## Seminar information archive

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

### 2015/04/07

#### Tuesday Seminar on Topology

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

Potential functions for Grassmannians (JAPANESE)

**Kazushi Ueda**(The University of Tokyo)Potential functions for Grassmannians (JAPANESE)

[ Abstract ]

Potential functions are Floer-theoretic invariants

obtained by counting Maslov index 2 disks

with Lagrangian boundary conditions.

In the talk, we will discuss our joint work

with Yanki Lekili and Yuichi Nohara

on Lagrangian torus fibrations on the Grassmannian

of 2-planes in an n-space,

the potential functions of their Lagrangian torus fibers,

and their relation with mirror symmetry for Grassmannians.

Potential functions are Floer-theoretic invariants

obtained by counting Maslov index 2 disks

with Lagrangian boundary conditions.

In the talk, we will discuss our joint work

with Yanki Lekili and Yuichi Nohara

on Lagrangian torus fibrations on the Grassmannian

of 2-planes in an n-space,

the potential functions of their Lagrangian torus fibers,

and their relation with mirror symmetry for Grassmannians.

#### Lie Groups and Representation Theory

16:30-18:00 Room #122 (Graduate School of Math. Sci. Bldg.)

Branching laws and elliptic boundary value problems

(English)

**Bent Orsted**(Aarhus University)Branching laws and elliptic boundary value problems

(English)

[ Abstract ]

Classically the Poisson transform relates harmonic functions in the complex upper half plane to their boundary values on the real axis. In

some recent work by Caffarelli et al. some new generalizations of this appears in connection with the fractional Laplacian. In this lecture we

shall explain how the symmetry-breaking operators introduced by T. Kobayashi for studying branching laws may shed new light on the situation for elliptic boundary value problems. This is based on joint work with J. M\"o{}llers and G. Zhang.

Classically the Poisson transform relates harmonic functions in the complex upper half plane to their boundary values on the real axis. In

some recent work by Caffarelli et al. some new generalizations of this appears in connection with the fractional Laplacian. In this lecture we

shall explain how the symmetry-breaking operators introduced by T. Kobayashi for studying branching laws may shed new light on the situation for elliptic boundary value problems. This is based on joint work with J. M\"o{}llers and G. Zhang.

### 2015/04/06

#### Seminar on Geometric Complex Analysis

10:30-12:00 Room #126 (Graduate School of Math. Sci. Bldg.)

Analytic torsion for K3 surfaces with involution (Japanese)

**Ken-ichi Yoshikawa**(Kyoto Univ.)Analytic torsion for K3 surfaces with involution (Japanese)

[ Abstract ]

In 2004, I introduced a holomorphic torsion invariant for 2-elementary K3 surfaces, i.e., K3 surfaces with involution. In the talk, I will report a recent progress in this invariant. Namely, for all possible deformation types, the holomorphic torsion invariant viewed as a function on the moduli space, is expressed as the product of an explicit Borcherds lift and an explicit Siegel modular form. If time permits, I will interpret the result in terms of the BCOV invariant, i.e., the genus-one string amplitude in B-model, for Calabi-Yau threefolds of Borcea-Voisin. This is a joint work with Shouhei Ma.

In 2004, I introduced a holomorphic torsion invariant for 2-elementary K3 surfaces, i.e., K3 surfaces with involution. In the talk, I will report a recent progress in this invariant. Namely, for all possible deformation types, the holomorphic torsion invariant viewed as a function on the moduli space, is expressed as the product of an explicit Borcherds lift and an explicit Siegel modular form. If time permits, I will interpret the result in terms of the BCOV invariant, i.e., the genus-one string amplitude in B-model, for Calabi-Yau threefolds of Borcea-Voisin. This is a joint work with Shouhei Ma.

### 2015/03/24

#### Tuesday Seminar on Topology

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

Knots and Mirror Symmetry (ENGLISH)

**Mina Aganagic**(University of California, Berkeley)Knots and Mirror Symmetry (ENGLISH)

[ Abstract ]

I will describe two conjectures relating knot theory and mirror symmetry. One can associate, to every knot K, one a Calabi-Yau manifold Y(K), which depends on the homotopy type of the knot only. The first conjecture is that Y(K) arises by a generalization of SYZ mirror symmetry, as mirror to the conifold, O(-1)+O(-1)->P^1. The second conjecture is that topological string provides a quantization of Y(K) which leads to quantum HOMFLY invariants of the knot. The conjectures are based on joint work with C. Vafa and also with T.Ekholm, L. Ng.

I will describe two conjectures relating knot theory and mirror symmetry. One can associate, to every knot K, one a Calabi-Yau manifold Y(K), which depends on the homotopy type of the knot only. The first conjecture is that Y(K) arises by a generalization of SYZ mirror symmetry, as mirror to the conifold, O(-1)+O(-1)->P^1. The second conjecture is that topological string provides a quantization of Y(K) which leads to quantum HOMFLY invariants of the knot. The conjectures are based on joint work with C. Vafa and also with T.Ekholm, L. Ng.

#### Lie Groups and Representation Theory

18:00-19:30 Room #126 (Graduate School of Math. Sci. Bldg.)

A Gysin formula for Hall-Littlewood polynomials

**Piotr Pragacz**(Institute of Mathematics, Polish Academy of Sciences)A Gysin formula for Hall-Littlewood polynomials

[ Abstract ]

Schubert calculus on Grassmannians is governed by Schur S-functions, the one on Lagrangian Grassmannians by Schur Q-functions. There were several attempts to give a unifying approach to both situations.

We propose to use Hall-Littlewood symmetric polynomials. They appeared implicitly in Hall's study of the combinatorial lattice structure of finite abelian p-groups and in Green's calculations of the characters of GL(n) over finite fields; they appeared explicitly in the work of Littlewood on some problems in representation theory.

With the projection in a Grassmann bundle, there is associated its Gysin map, induced by pushing forward cycles (topologists call it "integration along fibers").

We state and prove a Gysin formula for HL-polynomials in these bundles. We discuss its two specializations, giving better insights to previously known formulas for Schur S- and P-functions.

Schubert calculus on Grassmannians is governed by Schur S-functions, the one on Lagrangian Grassmannians by Schur Q-functions. There were several attempts to give a unifying approach to both situations.

We propose to use Hall-Littlewood symmetric polynomials. They appeared implicitly in Hall's study of the combinatorial lattice structure of finite abelian p-groups and in Green's calculations of the characters of GL(n) over finite fields; they appeared explicitly in the work of Littlewood on some problems in representation theory.

With the projection in a Grassmann bundle, there is associated its Gysin map, induced by pushing forward cycles (topologists call it "integration along fibers").

We state and prove a Gysin formula for HL-polynomials in these bundles. We discuss its two specializations, giving better insights to previously known formulas for Schur S- and P-functions.

### 2015/03/20

#### Numerical Analysis Seminar

13:30-15:00 Room #122 (Graduate School of Math. Sci. Bldg.)

Asymmetric Auctions (English)

**Gadi Fibich**(Tel Aviv University)Asymmetric Auctions (English)

[ Abstract ]

Auctions are central to the modern economy, both on-line and off-line. A fundamental result in auction theory is that when bidders are symmetric (identical), then under quite general conditions, all auctions are revenue equivalent. While it is known that this result does not hold when bidders are asymmetric, the effect of bidders' asymmetry is poorly understood, since asymmetric auctions are much harder to analyze.

In this talk I will discuss the mathematical theory of asymmetric auctions. I will focus on asymmetric first-price auctions, where the mathematical model is given by a nonstandard system of $n$ nonlinear ordinary differential equations, with $2n$ boundary conditions and a free boundary. I will present various analytic and numerical approaches for this system. Then I will present some recent results on asymptotic revenue equivalence of asymmetric auctions.

Joint work with A. Gavious and N. Gavish.

Auctions are central to the modern economy, both on-line and off-line. A fundamental result in auction theory is that when bidders are symmetric (identical), then under quite general conditions, all auctions are revenue equivalent. While it is known that this result does not hold when bidders are asymmetric, the effect of bidders' asymmetry is poorly understood, since asymmetric auctions are much harder to analyze.

In this talk I will discuss the mathematical theory of asymmetric auctions. I will focus on asymmetric first-price auctions, where the mathematical model is given by a nonstandard system of $n$ nonlinear ordinary differential equations, with $2n$ boundary conditions and a free boundary. I will present various analytic and numerical approaches for this system. Then I will present some recent results on asymptotic revenue equivalence of asymmetric auctions.

Joint work with A. Gavious and N. Gavish.

### 2015/03/19

#### FMSP Lectures

9:00-11:00 Room #大講義室 (Graduate School of Math. Sci. Bldg.)

Critical metrics for quadratic Riemannian functionals in dimension four (ENGLISH)

https://sites.google.com/site/princetontokyo/mini-courses

Hessian type equations on compact Kähler manifolds (ENGLISH)

https://sites.google.com/site/princetontokyo/mini-courses

**Matthew Gursky**(Univ. Nortre Dame) 9:00-9:50Critical metrics for quadratic Riemannian functionals in dimension four (ENGLISH)

[ Abstract ]

In these lectures I will give an overview of a proof of existence, via gluing methods, of metrics which are critical points of quadratic Riemannian functionals. This is a joint project with J. Viaclovsky.

These are functionals on the space of metrics which are given by integrals of quadratic polynomials in the curvature tensor. Our approach is to construct these metrics on connected sums of Einstein four-manifolds, specifically the Fubini-Study metric on CP2 and the product metric on S2 X S2. Using these metrics in various gluing configurations, toric-invariant critical metrics are found on connected sums for a specific quadratic functional, which depends on the global geometry of the factors.

I will also explain some recent work which attempts to understand the moduli space of critical metrics.

[ Reference URL ]In these lectures I will give an overview of a proof of existence, via gluing methods, of metrics which are critical points of quadratic Riemannian functionals. This is a joint project with J. Viaclovsky.

These are functionals on the space of metrics which are given by integrals of quadratic polynomials in the curvature tensor. Our approach is to construct these metrics on connected sums of Einstein four-manifolds, specifically the Fubini-Study metric on CP2 and the product metric on S2 X S2. Using these metrics in various gluing configurations, toric-invariant critical metrics are found on connected sums for a specific quadratic functional, which depends on the global geometry of the factors.

I will also explain some recent work which attempts to understand the moduli space of critical metrics.

https://sites.google.com/site/princetontokyo/mini-courses

**Gábor Székelyhidi**(Univ. Nortre Dame) 10:10-11:00Hessian type equations on compact Kähler manifolds (ENGLISH)

[ Abstract ]

I will discuss a priori estimates for a general class of nonlinear equations on compact Kähler manifolds. This unifies and generalizes several previous works on specific equations, such as the complex Monge-Ampère, Hessian, and inverse Hessian equations.

[ Reference URL ]I will discuss a priori estimates for a general class of nonlinear equations on compact Kähler manifolds. This unifies and generalizes several previous works on specific equations, such as the complex Monge-Ampère, Hessian, and inverse Hessian equations.

https://sites.google.com/site/princetontokyo/mini-courses

### 2015/03/18

#### FMSP Lectures

9:00-11:00 Room #大講義室 (Graduate School of Math. Sci. Bldg.)

Critical metrics for quadratic Riemannian functionals in dimension four (ENGLISH)

https://sites.google.com/site/princetontokyo/mini-courses

Hessian type equations on compact Kähler manifolds (ENGLISH)

https://sites.google.com/site/princetontokyo/mini-courses

**Matthew Gursky**(Univ. Nortre Dame) 9:00-9:50Critical metrics for quadratic Riemannian functionals in dimension four (ENGLISH)

[ Abstract ]

In these lectures I will give an overview of a proof of existence, via gluing methods, of metrics which are critical points of quadratic Riemannian functionals. This is a joint project with J. Viaclovsky.

These are functionals on the space of metrics which are given by integrals of quadratic polynomials in the curvature tensor. Our approach is to construct these metrics on connected sums of Einstein four-manifolds, specifically the Fubini-Study metric on CP2 and the product metric on S2 X S2. Using these metrics in various gluing configurations, toric-invariant critical metrics are found on connected sums for a specific quadratic functional, which depends on the global geometry of the factors.

I will also explain some recent work which attempts to understand the moduli space of critical metrics.

[ Reference URL ]In these lectures I will give an overview of a proof of existence, via gluing methods, of metrics which are critical points of quadratic Riemannian functionals. This is a joint project with J. Viaclovsky.

These are functionals on the space of metrics which are given by integrals of quadratic polynomials in the curvature tensor. Our approach is to construct these metrics on connected sums of Einstein four-manifolds, specifically the Fubini-Study metric on CP2 and the product metric on S2 X S2. Using these metrics in various gluing configurations, toric-invariant critical metrics are found on connected sums for a specific quadratic functional, which depends on the global geometry of the factors.

I will also explain some recent work which attempts to understand the moduli space of critical metrics.

https://sites.google.com/site/princetontokyo/mini-courses

**Gábor Székelyhidi**(Univ. Nortre Dame) 10:10-11:00Hessian type equations on compact Kähler manifolds (ENGLISH)

[ Abstract ]

I will discuss a priori estimates for a general class of nonlinear equations on compact Kähler manifolds. This unifies and generalizes several previous works on specific equations, such as the complex Monge-Ampère, Hessian, and inverse Hessian equations.

[ Reference URL ]I will discuss a priori estimates for a general class of nonlinear equations on compact Kähler manifolds. This unifies and generalizes several previous works on specific equations, such as the complex Monge-Ampère, Hessian, and inverse Hessian equations.

https://sites.google.com/site/princetontokyo/mini-courses

### 2015/03/17

#### FMSP Lectures

13:30-15:00, 15:30-17:30 Room #Balcony A, Kavli IPMU (Graduate School of Math. Sci. Bldg.)

Toric mirror symmetry via shift operators (ENGLISH)

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

**Hiroshi Iritani**(Kyoto University)Toric mirror symmetry via shift operators (ENGLISH)

[ Abstract ]

Recently, shift operator for equivariant quantum cohomology

has been introduced in the work of Braverman, Maulik, Okounkov and

Pandharipande. This can be viewed as an equivariant lift of the

Seidel representation, and intertwines quantum connections with

different equivariant parameters.

In this series of talks, I will explain that shift operators essentially

"reconstruct" mirrors of toric varieties. More precisely we obtain the

following from basic properties of shift operators:

1. Givental's mirror theorem.

2. Landau-Ginzburg potential and primitive form.

3. Extended I-functions.

We will also see that the Gamma integral structure arises as

a solution to the difference equation defined by shift operators.

[ Reference URL ]Recently, shift operator for equivariant quantum cohomology

has been introduced in the work of Braverman, Maulik, Okounkov and

Pandharipande. This can be viewed as an equivariant lift of the

Seidel representation, and intertwines quantum connections with

different equivariant parameters.

In this series of talks, I will explain that shift operators essentially

"reconstruct" mirrors of toric varieties. More precisely we obtain the

following from basic properties of shift operators:

1. Givental's mirror theorem.

2. Landau-Ginzburg potential and primitive form.

3. Extended I-functions.

We will also see that the Gamma integral structure arises as

a solution to the difference equation defined by shift operators.

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

#### FMSP Lectures

9:00-11:00 Room #大講義室 (Graduate School of Math. Sci. Bldg.)

Critical metrics for quadratic Riemannian functionals in dimension four (ENGLISH)

https://sites.google.com/site/princetontokyo/mini-courses

Hessian type equations on compact Kähler manifolds (ENGLISH)

https://sites.google.com/site/princetontokyo/mini-courses

**Matthew Gursky**(Univ. Nortre Dame) 9:00-9:50Critical metrics for quadratic Riemannian functionals in dimension four (ENGLISH)

[ Abstract ]

In these lectures I will give an overview of a proof of existence, via gluing methods, of metrics which are critical points of quadratic Riemannian functionals. This is a joint project with J. Viaclovsky.

These are functionals on the space of metrics which are given by integrals of quadratic polynomials in the curvature tensor. Our approach is to construct these metrics on connected sums of Einstein four-manifolds, specifically the Fubini-Study metric on CP2 and the product metric on S2 X S2. Using these metrics in various gluing configurations, toric-invariant critical metrics are found on connected sums for a specific quadratic functional, which depends on the global geometry of the factors.

I will also explain some recent work which attempts to understand the moduli space of critical metrics.

[ Reference URL ]In these lectures I will give an overview of a proof of existence, via gluing methods, of metrics which are critical points of quadratic Riemannian functionals. This is a joint project with J. Viaclovsky.

These are functionals on the space of metrics which are given by integrals of quadratic polynomials in the curvature tensor. Our approach is to construct these metrics on connected sums of Einstein four-manifolds, specifically the Fubini-Study metric on CP2 and the product metric on S2 X S2. Using these metrics in various gluing configurations, toric-invariant critical metrics are found on connected sums for a specific quadratic functional, which depends on the global geometry of the factors.

I will also explain some recent work which attempts to understand the moduli space of critical metrics.

https://sites.google.com/site/princetontokyo/mini-courses

**Gábor Székelyhidi**(Univ. Nortre Dame) 10:10-11:00Hessian type equations on compact Kähler manifolds (ENGLISH)

[ Abstract ]

I will discuss a priori estimates for a general class of nonlinear equations on compact Kähler manifolds. This unifies and generalizes several previous works on specific equations, such as the complex Monge-Ampère, Hessian, and inverse Hessian equations.

[ Reference URL ]I will discuss a priori estimates for a general class of nonlinear equations on compact Kähler manifolds. This unifies and generalizes several previous works on specific equations, such as the complex Monge-Ampère, Hessian, and inverse Hessian equations.

https://sites.google.com/site/princetontokyo/mini-courses

### 2015/03/16

#### FMSP Lectures

13:30-15:00, 15:30-17:30 Room #Balcony A, Kavli IPMU (Graduate School of Math. Sci. Bldg.)

Toric mirror symmetry via shift operators (ENGLISH)

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

**Hiroshi Iritani**(Kyoto University)Toric mirror symmetry via shift operators (ENGLISH)

[ Abstract ]

Recently, shift operator for equivariant quantum cohomology

has been introduced in the work of Braverman, Maulik, Okounkov and

Pandharipande. This can be viewed as an equivariant lift of the

Seidel representation, and intertwines quantum connections with

different equivariant parameters.

In this series of talks, I will explain that shift operators essentially

"reconstruct" mirrors of toric varieties. More precisely we obtain the

following from basic properties of shift operators:

1. Givental's mirror theorem.

2. Landau-Ginzburg potential and primitive form.

3. Extended I-functions.

We will also see that the Gamma integral structure arises as

a solution to the difference equation defined by shift operators.

[ Reference URL ]Recently, shift operator for equivariant quantum cohomology

has been introduced in the work of Braverman, Maulik, Okounkov and

Pandharipande. This can be viewed as an equivariant lift of the

Seidel representation, and intertwines quantum connections with

different equivariant parameters.

In this series of talks, I will explain that shift operators essentially

"reconstruct" mirrors of toric varieties. More precisely we obtain the

following from basic properties of shift operators:

1. Givental's mirror theorem.

2. Landau-Ginzburg potential and primitive form.

3. Extended I-functions.

We will also see that the Gamma integral structure arises as

a solution to the difference equation defined by shift operators.

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

### 2015/03/13

#### Colloquium

14:00-15:00 Room #大講義室 (Graduate School of Math. Sci. Bldg.)

[ Reference URL ]

http://www.ms.u-tokyo.ac.jp/~takayuki/index-j.html

**Takayuki Oda**(Graduate School of Mathematical Sciences, University of Tokyo)[ Reference URL ]

http://www.ms.u-tokyo.ac.jp/~takayuki/index-j.html

#### Colloquium

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

(JAPANESE)

[ Reference URL ]

http://www.ms.u-tokyo.ac.jp/teacher/kusuoka.html

**Shigeo KUSUOKA**(Graduate School of Mathematical Sciences, University of Tokyo)(JAPANESE)

[ Reference URL ]

http://www.ms.u-tokyo.ac.jp/teacher/kusuoka.html

#### Colloquium

15:10-16:10 Room #大講義室 (Graduate School of Math. Sci. Bldg.)

(JAPANESE)

[ Reference URL ]

http://www.ms.u-tokyo.ac.jp/teacher/miyaoka.html

**Yoichi Miyaoka**(Graduate School of Mathematical Sciences, University of Tokyo)(JAPANESE)

[ Reference URL ]

http://www.ms.u-tokyo.ac.jp/teacher/miyaoka.html

### 2015/03/10

#### Tuesday Seminar on Topology

16:30-18:00 Room #056 (Graduate School of Math. Sci. Bldg.)

Arnold conjecture, Floer homology,

and augmentation ideals of finite groups.

(ENGLISH)

**Andrei Pajitnov**(Univ. de Nantes)Arnold conjecture, Floer homology,

and augmentation ideals of finite groups.

(ENGLISH)

[ Abstract ]

Let H be a generic time-dependent 1-periodic

Hamiltonian on a closed weakly monotone

symplectic manifold M. We construct a refined version

of the Floer chain complex associated to (M,H),

and use it to obtain new lower bounds for the number P(H)

of the 1-periodic orbits of the corresponding hamiltonian

vector field. We prove in particular that

if the fundamental group of M is finite

and solvable or simple, then P(H)

is not less than the minimal number

of generators of the fundamental group.

This is joint work with Kaoru Ono.

Let H be a generic time-dependent 1-periodic

Hamiltonian on a closed weakly monotone

symplectic manifold M. We construct a refined version

of the Floer chain complex associated to (M,H),

and use it to obtain new lower bounds for the number P(H)

of the 1-periodic orbits of the corresponding hamiltonian

vector field. We prove in particular that

if the fundamental group of M is finite

and solvable or simple, then P(H)

is not less than the minimal number

of generators of the fundamental group.

This is joint work with Kaoru Ono.

### 2015/02/24

#### thesis presentations

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

Covariance Estimation from Ultra-High-Frequency Date（超高頻度データに対する共分散推定） (JAPANESE)

**小池 祐太**(情報・システム研究機構 統計数理研究所)Covariance Estimation from Ultra-High-Frequency Date（超高頻度データに対する共分散推定） (JAPANESE)

### 2015/02/23

#### Operator Algebra Seminars

16:30-18:00 Room #122 (Graduate School of Math. Sci. Bldg.)

Classification of (2+1)-TQFTs and its applications to physics and quantum computation (English)

**Zhenghan Wang**(Microsoft Research Station Q)Classification of (2+1)-TQFTs and its applications to physics and quantum computation (English)

### 2015/02/19

#### Infinite Analysis Seminar Tokyo

13:30-17:00 Room #002 (Graduate School of Math. Sci. Bldg.)

Skein algebra and mapping class group (JAPANESE)

An extension of the LMO functor (JAPANESE)

**Shunsuke Tsuji**(Graduate School of Mathematical Sciences, the University of Tokyo) 13:30-15:00Skein algebra and mapping class group (JAPANESE)

[ Abstract ]

We define some filtrations of skein modules and the skein algebra on an oriented surface, and define the completed skein modules and

the completed skein algebra of the surface with respect to these filtration. We give an explicit formula for the action of the Dehn

twists on the completed skein modules in terms of the action of the completed skein algebra of the surface. As an application, we describe the action of the Johnson kernel on the completed skein modules.

We define some filtrations of skein modules and the skein algebra on an oriented surface, and define the completed skein modules and

the completed skein algebra of the surface with respect to these filtration. We give an explicit formula for the action of the Dehn

twists on the completed skein modules in terms of the action of the completed skein algebra of the surface. As an application, we describe the action of the Johnson kernel on the completed skein modules.

**Yuta Nozaki**(Graduate School of Mathematical Sciences, the University of Tokyo) 15:30-17:00An extension of the LMO functor (JAPANESE)

[ Abstract ]

Cheptea, Habiro and Massuyeau introduced the LMO functor as an extension of the LMO invariant of closed 3-manifolds. The LMO functor is “the monoidal category of Lagrangian cobordisms between surfaces with at most one boundary component” to “the monoidal category of top-substantial Jacobi diagrams”. In this talk, we extend the LMO functor to the case of any number of boundary components. Moreover, we explain that the internal degree d part of the extension is a finite-type invariant of degree d.

Cheptea, Habiro and Massuyeau introduced the LMO functor as an extension of the LMO invariant of closed 3-manifolds. The LMO functor is “the monoidal category of Lagrangian cobordisms between surfaces with at most one boundary component” to “the monoidal category of top-substantial Jacobi diagrams”. In this talk, we extend the LMO functor to the case of any number of boundary components. Moreover, we explain that the internal degree d part of the extension is a finite-type invariant of degree d.

#### Seminar on Probability and Statistics

16:30-17:40 Room #052 (Graduate School of Math. Sci. Bldg.)

TBA

**Dobrislav Dobrev**(Board of Governors of the Federal Reserve System, Division of International Finance)TBA

[ Abstract ]

TBA

TBA

### 2015/02/18

#### Number Theory Seminar

16:40-17:40 Room #056 (Graduate School of Math. Sci. Bldg.)

Wild ramification and $K(\pi, 1)$ spaces (English)

**Piotr Achinger**(University of California, Berkeley)Wild ramification and $K(\pi, 1)$ spaces (English)

[ Abstract ]

A smooth variety in characteristic zero is Zariski-locally a $K(\pi,1)$ space, i.e., has trivial higher homotopy groups. This fact is of crucial importance in Artin's proof that $\ell$-adic cohomology agrees with singular cohomology over $\mathbb{C}$. The characteristic $p$ variant of this is not known --- we do not even know whether the affine plane is a $K(\pi, 1)$ in positive characteristic! I will show how to reduce this question to a ``Bertini-type’' statement regarding wild ramification of $\ell$-adic local systems on affine spaces, which might be of independent interest. I will verify this statement in the special case of local systems of rank $1$ and speculate on how one might treat the general case.

A smooth variety in characteristic zero is Zariski-locally a $K(\pi,1)$ space, i.e., has trivial higher homotopy groups. This fact is of crucial importance in Artin's proof that $\ell$-adic cohomology agrees with singular cohomology over $\mathbb{C}$. The characteristic $p$ variant of this is not known --- we do not even know whether the affine plane is a $K(\pi, 1)$ in positive characteristic! I will show how to reduce this question to a ``Bertini-type’' statement regarding wild ramification of $\ell$-adic local systems on affine spaces, which might be of independent interest. I will verify this statement in the special case of local systems of rank $1$ and speculate on how one might treat the general case.

#### Numerical Analysis Seminar

14:30-16:00 Room #002 (Graduate School of Math. Sci. Bldg.)

Harnack inequalities for supersolutions of fully nonlinear elliptic difference and differential equations (日本語)

**Nao Hamamuki**(Hokkaido University)Harnack inequalities for supersolutions of fully nonlinear elliptic difference and differential equations (日本語)

#### Numerical Analysis Seminar

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

Precise and fast computation of elliptic integrals and elliptic functions (日本語)

**Toshio Fukushima**(National Astronomical Observatory)Precise and fast computation of elliptic integrals and elliptic functions (日本語)

[ Abstract ]

Summarized is the recent progress of the methods to compute (i) Legendre's normal form complete elliptic integrals of all three kinds, $K(m)$, $E(m)$, and $\Pi(n|m)$, (ii) Legendre's normal form incomplete elliptic integrals of all three kinds, $F(\phi|m)$, $E(\phi|m)$, and $\Pi(\phi,n|m)$, (iii) Jacobian elliptic functions, $\mathrm{sn}(u|m)$, $\mathrm{cn}(u|m)$, $\mathrm{dn}(u|m)$, and $\mathrm{am}(u|m)$, (iv) the inverse functions of $K(m)$ and $E(m)$, $m_K(K)$ and $m_E(E)$, (v) the inverse of a general incomplete elliptic integral in Jacobi's form, $G(\mathrm{am}(u|m),n|m)$, with respect to $u$, and (vi) the partial derivatives of $\mathrm{sn}(u|m)$, $\mathrm{cn}(u|m)$, $dn(u|m)$, $E(\mathrm{am}(u|m)|m)$, and $\Pi(\mathrm{am}(u|m),n|m)$ with respect to $u$ and those of $F(\phi|m)$, $E(\phi|m)$, and $\Pi(\phi,n|m)$ with respect to $\phi$. In order to avoid the information loss when $n\ll 1$ and/or $m \ll 1$, focused are the associate incomplete elliptc integrals defined as $B(\phi|m)=[E(\phi|m)-(1-m)F(\phi|m)]/m$, $D(\phi|m)=[F(\phi|m)-E(\phi|m)]/m$, and $J(\phi,n|m)=[\Pi(\phi,n|m)-F(\phi|m)]/n$, and their complete versions, $B(m)=[E(m)-(1-m)K(m)]/m$, $D(m)=[K(m)-E(m)]/m$, and $J(n|m)=[\Pi(n|m)-K(m)]/n$. The main techniques used are (i) the piecewise approximation for single variable functions as $K(m)$, and (ii) the combination of repeated usage of the half and double argument transformations and the truncated Maclaurin series expansions with respect to $u = F(\phi|m)$. The new methods are of the full double precision accuracy without any chance of cancellation against small input arguments. They run significantly faster than the existing methods: (i) 2.5 times faster than Cody's Chebyshev polynomial approximations for $K(m)$ and $E(m)$, (ii) 2.5 times faster than Bulirsch's cel for $\Pi(n|m)$, (iii) slightly faster than Bulirsch's el1 for $F(\phi|m)$, (iv) 3.5 times faster than Carlson's $R_D$ for $E(\phi|m)$, (v) 3.5 times faster than Carlson's $R_C$, $R_D$, $R_F$, and $R_J$ for $\Pi(\phi,n|m)$, and (vi) 1.5 times faster than Bulirsch's \texttt{sncndn} for $\mathrm{sn}(u|m)$, $\mathrm{cn}(u|m)$, and $\mathrm{dn}(u|m)$.

Summarized is the recent progress of the methods to compute (i) Legendre's normal form complete elliptic integrals of all three kinds, $K(m)$, $E(m)$, and $\Pi(n|m)$, (ii) Legendre's normal form incomplete elliptic integrals of all three kinds, $F(\phi|m)$, $E(\phi|m)$, and $\Pi(\phi,n|m)$, (iii) Jacobian elliptic functions, $\mathrm{sn}(u|m)$, $\mathrm{cn}(u|m)$, $\mathrm{dn}(u|m)$, and $\mathrm{am}(u|m)$, (iv) the inverse functions of $K(m)$ and $E(m)$, $m_K(K)$ and $m_E(E)$, (v) the inverse of a general incomplete elliptic integral in Jacobi's form, $G(\mathrm{am}(u|m),n|m)$, with respect to $u$, and (vi) the partial derivatives of $\mathrm{sn}(u|m)$, $\mathrm{cn}(u|m)$, $dn(u|m)$, $E(\mathrm{am}(u|m)|m)$, and $\Pi(\mathrm{am}(u|m),n|m)$ with respect to $u$ and those of $F(\phi|m)$, $E(\phi|m)$, and $\Pi(\phi,n|m)$ with respect to $\phi$. In order to avoid the information loss when $n\ll 1$ and/or $m \ll 1$, focused are the associate incomplete elliptc integrals defined as $B(\phi|m)=[E(\phi|m)-(1-m)F(\phi|m)]/m$, $D(\phi|m)=[F(\phi|m)-E(\phi|m)]/m$, and $J(\phi,n|m)=[\Pi(\phi,n|m)-F(\phi|m)]/n$, and their complete versions, $B(m)=[E(m)-(1-m)K(m)]/m$, $D(m)=[K(m)-E(m)]/m$, and $J(n|m)=[\Pi(n|m)-K(m)]/n$. The main techniques used are (i) the piecewise approximation for single variable functions as $K(m)$, and (ii) the combination of repeated usage of the half and double argument transformations and the truncated Maclaurin series expansions with respect to $u = F(\phi|m)$. The new methods are of the full double precision accuracy without any chance of cancellation against small input arguments. They run significantly faster than the existing methods: (i) 2.5 times faster than Cody's Chebyshev polynomial approximations for $K(m)$ and $E(m)$, (ii) 2.5 times faster than Bulirsch's cel for $\Pi(n|m)$, (iii) slightly faster than Bulirsch's el1 for $F(\phi|m)$, (iv) 3.5 times faster than Carlson's $R_D$ for $E(\phi|m)$, (v) 3.5 times faster than Carlson's $R_C$, $R_D$, $R_F$, and $R_J$ for $\Pi(\phi,n|m)$, and (vi) 1.5 times faster than Bulirsch's \texttt{sncndn} for $\mathrm{sn}(u|m)$, $\mathrm{cn}(u|m)$, and $\mathrm{dn}(u|m)$.

### 2015/02/10

#### thesis presentations

9:30-10:45 Room #118 (Graduate School of Math. Sci. Bldg.)

On a new geometric construction of a family of Galois representations associated to modular forms

（保型形式に付随するガロア表現の族の新たな幾何的構成について） (JAPANESE)

**三原 朋樹**(東京大学大学院数理科学研究科)On a new geometric construction of a family of Galois representations associated to modular forms

（保型形式に付随するガロア表現の族の新たな幾何的構成について） (JAPANESE)

#### thesis presentations

11:00-12:15 Room #118 (Graduate School of Math. Sci. Bldg.)

VISIBLE ACTIONS OF REDUCTIVE ALGEBRAIC GROUPS ON COMPLEX ALGEBRAIC VARIETIES（簡約代数群の複素代数多様体への可視的作用について） (JAPANESE)

**田中 雄一郎**(東京大学大学院数理科学研究科)VISIBLE ACTIONS OF REDUCTIVE ALGEBRAIC GROUPS ON COMPLEX ALGEBRAIC VARIETIES（簡約代数群の複素代数多様体への可視的作用について） (JAPANESE)

#### thesis presentations

13:00-14:15 Room #118 (Graduate School of Math. Sci. Bldg.)

Autonomous limit of 4-dimensional Painlev´e-type equations and singular fibers of spectral curve fibrations（4次元Painlev´e 型方程式の自励極限とスペクトラル曲線ファイブレーションの特異ファイバー） (JAPANESE)

**中村 あかね**(東京大学大学院数理科学研究科)Autonomous limit of 4-dimensional Painlev´e-type equations and singular fibers of spectral curve fibrations（4次元Painlev´e 型方程式の自励極限とスペクトラル曲線ファイブレーションの特異ファイバー） (JAPANESE)

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