## Seminar information archive

Seminar information archive ～02/18｜Today's seminar 02/19 | Future seminars 02/20～

### 2015/11/18

#### FMSP Lectures

15:00-16:00,16:30-17:00 Room #大講義室 (Graduate School of Math. Sci. Bldg.)

Crossroads of symplectic rigidity and flexibility (ENGLISH)

http://faculty.ms.u-tokyo.ac.jp/Eliashberg201511.html

**Yakov Eliashberg**(Stanford University)Crossroads of symplectic rigidity and flexibility (ENGLISH)

[ Abstract ]

The development of flexible and rigid sides of symplectic and contact topology towards each other shaped this subject since its inception, and continues shaping its modern development.

In the series of lectures I will discuss the history of this struggle, as well as describe recent breakthroughs on the flexible side.

[ Reference URL ]The development of flexible and rigid sides of symplectic and contact topology towards each other shaped this subject since its inception, and continues shaping its modern development.

In the series of lectures I will discuss the history of this struggle, as well as describe recent breakthroughs on the flexible side.

http://faculty.ms.u-tokyo.ac.jp/Eliashberg201511.html

#### FMSP Lectures

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

Discretising systematically integrable systems (ENGLISH)

http://fmsp.ms.u-tokyo.ac.jp/FMSPLectures_Ramani1118.pdf

**Alfred Ramani**(Ecole Polytechnique)Discretising systematically integrable systems (ENGLISH)

[ Abstract ]

We present various methods for discretising integrable systerms inspired by the works of Hirota and Mickens. We apply these methods to the systematical discretisation of Painlevé equations.

[ Reference URL ]We present various methods for discretising integrable systerms inspired by the works of Hirota and Mickens. We apply these methods to the systematical discretisation of Painlevé equations.

http://fmsp.ms.u-tokyo.ac.jp/FMSPLectures_Ramani1118.pdf

#### Mathematical Biology Seminar

14:55-16:40 Room #128 (Graduate School of Math. Sci. Bldg.)

Spatial population dynamics as a point pattern dynamics (JAPANESE)

http://www.ics.nara-wu.ac.jp/jp/staff/takasu.html

**Fugo TAKASU**(Nara Women's University)Spatial population dynamics as a point pattern dynamics (JAPANESE)

[ Abstract ]

Spatial population dynamics has been conventionally described as

dynamical system where population size (or population density) changes

with time over space as a continuous "real-valued" variable; these are

often given as partial differential equations as reaction-diffusion

models. In this approach, we implicitly assume infinitely large

population thereby population size changes smoothly and

deterministically. In reality, however, a population is a collection of

a certain number of individuals each of which gives birth or dies with

some stochasticity in a space and the population size as the number of

individuals is "integer-valued". In this talk, I introduce an approach

to reconstruct conventional spatial population dynamics in terms of

point pattern dynamics as a stochastic process. I discuss how to

mathematically describe such spatial stochastic processes using the

moments of increasing order of dimension; densities of points, pairs,

and triplets, etc. are described by integro-differential equations.

Quantification of a point pattern is the key issue here. As examples, I

introduce spatial epidemic SIS and SIR models as point pattern dynamics;

each individual has a certain "mark" depending on its health status; a

snapshot of individuals’ distribution over space is represented by a

marked point pattern and this marked point pattern dynamically changes

with time.

[ Reference URL ]Spatial population dynamics has been conventionally described as

dynamical system where population size (or population density) changes

with time over space as a continuous "real-valued" variable; these are

often given as partial differential equations as reaction-diffusion

models. In this approach, we implicitly assume infinitely large

population thereby population size changes smoothly and

deterministically. In reality, however, a population is a collection of

a certain number of individuals each of which gives birth or dies with

some stochasticity in a space and the population size as the number of

individuals is "integer-valued". In this talk, I introduce an approach

to reconstruct conventional spatial population dynamics in terms of

point pattern dynamics as a stochastic process. I discuss how to

mathematically describe such spatial stochastic processes using the

moments of increasing order of dimension; densities of points, pairs,

and triplets, etc. are described by integro-differential equations.

Quantification of a point pattern is the key issue here. As examples, I

introduce spatial epidemic SIS and SIR models as point pattern dynamics;

each individual has a certain "mark" depending on its health status; a

snapshot of individuals’ distribution over space is represented by a

marked point pattern and this marked point pattern dynamically changes

with time.

http://www.ics.nara-wu.ac.jp/jp/staff/takasu.html

#### Seminar on Probability and Statistics

17:00-18:10 Room #056 (Graduate School of Math. Sci. Bldg.)

Order flow intensities for limit order book modelling

**Ioane Muni Toke**(University of New Caledonia)Order flow intensities for limit order book modelling

[ Abstract ]

Limit order books are at the core of electronic financial markets. Mathematical models of limit order books use point processes to model the arrival of limit, market and cancellation orders in the order book, but it is not clear what a "good" parametric model for the intensities of these point processes should be.

In the first part of the talk, we show that despite their simplicity basic Poisson processes can be used to accurately model a few features of the order book that more advanced models reproduce with volume-dependent intensities.

In the second part of the talk we present ongoing investigations in a more advanced statistical modelling of these order flow intensities using in particular normal mixture distributions and exponential models.

Limit order books are at the core of electronic financial markets. Mathematical models of limit order books use point processes to model the arrival of limit, market and cancellation orders in the order book, but it is not clear what a "good" parametric model for the intensities of these point processes should be.

In the first part of the talk, we show that despite their simplicity basic Poisson processes can be used to accurately model a few features of the order book that more advanced models reproduce with volume-dependent intensities.

In the second part of the talk we present ongoing investigations in a more advanced statistical modelling of these order flow intensities using in particular normal mixture distributions and exponential models.

### 2015/11/17

#### Tuesday Seminar on Topology

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

Topology of some three-dimensional singularities related to algebraic geometry (ENGLISH)

**Atsuko Katanaga**(Shinshu University)Topology of some three-dimensional singularities related to algebraic geometry (ENGLISH)

[ Abstract ]

In this talk, we deal with hypersurface isolated singularities. First, we will recall

some topological results of singularities. Next, we will sketch the classification of

singularities in algebraic geometry. Finally, we will focus on the three-dimensional

case and discuss some results obtained so far.

In this talk, we deal with hypersurface isolated singularities. First, we will recall

some topological results of singularities. Next, we will sketch the classification of

singularities in algebraic geometry. Finally, we will focus on the three-dimensional

case and discuss some results obtained so far.

#### Number Theory Seminar

18:00-19:00 Room #117 (Graduate School of Math. Sci. Bldg.)

The Tamagawa number formula over function fields. (English)

**Dennis Gaitsgory**(Harvard University & IHES)The Tamagawa number formula over function fields. (English)

[ Abstract ]

Let G be a semi-simple and simply connected group and X an algebraic curve. We consider $Bun_G(X)$, the moduli space of G-bundles on X. In their celebrated paper, Atiyah and Bott gave a formula for the cohomology of $Bun_G$, namely $H^*(Bun_G)=Sym(H_*(X)\otimes V)$, where V is the space of generators for $H^*_G(pt)$. When we take our ground field to be a finite field, the Atiyah-Bott formula implies the Tamagawa number conjecture for the function field of X.

The caveat here is that the A-B proof uses the interpretation of $Bun_G$ as the space of connection forms modulo gauge transformations, and thus only works over complex numbers (but can be extend to any field of characteristic zero). In the talk we will outline an algebro-geometric proof that works over any ground field. As its main geometric ingredient, it uses the fact that the space of rational maps from X to G is homologically contractible. Because of the nature of the latter statement, the proof necessarily uses tools from higher category theory. So, it can be regarded as an example how the latter can be used to prove something concrete: a construction at the level of 2-categories leads to an equality of numbers.

Let G be a semi-simple and simply connected group and X an algebraic curve. We consider $Bun_G(X)$, the moduli space of G-bundles on X. In their celebrated paper, Atiyah and Bott gave a formula for the cohomology of $Bun_G$, namely $H^*(Bun_G)=Sym(H_*(X)\otimes V)$, where V is the space of generators for $H^*_G(pt)$. When we take our ground field to be a finite field, the Atiyah-Bott formula implies the Tamagawa number conjecture for the function field of X.

The caveat here is that the A-B proof uses the interpretation of $Bun_G$ as the space of connection forms modulo gauge transformations, and thus only works over complex numbers (but can be extend to any field of characteristic zero). In the talk we will outline an algebro-geometric proof that works over any ground field. As its main geometric ingredient, it uses the fact that the space of rational maps from X to G is homologically contractible. Because of the nature of the latter statement, the proof necessarily uses tools from higher category theory. So, it can be regarded as an example how the latter can be used to prove something concrete: a construction at the level of 2-categories leads to an equality of numbers.

### 2015/11/16

#### Seminar on Geometric Complex Analysis

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

Towards the complex geometry of Teichmuller space with extremal length (English)

**Hideki Miyachi**(Osaka University)Towards the complex geometry of Teichmuller space with extremal length (English)

[ Abstract ]

In this talk, in aiming for studying a relation between the topological aspect and the complex analytical aspect of Teichmuller space, I will discuss a complex analytic property of extremal length functions. More precisely, I will give a concrete formula of the Levi form of the extremal length functions for ``generic” measured foliations and show that the reciprocal of the extremal length function is plurisuperharmonic. As a corollary, I will give alternate proofs of S. Krushkal results that the distance function for the Teichmuller distance is plurisubharmonic, and Teichmuller space is hyperconvex. If time permits, I will give a topological description of the Levi form with using the Thurston's symplectic form.

In this talk, in aiming for studying a relation between the topological aspect and the complex analytical aspect of Teichmuller space, I will discuss a complex analytic property of extremal length functions. More precisely, I will give a concrete formula of the Levi form of the extremal length functions for ``generic” measured foliations and show that the reciprocal of the extremal length function is plurisuperharmonic. As a corollary, I will give alternate proofs of S. Krushkal results that the distance function for the Teichmuller distance is plurisubharmonic, and Teichmuller space is hyperconvex. If time permits, I will give a topological description of the Levi form with using the Thurston's symplectic form.

#### FMSP Lectures

15:00-16:00,16:30-17:00 Room #大講義室 (Graduate School of Math. Sci. Bldg.)

Crossroads of symplectic rigidity and flexibility (ENGLISH)

http://faculty.ms.u-tokyo.ac.jp/Eliashberg201511.html

**Yakov Eliashberg**(Stanford University)Crossroads of symplectic rigidity and flexibility (ENGLISH)

[ Abstract ]

The development of flexible and rigid sides of symplectic and contact topology towards each other shaped this subject since its inception, and continues shaping its modern development.

In the series of lectures I will discuss the history of this struggle, as well as describe recent breakthroughs on the flexible side.

[ Reference URL ]The development of flexible and rigid sides of symplectic and contact topology towards each other shaped this subject since its inception, and continues shaping its modern development.

In the series of lectures I will discuss the history of this struggle, as well as describe recent breakthroughs on the flexible side.

http://faculty.ms.u-tokyo.ac.jp/Eliashberg201511.html

#### Algebraic Geometry Seminar

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

Counting curves on surface in Calabi-Yau threefolds and the proof of S-duality modularity conjecture (English)

**Artan Sheshmani**(IPMU/ Ohio State University)Counting curves on surface in Calabi-Yau threefolds and the proof of S-duality modularity conjecture (English)

[ Abstract ]

I will talk about recent joint works with Amin Gholampour, Richard Thomas and Yukinobu Toda, on an algebraic-geometric proof of the S-duality conjecture in superstring theory, made formerly by physicists Gaiotto, Strominger, Yin, regarding the modularity of DT invariants of sheaves supported on hyperplane sections of the quintic Calabi-Yau threefold. Our strategy is to first use degeneration and localization techniques to reduce the threefold theory to a certain intersection theory over the relative Hilbert scheme of points on surfaces and then prove modularity; More precisely, together with Gholampour we have proven that the generating series, associated to the top intersection numbers of the Hilbert scheme of points, relative to an effective divisor, on a smooth quasi-projective surface is a modular form. This is a generalization of the result of Okounkov-Carlsson, where they used representation theory and the machinery of vertex operators to prove this statement for absolute Hilbert schemes. These intersection numbers eventually, together with the generating series of Noether-Lefschetz numbers as I will explain, will provide the ingredients to achieve a complete algebraic-geometric proof of S-duality modularity conjecture.

I will talk about recent joint works with Amin Gholampour, Richard Thomas and Yukinobu Toda, on an algebraic-geometric proof of the S-duality conjecture in superstring theory, made formerly by physicists Gaiotto, Strominger, Yin, regarding the modularity of DT invariants of sheaves supported on hyperplane sections of the quintic Calabi-Yau threefold. Our strategy is to first use degeneration and localization techniques to reduce the threefold theory to a certain intersection theory over the relative Hilbert scheme of points on surfaces and then prove modularity; More precisely, together with Gholampour we have proven that the generating series, associated to the top intersection numbers of the Hilbert scheme of points, relative to an effective divisor, on a smooth quasi-projective surface is a modular form. This is a generalization of the result of Okounkov-Carlsson, where they used representation theory and the machinery of vertex operators to prove this statement for absolute Hilbert schemes. These intersection numbers eventually, together with the generating series of Noether-Lefschetz numbers as I will explain, will provide the ingredients to achieve a complete algebraic-geometric proof of S-duality modularity conjecture.

### 2015/11/14

#### Harmonic Analysis Komaba Seminar

13:00-18:00 Room #128 (Graduate School of Math. Sci. Bldg.)

### 2015/11/13

#### FMSP Lectures

15:00-16:00,16:30-17:30 Room #大講義室 (Graduate School of Math. Sci. Bldg.)

Crossroads of symplectic rigidity and flexibility (ENGLISH)

http://faculty.ms.u-tokyo.ac.jp/Eliashberg201511.html

**Yakov Eliashberg**(Stanford University)Crossroads of symplectic rigidity and flexibility (ENGLISH)

[ Abstract ]

The development of flexible and rigid sides of symplectic and contact topology towards each other shaped this subject since its inception, and continues shaping its modern development.

In the series of lectures I will discuss the history of this struggle, as well as describe recent breakthroughs on the flexible side.

[ Reference URL ]The development of flexible and rigid sides of symplectic and contact topology towards each other shaped this subject since its inception, and continues shaping its modern development.

In the series of lectures I will discuss the history of this struggle, as well as describe recent breakthroughs on the flexible side.

http://faculty.ms.u-tokyo.ac.jp/Eliashberg201511.html

#### Geometry Colloquium

10:00-11:30 Room #126 (Graduate School of Math. Sci. Bldg.)

Closed random mapping tori are asymmetric

(Japanese)

**MASAI, Hideyoshi**(The University of Tokyo)Closed random mapping tori are asymmetric

(Japanese)

[ Abstract ]

We consider random walks on the mapping class group of closed surfaces and mapping tori of such random mapping classes. It has been shown that such random mapping tori admit hyperbolic structure, and hence their symmetry groups are finite groups. In this talk we prove that the symmetry group of random mapping tori are trivial.

We consider random walks on the mapping class group of closed surfaces and mapping tori of such random mapping classes. It has been shown that such random mapping tori admit hyperbolic structure, and hence their symmetry groups are finite groups. In this talk we prove that the symmetry group of random mapping tori are trivial.

### 2015/11/10

#### Tuesday Seminar on Topology

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

Topological T-duality for "Real" circle bundle (JAPANESE)

**Kiyonori Gomi**(Shinshu University)Topological T-duality for "Real" circle bundle (JAPANESE)

[ Abstract ]

Topological T-duality originates from T-duality in superstring theory,

and is first studied by Bouwkneght, Evslin and Mathai. The duality

basically consists of two parts: The first part is that, for any pair

of a principal circle bundle with `H-flux', there is another `T-dual'

pair on the same base space. The second part states that the twisted

K-groups of the total spaces of principal circle bundles in duality

are isomorphic under degree shift. This is the most simple topological

T-duality following Bunke and Schick, and there are a number of

generalizations. The generalization I will talk about is a topological

T-duality for "Real" circle bundles, motivated by T-duality in type II

orbifold string theory. In this duality, a variant of Z_2-equivariant

K-theory appears.

Topological T-duality originates from T-duality in superstring theory,

and is first studied by Bouwkneght, Evslin and Mathai. The duality

basically consists of two parts: The first part is that, for any pair

of a principal circle bundle with `H-flux', there is another `T-dual'

pair on the same base space. The second part states that the twisted

K-groups of the total spaces of principal circle bundles in duality

are isomorphic under degree shift. This is the most simple topological

T-duality following Bunke and Schick, and there are a number of

generalizations. The generalization I will talk about is a topological

T-duality for "Real" circle bundles, motivated by T-duality in type II

orbifold string theory. In this duality, a variant of Z_2-equivariant

K-theory appears.

### 2015/11/09

#### Algebraic Geometry Seminar

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

3-dimensional McKay correspondence (English)

**Yukari Ito**(Nagoya University)3-dimensional McKay correspondence (English)

[ Abstract ]

The original McKay correspondence is a relation between group theory of a finite subgroup G of SL(2,C) and geometry of the minimal resolution of the quotient singularity by G, and was generalized several ways. In particular, 3-dimensional generalization was extended to derived categorical eqivalence and the G-Hilbert scheme was useful to explain the correspondence. However, most results hold only for abelian subgroups. In this talk, I would like to introduce an iterated G-Hilbert scheme and show more geometrical McKay correspondence for non-abelian subgroups.

The original McKay correspondence is a relation between group theory of a finite subgroup G of SL(2,C) and geometry of the minimal resolution of the quotient singularity by G, and was generalized several ways. In particular, 3-dimensional generalization was extended to derived categorical eqivalence and the G-Hilbert scheme was useful to explain the correspondence. However, most results hold only for abelian subgroups. In this talk, I would like to introduce an iterated G-Hilbert scheme and show more geometrical McKay correspondence for non-abelian subgroups.

### 2015/11/05

#### Algebraic Geometry Seminar

15:30-17:00 Room #126 (Graduate School of Math. Sci. Bldg.)

Compact moduli of marked noncommutative del Pezzo surfaces via quivers (English)

**Shinnosuke Okawa**(Osaka University)Compact moduli of marked noncommutative del Pezzo surfaces via quivers (English)

[ Abstract ]

I will introduce certain GIT construction via quivers of compactified moduli spaces of marked noncommutative del Pezzo surfaces. For projective plane, quadric surface, and those of degree 3, 2, 1, we obtain projective toric varieties of dimension 2, 3, 8, 9, 10, respectively. Then I will discuss relations with deformation theory of abelian categories, blow-up of noncommutative projective planes, and three-block exceptional collections due to Karpov and Nogin. This talk is based on joint works in progress with Tarig Abdelgadir and Kazushi Ueda.

I will introduce certain GIT construction via quivers of compactified moduli spaces of marked noncommutative del Pezzo surfaces. For projective plane, quadric surface, and those of degree 3, 2, 1, we obtain projective toric varieties of dimension 2, 3, 8, 9, 10, respectively. Then I will discuss relations with deformation theory of abelian categories, blow-up of noncommutative projective planes, and three-block exceptional collections due to Karpov and Nogin. This talk is based on joint works in progress with Tarig Abdelgadir and Kazushi Ueda.

#### Applied Analysis

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

The effect of a line with fast diffusion on Fisher-KPP propagation (ENGLISH)

**Henri Berestycki**(EHESS)The effect of a line with fast diffusion on Fisher-KPP propagation (ENGLISH)

[ Abstract ]

I will present a system of equations describing the effect of inclusion of a line (the "road") with fast diffusion on biological invasions in the plane. Outside of the road, the propagation is of the classical Fisher-KPP type. We find that past a certain precise threshold for the ratio of diffusivity coefficients, the presence of the road enhances the speed of global propagation. I will discuss several further effects such as transport or reaction on the road. I will also discuss the influence of various parameters on the asymptotic behaviour of the invasion speed and shape. I report here on results from a series of joint works with Jean-Michel Roquejoffre and Luca Rossi.

I will present a system of equations describing the effect of inclusion of a line (the "road") with fast diffusion on biological invasions in the plane. Outside of the road, the propagation is of the classical Fisher-KPP type. We find that past a certain precise threshold for the ratio of diffusivity coefficients, the presence of the road enhances the speed of global propagation. I will discuss several further effects such as transport or reaction on the road. I will also discuss the influence of various parameters on the asymptotic behaviour of the invasion speed and shape. I report here on results from a series of joint works with Jean-Michel Roquejoffre and Luca Rossi.

### 2015/11/02

#### Seminar on Geometric Complex Analysis

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

A class of non-Kahler manifolds (Japanese)

**Shimobe Hirokazu**(Osaka Univ.)A class of non-Kahler manifolds (Japanese)

[ Abstract ]

We consider a special case of compact complex manifolds which are said to be super strongly Gauduchon manifolds. A super strongly Gauduchon manifold is a complex manifold with a super strongly Gauduchon metric. We mainly consider non-Kähler super strongly Gauduchon manifolds. We give a cohomological condition for a compact complex manifold to have a super strongly Gauduchon metric, and give examples of non-trivial super strongly Gauduchon manifolds from nil-manifolds. We also consider its stability under small deformations and proper modifications of super strongly Gauduchon manifolds.

We consider a special case of compact complex manifolds which are said to be super strongly Gauduchon manifolds. A super strongly Gauduchon manifold is a complex manifold with a super strongly Gauduchon metric. We mainly consider non-Kähler super strongly Gauduchon manifolds. We give a cohomological condition for a compact complex manifold to have a super strongly Gauduchon metric, and give examples of non-trivial super strongly Gauduchon manifolds from nil-manifolds. We also consider its stability under small deformations and proper modifications of super strongly Gauduchon manifolds.

#### Tokyo Probability Seminar

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

Concentrations for the travel cost of the simple random walk in random potentials

**Naoki Kubota**(College of Science and Tenology, Nihon University)Concentrations for the travel cost of the simple random walk in random potentials

### 2015/10/30

#### FMSP Lectures

15:00-16:15 Room #128 (Graduate School of Math. Sci. Bldg.)

Asymptotic behaviour of a nonlocal logistic equation (ENGLISH)

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

**Arnaud Ducrot**(University of Bordeaux)Asymptotic behaviour of a nonlocal logistic equation (ENGLISH)

[ Abstract ]

In this talk we consider a nonlocal logistic equation endowed with periodic boundary conditions modelling the motion of cells. This equation takes into account birth and death process using a simple logistic effect while the motion of particles follows a nonlocal Darcy law with a smooth kernel.

We first investigate the well-posedness of the problem before investigating the long time behaviour of the solutions. The lack of asymptotic compactness of the semiflow is overcome by using a Young measure framework. Using a suitable energy functional, we

establish the convergence of the solutions with respect to the Young measure topology.

[ Reference URL ]In this talk we consider a nonlocal logistic equation endowed with periodic boundary conditions modelling the motion of cells. This equation takes into account birth and death process using a simple logistic effect while the motion of particles follows a nonlocal Darcy law with a smooth kernel.

We first investigate the well-posedness of the problem before investigating the long time behaviour of the solutions. The lack of asymptotic compactness of the semiflow is overcome by using a Young measure framework. Using a suitable energy functional, we

establish the convergence of the solutions with respect to the Young measure topology.

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

#### FMSP Lectures

16:30-17:45 Room #128 (Graduate School of Math. Sci. Bldg.)

How should a drop of liquid on a smooth curved surface move in zero gravity? (ENGLISH)

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

**Peter Bates**(Michigan State University)How should a drop of liquid on a smooth curved surface move in zero gravity? (ENGLISH)

[ Abstract ]

Questions such as this may be formulated as questions regarding solutions to nonlinear evolutionary partial differential equations having a small coefficient on the leading order derivative term. Evolutionary partial differential equations may be regarded as (semi-) dynamical systems in an infinite-dimensional space. An abstract theorem is proved giving the existence of an invariant manifold for a semi-dynamical system when an approximately invariant manifold exists with a certain topological nondegeneracy condition in a neighborhood. This is then used to prove the existence of eternal solutions to the nonlinear PDE and answer the question about the motion of a droplet on a curved manifold. The abstract theorem extends fundamental work of Hirsch-Pugh-Shub and Fenichel on the perturbation of invariant manifolds from the 1970's to infinite-dimensional semi-dynamical systems.

This represents joint work with Kening Lu and Chongchun Zeng.

[ Reference URL ]Questions such as this may be formulated as questions regarding solutions to nonlinear evolutionary partial differential equations having a small coefficient on the leading order derivative term. Evolutionary partial differential equations may be regarded as (semi-) dynamical systems in an infinite-dimensional space. An abstract theorem is proved giving the existence of an invariant manifold for a semi-dynamical system when an approximately invariant manifold exists with a certain topological nondegeneracy condition in a neighborhood. This is then used to prove the existence of eternal solutions to the nonlinear PDE and answer the question about the motion of a droplet on a curved manifold. The abstract theorem extends fundamental work of Hirsch-Pugh-Shub and Fenichel on the perturbation of invariant manifolds from the 1970's to infinite-dimensional semi-dynamical systems.

This represents joint work with Kening Lu and Chongchun Zeng.

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

### 2015/10/28

#### Mathematical Biology Seminar

14:55-16:40 Room #128演習室 (Graduate School of Math. Sci. Bldg.)

Application of stochastic control theory to r/K selection theory affiliation (JAPANESE)

**Ryo Oizumi**(Ministry of Health, Labour and Welfare)Application of stochastic control theory to r/K selection theory affiliation (JAPANESE)

#### Operator Algebra Seminars

16:45-18:15 Room #118 (Graduate School of Math. Sci. Bldg.)

Uniformly locally finite metric spaces and Folner type conditions

**Hiroki Sako**(Niigata University)Uniformly locally finite metric spaces and Folner type conditions

### 2015/10/27

#### Number Theory Seminar

18:00-19:00 Room #002 (Graduate School of Math. Sci. Bldg.)

On the period conjecture of Gross-Deligne for fibrations (English)

**Masanori Asakura**(Hokkaido University)On the period conjecture of Gross-Deligne for fibrations (English)

[ Abstract ]

The period conjecture of Gross-Deligne asserts that the periods of algebraic varieties with complex multiplication are products of values of the gamma function at rational numbers. This is proved for CM elliptic curves by Lerch-Chowla-Selberg, and for abelian varieties by Shimura-Deligne-Anderson. However the question in the general case is still open. In this talk, we verify an alternating variant of the period conjecture for the cohomology of fibrations with relative multiplication. The proof relies on the Saito-Terasoma product formula for epsilon factors of integrable regular singular connections and the Riemann-Roch-Hirzebruch theorem. This is a joint work with Javier Fresan.

The period conjecture of Gross-Deligne asserts that the periods of algebraic varieties with complex multiplication are products of values of the gamma function at rational numbers. This is proved for CM elliptic curves by Lerch-Chowla-Selberg, and for abelian varieties by Shimura-Deligne-Anderson. However the question in the general case is still open. In this talk, we verify an alternating variant of the period conjecture for the cohomology of fibrations with relative multiplication. The proof relies on the Saito-Terasoma product formula for epsilon factors of integrable regular singular connections and the Riemann-Roch-Hirzebruch theorem. This is a joint work with Javier Fresan.

#### Tuesday Seminar on Topology

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

Heegaard Floer homology for graphs (JAPANESE)

**Yuanyuan Bao**(The University of Tokyo)Heegaard Floer homology for graphs (JAPANESE)

[ Abstract ]

Ozsváth and Szabó defined the Heegaard Floer homology (HF) for a closed oriented 3-manifold. The definition was then generalized to links embedded in a 3-manifold and the manifolds with boundary (sutured and bordered manifolds). In the case of links, there is a beautiful combinatorial way to rewrite the original definition of HF, which was defined on a Heegaard diagram of the given link, by using grid diagram. For a balanced bipartite graph, we defined its Heegaard diagram and the HF for it. Around the same time, Harvey and O’Donnol defined the combinatorial HF for transverse graphs (see the definition in [arXiv:1506.04785v1]). In this talk, we compare these two methods.

Ozsváth and Szabó defined the Heegaard Floer homology (HF) for a closed oriented 3-manifold. The definition was then generalized to links embedded in a 3-manifold and the manifolds with boundary (sutured and bordered manifolds). In the case of links, there is a beautiful combinatorial way to rewrite the original definition of HF, which was defined on a Heegaard diagram of the given link, by using grid diagram. For a balanced bipartite graph, we defined its Heegaard diagram and the HF for it. Around the same time, Harvey and O’Donnol defined the combinatorial HF for transverse graphs (see the definition in [arXiv:1506.04785v1]). In this talk, we compare these two methods.

#### Tuesday Seminar on Topology

15:00-16:30 Room #056 (Graduate School of Math. Sci. Bldg.)

The unfolded Seiberg-Witten-Floer spectrum and its applications

(ENGLISH)

**Jianfeng Lin**(UCLA)The unfolded Seiberg-Witten-Floer spectrum and its applications

(ENGLISH)

[ Abstract ]

Following Furuta's idea of finite dimensional approximation in

the Seiberg-Witten theory, Manolescu defined the Seiberg-Witten-Floer

stable homotopy type for rational homology three-spheres in 2003. In

this talk, I will explain how to construct similar invariants for a

general three-manifold and discuss some applications of these new

invariants. This is a joint work with Tirasan Khandhawit and Hirofumi

Sasahira.

Following Furuta's idea of finite dimensional approximation in

the Seiberg-Witten theory, Manolescu defined the Seiberg-Witten-Floer

stable homotopy type for rational homology three-spheres in 2003. In

this talk, I will explain how to construct similar invariants for a

general three-manifold and discuss some applications of these new

invariants. This is a joint work with Tirasan Khandhawit and Hirofumi

Sasahira.

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