## Seminar information archive

Seminar information archive ～05/21｜Today's seminar 05/22 | Future seminars 05/23～

### 2019/06/11

#### Tuesday Seminar of Analysis

16:50-18:20 Room #128 (Graduate School of Math. Sci. Bldg.)

Solutions to two conjectures in branched transport: stability and regularity of optimal paths (English)

**Antonio De Rosa**(Courant Institute of Mathematical Sciences)Solutions to two conjectures in branched transport: stability and regularity of optimal paths (English)

[ Abstract ]

Models involving branched structures are employed to describe several supply-demand systems such as the structure of the nerves of a leaf, the system of roots of a tree and the nervous or cardiovascular systems. The transportation cost in these models is proportional to a concave power $\alpha \in (0,1)$ of the intensity of the flow. We focus on the stability of the optimal transports, with respect to variations of the source and target measures. The stability was known when $\alpha$ is bigger than a critical threshold, but we prove it for every exponent $\alpha \in (0,1)$ and we provide a counterexample for $\alpha=0$. Thus we completely solve a conjecture of the book Optimal transportation networks by Bernot, Caselles and Morel. Moreover the robustness of our proof allows us to get the stability for more general lower semicontinuous functional. Furthermore, we prove the stability for the mailing problem, which was completely open in the literature, solving another conjecture of the aforementioned book. We use the latter result to show the regularity of the optimal networks. (Joint works with Maria Colombo and Andrea Marchese)

Models involving branched structures are employed to describe several supply-demand systems such as the structure of the nerves of a leaf, the system of roots of a tree and the nervous or cardiovascular systems. The transportation cost in these models is proportional to a concave power $\alpha \in (0,1)$ of the intensity of the flow. We focus on the stability of the optimal transports, with respect to variations of the source and target measures. The stability was known when $\alpha$ is bigger than a critical threshold, but we prove it for every exponent $\alpha \in (0,1)$ and we provide a counterexample for $\alpha=0$. Thus we completely solve a conjecture of the book Optimal transportation networks by Bernot, Caselles and Morel. Moreover the robustness of our proof allows us to get the stability for more general lower semicontinuous functional. Furthermore, we prove the stability for the mailing problem, which was completely open in the literature, solving another conjecture of the aforementioned book. We use the latter result to show the regularity of the optimal networks. (Joint works with Maria Colombo and Andrea Marchese)

### 2019/06/06

#### Information Mathematics Seminar

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

Lattice Cryptography (Japanese)

**Tatsuaki Okamoto**(NTT)Lattice Cryptography (Japanese)

[ Abstract ]

Properties and a construction of lattice cryptography.

Properties and a construction of lattice cryptography.

### 2019/06/05

#### Number Theory Seminar

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

Duality of Drinfeld modules and P-adic properties of Drinfeld modular forms (English)

**Shin Hattori**(Tokyo City University)Duality of Drinfeld modules and P-adic properties of Drinfeld modular forms (English)

[ Abstract ]

Let p be a rational prime, q>1 a p-power and P a non-constant irreducible polynomial in F_q[t]. The notion of Drinfeld modular form is an analogue over F_q(t) of that of elliptic modular form. Numerical computations suggest that Drinfeld modular forms enjoy some P-adic structures comparable to the elliptic analogue, while at present their P-adic properties are less well understood than the p-adic elliptic case. In 1990s, Taguchi established duality theories for Drinfeld modules and also for a certain class of finite flat group schemes called finite v-modules. Using the duality for the latter, we can define a function field analogue of the Hodge-Tate map. In this talk, I will explain how the Taguchi's theory and our Hodge-Tate map yield results on Drinfeld modular forms which are classical to elliptic modular forms e.g. P-adic congruences of Fourier coefficients imply p-adic congruences of weights.

Let p be a rational prime, q>1 a p-power and P a non-constant irreducible polynomial in F_q[t]. The notion of Drinfeld modular form is an analogue over F_q(t) of that of elliptic modular form. Numerical computations suggest that Drinfeld modular forms enjoy some P-adic structures comparable to the elliptic analogue, while at present their P-adic properties are less well understood than the p-adic elliptic case. In 1990s, Taguchi established duality theories for Drinfeld modules and also for a certain class of finite flat group schemes called finite v-modules. Using the duality for the latter, we can define a function field analogue of the Hodge-Tate map. In this talk, I will explain how the Taguchi's theory and our Hodge-Tate map yield results on Drinfeld modular forms which are classical to elliptic modular forms e.g. P-adic congruences of Fourier coefficients imply p-adic congruences of weights.

### 2019/06/04

#### PDE Real Analysis Seminar

10:30-11:30 Room #056 (Graduate School of Math. Sci. Bldg.)

Recent progresses in nonlinear potential theory (English)

**Giuseppe Mingione**(Università di Parma)Recent progresses in nonlinear potential theory (English)

[ Abstract ]

Nonlinear Potential Theory aims at studying the fine properties of solutions to nonlinear, potentially degenerate nonlinear elliptic and parabolic equations in terms of the regularity of the give data. A major model example is here given by the $p$-Laplacean equation

$$ -\operatorname{div}(|Du|^{p-2}Du) = \mu \quad\quad p > 1, $$

where $\mu$ is a Borel measure with finite total mass. When $p = 2$ we find the familiar case of the Poisson equation from which classical Potential Theory stems. Although many basic tools from the classical linear theory are not at hand - most notably: representation formulae via fundamental solutions - many of the classical information can be retrieved for solutions and their pointwise behaviour. In this talk I am going to give a survey of recent results in the field. Especially, I will explain the possibility of getting linear and nonlinear potential estimates for solutions to nonlinear elliptic and parabolic equations which are totally similar to those available in the linear case. I will also draw some parallels with what is nowadays called Nonlinear Calderón-Zygmund theory.

Nonlinear Potential Theory aims at studying the fine properties of solutions to nonlinear, potentially degenerate nonlinear elliptic and parabolic equations in terms of the regularity of the give data. A major model example is here given by the $p$-Laplacean equation

$$ -\operatorname{div}(|Du|^{p-2}Du) = \mu \quad\quad p > 1, $$

where $\mu$ is a Borel measure with finite total mass. When $p = 2$ we find the familiar case of the Poisson equation from which classical Potential Theory stems. Although many basic tools from the classical linear theory are not at hand - most notably: representation formulae via fundamental solutions - many of the classical information can be retrieved for solutions and their pointwise behaviour. In this talk I am going to give a survey of recent results in the field. Especially, I will explain the possibility of getting linear and nonlinear potential estimates for solutions to nonlinear elliptic and parabolic equations which are totally similar to those available in the linear case. I will also draw some parallels with what is nowadays called Nonlinear Calderón-Zygmund theory.

#### Tuesday Seminar on Topology

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

Gluck twist on branched twist spins (JAPANESE)

**Mizuki Fukuda**(Tokyo Gakugei University)Gluck twist on branched twist spins (JAPANESE)

[ Abstract ]

A branched twist spin is an embedded two sphere in the four sphere and it is defined as the set of singular points of a circle action on the four sphere. Gluck showed that the set of isotopy classes of diffeomorphisms on $S^1 \times S^2$ is isomorphic to $Z_2$, and an operation of removing a neighborhood of 2-knot from the four sphere and regluing it by the generator of $Z_2$ is called a Gluck twist. It is known by Pao that the Gluck twist along a branched twist spin does not change the four sphere. In this talk, we give an another proof of Pao’s result by using a decomposition of $S^4$ associated with the circle action, and we show that the set of branched twist spins does not change by the Gluck twist.

A branched twist spin is an embedded two sphere in the four sphere and it is defined as the set of singular points of a circle action on the four sphere. Gluck showed that the set of isotopy classes of diffeomorphisms on $S^1 \times S^2$ is isomorphic to $Z_2$, and an operation of removing a neighborhood of 2-knot from the four sphere and regluing it by the generator of $Z_2$ is called a Gluck twist. It is known by Pao that the Gluck twist along a branched twist spin does not change the four sphere. In this talk, we give an another proof of Pao’s result by using a decomposition of $S^4$ associated with the circle action, and we show that the set of branched twist spins does not change by the Gluck twist.

### 2019/05/29

#### Algebraic Geometry Seminar

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

Minimal log discrepancies of 3-dimensional non-canonical singularities (English)

**Chen Jiang**(Fudan/MSRI)Minimal log discrepancies of 3-dimensional non-canonical singularities (English)

[ Abstract ]

Canonical and terminal singularities, introduced by Reid, appear naturally in minimal model program and play important roles in the birational classification of higher dimensional algebraic varieties. Such singularities are well-understood in dimension 3, while the property of non-canonical singularities is still mysterious. We investigate the difference between canonical and non-canonical singularities via minimal log discrepancies (MLD). We show that there is a gap between MLD of 3-dimensional non-canonical singularities and that of 3-dimensional canonical singularities, which is predicted by a conjecture of Shokurov.

This result on local singularities has applications to global geometry of Calabi–Yau 3-folds. We show that the set of all non-canonical klt Calabi–Yau 3-folds are bounded modulo flops, and the global indices of all klt Calabi–Yau 3-folds are bounded from above.

Canonical and terminal singularities, introduced by Reid, appear naturally in minimal model program and play important roles in the birational classification of higher dimensional algebraic varieties. Such singularities are well-understood in dimension 3, while the property of non-canonical singularities is still mysterious. We investigate the difference between canonical and non-canonical singularities via minimal log discrepancies (MLD). We show that there is a gap between MLD of 3-dimensional non-canonical singularities and that of 3-dimensional canonical singularities, which is predicted by a conjecture of Shokurov.

This result on local singularities has applications to global geometry of Calabi–Yau 3-folds. We show that the set of all non-canonical klt Calabi–Yau 3-folds are bounded modulo flops, and the global indices of all klt Calabi–Yau 3-folds are bounded from above.

#### Number Theory Seminar

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

On supersingular loci of Shimura varieties for quaternion unitary groups of degree 2 (Japanese)

**Yasuhiro Oki**(University of Tokyo)On supersingular loci of Shimura varieties for quaternion unitary groups of degree 2 (Japanese)

### 2019/05/28

#### Seminar on Mathematics for various disciplines

10:30-11:30 Room #056 (Graduate School of Math. Sci. Bldg.)

#### Tuesday Seminar on Topology

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

Exotic four-manifolds via positive factorizations (ENGLISH)

**R. Inanc Baykur**(University of Massachusetts)Exotic four-manifolds via positive factorizations (ENGLISH)

[ Abstract ]

We will discuss several new ideas and techniques for producing positive Dehn twist factorizations of surface mapping classes, which yield novel constructions of various interesting four-manifolds, such as symplectic Calabi-Yau surfaces and exotic rational surfaces, via Lefschetz pencils.

We will discuss several new ideas and techniques for producing positive Dehn twist factorizations of surface mapping classes, which yield novel constructions of various interesting four-manifolds, such as symplectic Calabi-Yau surfaces and exotic rational surfaces, via Lefschetz pencils.

### 2019/05/27

#### Seminar on Geometric Complex Analysis

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

Gluing construction of K3 surfaces (Japanese)

**Takayuki Koike**(Osaka City Univ.)Gluing construction of K3 surfaces (Japanese)

[ Abstract ]

Arnol'd showed the uniqueness of the complex analytic structure of a small neighborhood of an elliptic curve embedded in a surface whose normal bundle satisfies "Diophantine condition" in the Picard variety. By applying this theorem, we construct a K3 surface by holomorphically patching two open complex surfaces obtained as the complements of tubular neighborhoods of anti-canonical curves of blow-ups of the projective planes at general nine points. Our construction has 19 complex dimensional degrees of freedom. For general parameters, the resulting K3 surface is neither Kummer nor projective. By the argument based on the concrete computation of the period map, we also investigate which points in the period domain correspond to K3 surfaces obtained by such construction. (Based on joint work with Takato Uehara)

Arnol'd showed the uniqueness of the complex analytic structure of a small neighborhood of an elliptic curve embedded in a surface whose normal bundle satisfies "Diophantine condition" in the Picard variety. By applying this theorem, we construct a K3 surface by holomorphically patching two open complex surfaces obtained as the complements of tubular neighborhoods of anti-canonical curves of blow-ups of the projective planes at general nine points. Our construction has 19 complex dimensional degrees of freedom. For general parameters, the resulting K3 surface is neither Kummer nor projective. By the argument based on the concrete computation of the period map, we also investigate which points in the period domain correspond to K3 surfaces obtained by such construction. (Based on joint work with Takato Uehara)

### 2019/05/24

#### Colloquium

15:30-16:30 Room #002 (Graduate School of Math. Sci. Bldg.)

### 2019/05/23

#### Information Mathematics Seminar

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

Problems of Bitcoin, other Cryptocurrencies, and Blochchains (Japanese)

**Tatsuaki Okamoto**Problems of Bitcoin, other Cryptocurrencies, and Blochchains (Japanese)

[ Abstract ]

Explanation of problems of bitcoin.

Explanation of problems of bitcoin.

### 2019/05/22

#### Algebraic Geometry Seminar

15:30-17:00 Room #122 (Graduate School of Math. Sci. Bldg.)

Bogomolov type vanishing on three-dimensional Mori fiber spaces in positive characteristic

**Tatsuro Kawakami**(Tokyo)Bogomolov type vanishing on three-dimensional Mori fiber spaces in positive characteristic

[ Abstract ]

In characteristic zero, cotangent bundle of n(>1)-dimensional smooth projective varieties does not contain a big line bundle. This is a part of Bogomolov vanishing and this vanishing plays an important role in the proof of Miyaoka-Yau inequality. In positive characteristic, it is known that Bogomolov vanishing does not hold. There exists a general type surface whose cotangent bundle contains an ample line bundle. So, it is natural to ask when Bogomolov type vanishing holds in positive characteristic. In this talk, I discuss Bogomolov type vanishing on three-dimensional Mori fiber spaces in positive characteristic.

In characteristic zero, cotangent bundle of n(>1)-dimensional smooth projective varieties does not contain a big line bundle. This is a part of Bogomolov vanishing and this vanishing plays an important role in the proof of Miyaoka-Yau inequality. In positive characteristic, it is known that Bogomolov vanishing does not hold. There exists a general type surface whose cotangent bundle contains an ample line bundle. So, it is natural to ask when Bogomolov type vanishing holds in positive characteristic. In this talk, I discuss Bogomolov type vanishing on three-dimensional Mori fiber spaces in positive characteristic.

### 2019/05/21

#### Tuesday Seminar on Topology

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

On the dealternating number and the alternation number (ENGLISH)

**Maria de los Angeles Guevara**(Osaka City University)On the dealternating number and the alternation number (ENGLISH)

[ Abstract ]

Links can be divided into alternating and non-alternating depending on if they possess an alternating diagram or not. After the proof of the Tait flype conjecture on alternating links, it became an important question to ask how a non-alternating link is “close to” alternating links. The dealternating and alternation numbers, which are invariants introduced by C. Adams et al. and A. Kawauchi, respectively, can deal with this question. By definitions, for any link, its alternation number is less than or equal to its dealternating number. It is known that in general the equality does not hold. However, in general, it is not easy to show a gap between these invariants. In this seminar, we will show some results regarding these invariants. In particular, for each pair of positive integers, we will construct infinitely many knots, which have dealternating and alternation numbers determined for these integers. Therefore, an arbitrary gap between the values of these invariants will be obtained.

Links can be divided into alternating and non-alternating depending on if they possess an alternating diagram or not. After the proof of the Tait flype conjecture on alternating links, it became an important question to ask how a non-alternating link is “close to” alternating links. The dealternating and alternation numbers, which are invariants introduced by C. Adams et al. and A. Kawauchi, respectively, can deal with this question. By definitions, for any link, its alternation number is less than or equal to its dealternating number. It is known that in general the equality does not hold. However, in general, it is not easy to show a gap between these invariants. In this seminar, we will show some results regarding these invariants. In particular, for each pair of positive integers, we will construct infinitely many knots, which have dealternating and alternation numbers determined for these integers. Therefore, an arbitrary gap between the values of these invariants will be obtained.

### 2019/05/20

#### Seminar on Geometric Complex Analysis

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

Cohomology and normal reduction numbers of normal surface singularities (Japanese)

**Tomohiro Okuma**(Yamagata Univ.)Cohomology and normal reduction numbers of normal surface singularities (Japanese)

[ Abstract ]

The normal reduction number of a normal surface singularity relates the maximal degree of the generators of associated graded algebra for certain line bundles on resolution spaces. We show fundamental properties of this invariant and formulas for some special cases. This talk is based on the joint work with Kei-ichi Watanabe and Ken-ichi Yoshida.

The normal reduction number of a normal surface singularity relates the maximal degree of the generators of associated graded algebra for certain line bundles on resolution spaces. We show fundamental properties of this invariant and formulas for some special cases. This talk is based on the joint work with Kei-ichi Watanabe and Ken-ichi Yoshida.

### 2019/05/16

#### Information Mathematics Seminar

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

Bitcoin: Revolution of Electronic Money (Japanese)

**Tatsuaki Okamoto**(NTT)Bitcoin: Revolution of Electronic Money (Japanese)

[ Abstract ]

Explanation of the system of bitcoin

Explanation of the system of bitcoin

### 2019/05/15

#### FMSP Lectures

17:30-18:30 Room #122 (Graduate School of Math. Sci. Bldg.)

'Oumuamua, the Gömböc and the Pebbles of Mars (ENGLISH)

http://fmsp.ms.u-tokyo.ac.jp/FMSPLectures_Domokos.pdf

**Gábor Domokos**(Hungarian Academy of Sciences/Budapest University of Technology and Economics)'Oumuamua, the Gömböc and the Pebbles of Mars (ENGLISH)

[ Abstract ]

In this talk I will concentrate on two examples from planetary science, which made the headlines in recent years to highlight the power and significance of nonlinear geometric partial differential equations (PDEs) explaining puzzles presented by Nature. One key link between PDE theory of shape evolution and natural phenomena is the Gömböc, the first mono-monostatic object whose existence was first conjectured by V.I. Arnold in 1995. I will explain the connection and illustrate the process how mathematical models of Nature may be identified.

[ Reference URL ]In this talk I will concentrate on two examples from planetary science, which made the headlines in recent years to highlight the power and significance of nonlinear geometric partial differential equations (PDEs) explaining puzzles presented by Nature. One key link between PDE theory of shape evolution and natural phenomena is the Gömböc, the first mono-monostatic object whose existence was first conjectured by V.I. Arnold in 1995. I will explain the connection and illustrate the process how mathematical models of Nature may be identified.

http://fmsp.ms.u-tokyo.ac.jp/FMSPLectures_Domokos.pdf

#### FMSP Lectures

15:00-17:20 Room #122 (Graduate School of Math. Sci. Bldg.)

Part 1 : Categorical analogues of surface singularities

Part 2 : Prismatic Homology (ENGLISH)

http://fmsp.ms.u-tokyo.ac.jp/FMSPLectures_Carter.pdf

**J. Scott Carter**(University of South Alabama / Osaka City University)Part 1 : Categorical analogues of surface singularities

Part 2 : Prismatic Homology (ENGLISH)

[ Abstract ]

Part 1 :

Isotopy classes of surfaces that are embedded in 3-space can be described as a free 4-category that has one object and one weakly invertible arrow. That description coincides with a fundamental higher homotopy group. The surface singularities that correspond to cusps and optimal points on folds can be used to develop categorical analogues of swallow-tails and horizontal cusps. In this talk, the 4-category will be constructed from the ground up, and the general structure will be described.

Part 2 :

A qualgebra is a set that has two binary operations whose relationships to each other are similar to the relations between group multiplication and conjugation. The axioms themselves are described in terms of isotopies of knotted trivalent graphs and the handle-body knots that are represented. The moves naturally live in prisms. By using a generalization of the tensor product of chain complexes, a homology theory is presented that encapsulates these axioms and the higher order relations between them. We show how to use this homology theory to give a solution a system of tensor equations related to the Yang-Baxter relation.

[ Reference URL ]Part 1 :

Isotopy classes of surfaces that are embedded in 3-space can be described as a free 4-category that has one object and one weakly invertible arrow. That description coincides with a fundamental higher homotopy group. The surface singularities that correspond to cusps and optimal points on folds can be used to develop categorical analogues of swallow-tails and horizontal cusps. In this talk, the 4-category will be constructed from the ground up, and the general structure will be described.

Part 2 :

A qualgebra is a set that has two binary operations whose relationships to each other are similar to the relations between group multiplication and conjugation. The axioms themselves are described in terms of isotopies of knotted trivalent graphs and the handle-body knots that are represented. The moves naturally live in prisms. By using a generalization of the tensor product of chain complexes, a homology theory is presented that encapsulates these axioms and the higher order relations between them. We show how to use this homology theory to give a solution a system of tensor equations related to the Yang-Baxter relation.

http://fmsp.ms.u-tokyo.ac.jp/FMSPLectures_Carter.pdf

#### Algebraic Geometry Seminar

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

On quasi-log canonical pairs

(Japanese)

**Osamu Fujino**(Osaka)On quasi-log canonical pairs

(Japanese)

[ Abstract ]

The notion of quasi-log canonical pairs was introduced by Florin Ambro. It is a kind of generalizations of that of log canonical pairs. Now we know that quasi-log canonical pairs are ubiquitous in the theory of minimal models. In this talk, I will explain some basic properties and examples of quasi-log canonical pairs. I will also discuss some new developments around quasi-log canonical pairs. Some parts are joint works with Haidong Liu.

The notion of quasi-log canonical pairs was introduced by Florin Ambro. It is a kind of generalizations of that of log canonical pairs. Now we know that quasi-log canonical pairs are ubiquitous in the theory of minimal models. In this talk, I will explain some basic properties and examples of quasi-log canonical pairs. I will also discuss some new developments around quasi-log canonical pairs. Some parts are joint works with Haidong Liu.

#### Operator Algebra Seminars

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

Combinatorial aspects of Borel functions

**Takayuki Kihara**(Nagoya University)Combinatorial aspects of Borel functions

### 2019/05/14

#### Tuesday Seminar on Topology

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

Diagrammatic Algebra (ENGLISH)

**J. Scott Carter**(University of South Alabama, Osaka City University)Diagrammatic Algebra (ENGLISH)

[ Abstract ]

Three main ideas will be explored. First, a higher dimensional category (a category that has arrows, double arrows, triple arrows, and quadruple arrows) that is based upon the axioms of a Frobenius algebra will be outlined. Then these structures will be promoted into one higher dimension so that braiding can be introduced. Second, relationships between braiding and multiplication will be studied from a homological perspective. Third, the next order relations will be used to formulate a system of abstract tensor equations that are analogous to the Yang-Baxter relation. In this way, a broad outline of the notion of diagrammatic algebra will be presented.

Three main ideas will be explored. First, a higher dimensional category (a category that has arrows, double arrows, triple arrows, and quadruple arrows) that is based upon the axioms of a Frobenius algebra will be outlined. Then these structures will be promoted into one higher dimension so that braiding can be introduced. Second, relationships between braiding and multiplication will be studied from a homological perspective. Third, the next order relations will be used to formulate a system of abstract tensor equations that are analogous to the Yang-Baxter relation. In this way, a broad outline of the notion of diagrammatic algebra will be presented.

### 2019/05/13

#### Seminar on Geometric Complex Analysis

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

Some Bonnet--Myers Type Theorems for Transverse Ricci Solitons on Complete Sasaki Manifolds (Japanese)

**Homare Tadano**(Tokyo Univ. of Science)Some Bonnet--Myers Type Theorems for Transverse Ricci Solitons on Complete Sasaki Manifolds (Japanese)

[ Abstract ]

The aim of this talk is to discuss the compactness of complete Ricci solitons and its generalizations. Ricci solitons were introduced by R. Hamilton in 1982 and are natural generalizations of Einstein manifolds. They correspond to self-similar solutions to the Ricci flow and often arise as singularity models of the flow. The importance of Ricci solitons was demonstrated by G. Perelman, where they played crucial roles in his affirmative resolution of the Poincare conjecture.

In this talk, after we review basic facts on Ricci solitons, I would like to introduce some Bonnet--Myers type theorems for complete Ricci solitons. Our results generalize the previous Bonnet--Myers type theorems due to W. Ambrose (1957), J. Cheeger, M. Gromov, and M. Taylor (1982), M. Fernandez-Lopez and E. Garcia-Rio (2008), M. Limoncu (2010, 2012), Z. Qian (1997), Y. Soylu (2017), and G. Wei and W. Wylie (2009). Moreover, I would also like to extend such Bonnet--Myers type theorems to the case of transverse Ricci solitons on complete Sasaki manifolds. Our results generalize the previous Bonnet--Myers type theorems for complete Sasaki manifolds due to I. Hasegawa and M. Seino (1981) and Y. Nitta (2009).

The aim of this talk is to discuss the compactness of complete Ricci solitons and its generalizations. Ricci solitons were introduced by R. Hamilton in 1982 and are natural generalizations of Einstein manifolds. They correspond to self-similar solutions to the Ricci flow and often arise as singularity models of the flow. The importance of Ricci solitons was demonstrated by G. Perelman, where they played crucial roles in his affirmative resolution of the Poincare conjecture.

In this talk, after we review basic facts on Ricci solitons, I would like to introduce some Bonnet--Myers type theorems for complete Ricci solitons. Our results generalize the previous Bonnet--Myers type theorems due to W. Ambrose (1957), J. Cheeger, M. Gromov, and M. Taylor (1982), M. Fernandez-Lopez and E. Garcia-Rio (2008), M. Limoncu (2010, 2012), Z. Qian (1997), Y. Soylu (2017), and G. Wei and W. Wylie (2009). Moreover, I would also like to extend such Bonnet--Myers type theorems to the case of transverse Ricci solitons on complete Sasaki manifolds. Our results generalize the previous Bonnet--Myers type theorems for complete Sasaki manifolds due to I. Hasegawa and M. Seino (1981) and Y. Nitta (2009).

#### Numerical Analysis Seminar

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

Iterative refinement for symmetric eigenvalue problems (Japanese)

**Kensuke Aishima**(Hosei University)Iterative refinement for symmetric eigenvalue problems (Japanese)

### 2019/05/09

#### Information Mathematics Seminar

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

Advances in Theory of Cryptography (Japanese)

**Tatsuaki Okamoto**(NTT)Advances in Theory of Cryptography (Japanese)

[ Abstract ]

Introduction to ZK-SNARK and UC.

Introduction to ZK-SNARK and UC.

### 2019/05/08

#### Algebraic Geometry Seminar

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

On Minimal model theory for log canonical pairs with big boundary divisors

**Kenta Hashizume**(Tokyo)On Minimal model theory for log canonical pairs with big boundary divisors

[ Abstract ]

In 2010, Birkar--Cascini--Hacon--McKernan established the minimal model theory for Kawamata log terminal pairs with big boundary divisors, and a lot of theorems in the birational geometry are

proved by applying this result. It is expected that this result can be generalized to log canonical pairs. Currently, it is known that the minimal model theory for log canonical pairs can be reduced to the case of big boundary divisors. In this talk, we introduce a partial generalization of the result by Birkar--Cascini--Hacon--McKernan. Roughly speaking, we generalized their result to lc pairs with big boundary divisors having only small lc centers. We also explain another generalization, which is originally announced by Hu, and we discuss termination of log minimal model program in a spacial case. This is a joint work with Zhengyu Hu, and the work is in progress.

In 2010, Birkar--Cascini--Hacon--McKernan established the minimal model theory for Kawamata log terminal pairs with big boundary divisors, and a lot of theorems in the birational geometry are

proved by applying this result. It is expected that this result can be generalized to log canonical pairs. Currently, it is known that the minimal model theory for log canonical pairs can be reduced to the case of big boundary divisors. In this talk, we introduce a partial generalization of the result by Birkar--Cascini--Hacon--McKernan. Roughly speaking, we generalized their result to lc pairs with big boundary divisors having only small lc centers. We also explain another generalization, which is originally announced by Hu, and we discuss termination of log minimal model program in a spacial case. This is a joint work with Zhengyu Hu, and the work is in progress.

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