecology, demography, finance (stock and currency trading), cancer biology, genetics, fisheries, forestry, meteorology, agriculture, physics, cell biology, computer network engineering, and number theory. This survey talk will review some empirical and theoretical results and open problems about Taylor's law, including recently proved versions of Taylor's law for nonnegative stable laws with infinite mean.

https://www.rockefeller.edu/our-scientists/heads-of-laboratories/940-joel-e-cohen/

### 2019/06/28

#### Algebraic Geometry Seminar

15:30-17:00 Room #118 (Graduate School of Math. Sci. Bldg.)

Rational curves on prime Fano 3-folds (TBA)

**Sho Tanimoto**(Kumamoto)Rational curves on prime Fano 3-folds (TBA)

[ Abstract ]

One of important topics in algebraic geometry is the space of rational curves, e.g., the dimension and the number of components of the moduli spaces of rational curves on an algebraic variety X. One of interesting situations where this question is extensively studied is when X is a Fano variety since in this case X is rationally connected so that it does contain a lots of rational curves. In this talk I will talk about my joint work with Brian Lehmann which settles this problem for most Fano 3-folds of Picard rank 1, e.g., a general quartic 3-fold in P^4, and our approach is inspired by Manin’s conjecture which predicts the asymptotic formula for the counting function of rational points on a Fano variety. In particular we systematically use geometric invariants in Manin’s conjecture which have been studied by many mathematicians including Brian and me.

One of important topics in algebraic geometry is the space of rational curves, e.g., the dimension and the number of components of the moduli spaces of rational curves on an algebraic variety X. One of interesting situations where this question is extensively studied is when X is a Fano variety since in this case X is rationally connected so that it does contain a lots of rational curves. In this talk I will talk about my joint work with Brian Lehmann which settles this problem for most Fano 3-folds of Picard rank 1, e.g., a general quartic 3-fold in P^4, and our approach is inspired by Manin’s conjecture which predicts the asymptotic formula for the counting function of rational points on a Fano variety. In particular we systematically use geometric invariants in Manin’s conjecture which have been studied by many mathematicians including Brian and me.

#### Colloquium

15:30-16:30 Room #056 (Graduate School of Math. Sci. Bldg.)

### 2019/06/27

#### Information Mathematics Seminar

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

The Evolution of Elliptic Curve Cryptography (Japanese)

**Katsuyuki Takashima**(Mitsubishi Electric Co./Kyushu Univ.)The Evolution of Elliptic Curve Cryptography (Japanese)

[ Abstract ]

Explanation of the evolution of elliptic curve cryptography

Explanation of the evolution of elliptic curve cryptography

### 2019/06/26

#### Operator Algebra Seminars

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

### 2019/06/25

#### Tuesday Seminar on Topology

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

Geometry of symplectic log Calabi-Yau surfaces (ENGLISH)

**Tian-Jun Li**(University of Minnesota)Geometry of symplectic log Calabi-Yau surfaces (ENGLISH)

[ Abstract ]

This is a survey on the geometry of symplectic log Calabi-Yau surfaces, which are the symplectic analogues of Looijenga pairs. We address the classification up to symplectic deformation, the relations between symplectic circular sequences and anti-canonical sequences, contact trichotomy, and symplectic fillings. This is a joint work with Cheuk Yu Mak.

This is a survey on the geometry of symplectic log Calabi-Yau surfaces, which are the symplectic analogues of Looijenga pairs. We address the classification up to symplectic deformation, the relations between symplectic circular sequences and anti-canonical sequences, contact trichotomy, and symplectic fillings. This is a joint work with Cheuk Yu Mak.

### 2019/06/24

#### Seminar on Geometric Complex Analysis

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

A certain holomorphic invariant and its applications (Japanese)

**Atsushi Yamamori**(Kogakuin University)A certain holomorphic invariant and its applications (Japanese)

[ Abstract ]

In this talk, we first explain a Bergman geometric proof of inequivalence of the unit ball and the bidisk. In this proof, the homogeneity of the domains plays a substantial role. We next explain a recent attempt to extend our method for non-homogeneous cases.

In this talk, we first explain a Bergman geometric proof of inequivalence of the unit ball and the bidisk. In this proof, the homogeneity of the domains plays a substantial role. We next explain a recent attempt to extend our method for non-homogeneous cases.

### 2019/06/21

#### Information Mathematics Seminar

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

Functional Encryption (Japanese)

**Tatsuaki Okamoto**(NTT)Functional Encryption (Japanese)

[ Abstract ]

Explanation of various functional encryptions.

Explanation of various functional encryptions.

### 2019/06/20

#### Applied Analysis

16:00-17:30 Room #118 (Graduate School of Math. Sci. Bldg.)

#### Mathematical Biology Seminar

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

Diffusion limit for the partner model at the critical value (ENGLISH)

**Eric Foxall**(University of Alberta)Diffusion limit for the partner model at the critical value (ENGLISH)

[ Abstract ]

The partner model is a stochastic SIS model of infection spread over a dynamic network of monogamous partnerships. In previous work, Edwards, Foxall and van den Driessche identify a threshold in parameter space for spread of the infection and show the time to extinction of the infection is of order log(N) below the threshold, where N is population size, and grows exponentially in N above the

threshold. Later, Foxall shows the time to extinction at threshold is of order sqrt(N). Here we go further and derive a single-variable diffusion limit for the number of infectious individuals rescaled by sqrt(N) in both population and time, and show convergence in distribution of the rescaled extinction time. Since the model has effectively four variables and two relevant time scales, the proof features a succession of probability estimates to control trajectories, as well as an averaging result to contend with the fast partnership dynamics.

The partner model is a stochastic SIS model of infection spread over a dynamic network of monogamous partnerships. In previous work, Edwards, Foxall and van den Driessche identify a threshold in parameter space for spread of the infection and show the time to extinction of the infection is of order log(N) below the threshold, where N is population size, and grows exponentially in N above the

threshold. Later, Foxall shows the time to extinction at threshold is of order sqrt(N). Here we go further and derive a single-variable diffusion limit for the number of infectious individuals rescaled by sqrt(N) in both population and time, and show convergence in distribution of the rescaled extinction time. Since the model has effectively four variables and two relevant time scales, the proof features a succession of probability estimates to control trajectories, as well as an averaging result to contend with the fast partnership dynamics.

### 2019/06/19

#### Operator Algebra Seminars

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

Type classification of extreme quantized characters

**Ryosuke Sato**(Nagoya University)Type classification of extreme quantized characters

#### Algebraic Geometry Seminar

15:30-17:00 Room #118 (Graduate School of Math. Sci. Bldg.)

A pencil of Enriques surfaces with non-algebraic integral Hodge classes (TBA)

**Fumiaki Suzuki**(UIC)A pencil of Enriques surfaces with non-algebraic integral Hodge classes (TBA)

[ Abstract ]

The integral Hodge conjecture is the statement that the integral Hodge classes are algebraic on smooth complex projective varieties. It is known that the conjecture can fail in general. There are two types of counterexamples, ones with non-algebraic integral Hodge classes of torsion-type and of non-torsion type, the first of which were given by Atiyah-Hirzebruch and Kollar, respectively.

In this talk, we exhibit a pencil of Enriques surfaces defined over Q with non-algebraic integral Hodge classes of non-torsion type. This construction relates to certain questions concerning rational points of algebraic varieties.

This gives the first example of a threefold with the trivial Chow group of zero-cycles on which the integral Hodge conjecture fails. As an application, we construct a fourfold which gives the negative answer to a classical question on the universality of the Abel-Jacobi maps.

This is a joint work with John Christian Ottem.

The integral Hodge conjecture is the statement that the integral Hodge classes are algebraic on smooth complex projective varieties. It is known that the conjecture can fail in general. There are two types of counterexamples, ones with non-algebraic integral Hodge classes of torsion-type and of non-torsion type, the first of which were given by Atiyah-Hirzebruch and Kollar, respectively.

In this talk, we exhibit a pencil of Enriques surfaces defined over Q with non-algebraic integral Hodge classes of non-torsion type. This construction relates to certain questions concerning rational points of algebraic varieties.

This gives the first example of a threefold with the trivial Chow group of zero-cycles on which the integral Hodge conjecture fails. As an application, we construct a fourfold which gives the negative answer to a classical question on the universality of the Abel-Jacobi maps.

This is a joint work with John Christian Ottem.

### 2019/06/18

#### Seminar on Probability and Statistics

11:00-12:10 Room #052 (Graduate School of Math. Sci. Bldg.)

Gaussian and bootstrap approximations of high-dimensional U-statistics with applications and extensions ※変更の可能性あり

**Xiaohui Chen**(University of Illinois at Urbana–Champaign)Gaussian and bootstrap approximations of high-dimensional U-statistics with applications and extensions ※変更の可能性あり

[ Abstract ]

We shall first discuss the Gaussian approximation of high-dimensional and non-degenerate U-statistics of order two under the supremum norm. A two-step Gaussian approximation procedure that does not impose structural assumptions on the data distribution is proposed. Subject to mild moment conditions on the kernel, we establish the explicit rate of convergence that decays polynomially in sample size for a high-dimensional scaling limit, where the dimension can be much larger than the sample size. We also provide computable approximation methods for the quantiles of the maxima of centered U-statistics. Specifically, we provide a unified perspective for the empirical, the randomly reweighted, and the multiplier bootstraps as randomly reweighted quadratic forms, all asymptotically valid and inferentially first-order equivalent in high-dimensions.

The bootstrap methods are applied on statistical applications for high-dimensional non-Gaussian data including: (i) principled and data-dependent tuning parameter selection for regularized estimation of the covariance matrix and its related functionals; (ii) simultaneous inference for the covariance and rank correlation matrices. In particular, for the thresholded covariance matrix estimator with the bootstrap selected tuning parameter, we show that the Gaussian-like convergence rates can be achieved for heavy-tailed data, which are less conservative than those obtained by the Bonferroni technique that

ignores the dependency in the underlying data distribution. In addition, we also show that even for subgaussian distributions, error bounds of the bootstrapped thresholded covariance matrix estimator can be much tighter than those of the minimax estimator with a universal threshold.

Time permitting, we will discuss some extensions to the infinite-dimensional version (i.e., U-processes of increasing complexity) and to the randomized inference via the incomplete U-statistics whose computational cost can be made independent of the order.

We shall first discuss the Gaussian approximation of high-dimensional and non-degenerate U-statistics of order two under the supremum norm. A two-step Gaussian approximation procedure that does not impose structural assumptions on the data distribution is proposed. Subject to mild moment conditions on the kernel, we establish the explicit rate of convergence that decays polynomially in sample size for a high-dimensional scaling limit, where the dimension can be much larger than the sample size. We also provide computable approximation methods for the quantiles of the maxima of centered U-statistics. Specifically, we provide a unified perspective for the empirical, the randomly reweighted, and the multiplier bootstraps as randomly reweighted quadratic forms, all asymptotically valid and inferentially first-order equivalent in high-dimensions.

The bootstrap methods are applied on statistical applications for high-dimensional non-Gaussian data including: (i) principled and data-dependent tuning parameter selection for regularized estimation of the covariance matrix and its related functionals; (ii) simultaneous inference for the covariance and rank correlation matrices. In particular, for the thresholded covariance matrix estimator with the bootstrap selected tuning parameter, we show that the Gaussian-like convergence rates can be achieved for heavy-tailed data, which are less conservative than those obtained by the Bonferroni technique that

ignores the dependency in the underlying data distribution. In addition, we also show that even for subgaussian distributions, error bounds of the bootstrapped thresholded covariance matrix estimator can be much tighter than those of the minimax estimator with a universal threshold.

Time permitting, we will discuss some extensions to the infinite-dimensional version (i.e., U-processes of increasing complexity) and to the randomized inference via the incomplete U-statistics whose computational cost can be made independent of the order.

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