## Seminar information archive

Seminar information archive ～04/19｜Today's seminar 04/20 | Future seminars 04/21～

### 2015/08/28

#### Colloquium

16:50-17:50 Room #002 (Graduate School of Math. Sci. Bldg.)

On the development of Riemann surfaces and moduli (ENGLISH)

**Athanase Papadopoulos**(Institut de Recherche Mathématique Avancée, Université de Strasbourg et CNRS)On the development of Riemann surfaces and moduli (ENGLISH)

[ Abstract ]

I will describe a selection of major fundamental ideas in the theory

of Riemann surfaces and moduli, starting from the work of Riemann, and

ending with recent works.

I will describe a selection of major fundamental ideas in the theory

of Riemann surfaces and moduli, starting from the work of Riemann, and

ending with recent works.

### 2015/08/07

#### Seminar on Probability and Statistics

14:40-15:50 Room #052 (Graduate School of Math. Sci. Bldg.)

Effectiveness of time-varying minimum value at risk and expected shortfall hedging

**UBUKATA, Masato**(Kushiro Public University of Economics)Effectiveness of time-varying minimum value at risk and expected shortfall hedging

[ Abstract ]

This paper assesses the incremental value of time-varying minimum value at risk (VaR) and expected shortfall (ES) hedging strategies over unconditional hedging strategy. The conditional futures hedge ratios are calculated through estimation of multivariate volatility models under a skewed and leptokurtic distribution and Monte Carlo simulation for conditional skewness and kurtosis of hedged portfolio returns. We examine DCC-GJR models with or without encompassing realized covariance measure (RCM) from high-frequency data under a multivariate skewed Student's t-distribution. In the out-of-sample analysis with a daily rebalancing approach, the empirical results show that the conditional minimum VaR and ES hedging strategies outperform the unconditional hedging strategy. We find that the use of RCM improves the futures hedging performance for a short hedge, although the degree of improvement is small relative to that when switching from unconditional to conditional.

This paper assesses the incremental value of time-varying minimum value at risk (VaR) and expected shortfall (ES) hedging strategies over unconditional hedging strategy. The conditional futures hedge ratios are calculated through estimation of multivariate volatility models under a skewed and leptokurtic distribution and Monte Carlo simulation for conditional skewness and kurtosis of hedged portfolio returns. We examine DCC-GJR models with or without encompassing realized covariance measure (RCM) from high-frequency data under a multivariate skewed Student's t-distribution. In the out-of-sample analysis with a daily rebalancing approach, the empirical results show that the conditional minimum VaR and ES hedging strategies outperform the unconditional hedging strategy. We find that the use of RCM improves the futures hedging performance for a short hedge, although the degree of improvement is small relative to that when switching from unconditional to conditional.

#### Seminar on Probability and Statistics

13:20-14:30 Room #052 (Graduate School of Math. Sci. Bldg.)

ESTIMATION OF INTEGRATED QUADRATIC COVARIATION BETWEEN TWO ASSETS WITH ENDOGENOUS SAMPLING TIMES

**Yoann Potiron**(University of Chicago)ESTIMATION OF INTEGRATED QUADRATIC COVARIATION BETWEEN TWO ASSETS WITH ENDOGENOUS SAMPLING TIMES

[ Abstract ]

When estimating integrated covariation between two assets based on high-frequency data,simple assumptions are usually imposed on the relationship between the price processes and the observation times. In this paper, we introduce an endogenous 2-dimensional model and show that it is more general than the existing endogenous models of the literature. In addition, we establish a central limit theorem for the Hayashi-Yoshida estimator in this general endogenous model in the case where prices follow pure-diffusion processes.

When estimating integrated covariation between two assets based on high-frequency data,simple assumptions are usually imposed on the relationship between the price processes and the observation times. In this paper, we introduce an endogenous 2-dimensional model and show that it is more general than the existing endogenous models of the literature. In addition, we establish a central limit theorem for the Hayashi-Yoshida estimator in this general endogenous model in the case where prices follow pure-diffusion processes.

### 2015/07/30

#### thesis presentations

10:30-11:45 Room #128 (Graduate School of Math. Sci. Bldg.)

Theory and application of a meta lambda calculus with cross-level computation （レベル横断的計算機構を持つメタラムダ計算の理論と応用） (JAPANESE)

**飛澤 和則**(東京大学大学院数理科学研究科)Theory and application of a meta lambda calculus with cross-level computation （レベル横断的計算機構を持つメタラムダ計算の理論と応用） (JAPANESE)

### 2015/07/29

#### thesis presentations

16:00-17:15 Room #128 (Graduate School of Math. Sci. Bldg.)

Accelerating convergence and tractability of multivariate numerical integration when the L1-norms of the higher order derivatives of the integrand grow at most exponentially（被積分関数の高階偏微分のL1ノルムの増大度が高々指数的である場合の多次元数値積分の加速的な収束と計算容易性） (JAPANESE)

**鈴木 航介**(東京大学大学院数里科学研究科)Accelerating convergence and tractability of multivariate numerical integration when the L1-norms of the higher order derivatives of the integrand grow at most exponentially（被積分関数の高階偏微分のL1ノルムの増大度が高々指数的である場合の多次元数値積分の加速的な収束と計算容易性） (JAPANESE)

#### thesis presentations

17:30-18:45 Room #128 (Graduate School of Math. Sci. Bldg.)

Research on Walsh figure of merit for higher order convergent Quasi-Monte Carlo integration（高次収束準モンテカルロ積分のためのWalsh figure of meritの研究） (JAPANESE)

**芳木 武仁**(東京大学大学院数理科学研究科)Research on Walsh figure of merit for higher order convergent Quasi-Monte Carlo integration（高次収束準モンテカルロ積分のためのWalsh figure of meritの研究） (JAPANESE)

### 2015/07/28

#### Lectures

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

Convergence of some horocyclic deformations to the Gardiner-Masur

boundary of Teichmueller space. (ENGLISH)

**Vincent Alberge**(Université de Strasbourg)Convergence of some horocyclic deformations to the Gardiner-Masur

boundary of Teichmueller space. (ENGLISH)

[ Abstract ]

It is well known that a point of the Teichmueller space and a measured foliation determine an isometric embedding of the hyperbolic disc to the Teichmueller space equipped with the so-called Teichmueller metric. In this talk, we will consider the image by this embedding of a particular horocycle whose points will be called an horocyclic deformation. To be more precise, we will be interested in the closure of this subset in the Gardiner-Masur compactification. As the embedding of the disc does not admit a continuous extension to boundaries, we cannot say that the boundary of the set of horocyclic deformations consists of one point.

However, according to Miyachi's results, we will see that it is the case if the given foliation is either a simple closed curve or a uniquely ergodic foliation.

It is well known that a point of the Teichmueller space and a measured foliation determine an isometric embedding of the hyperbolic disc to the Teichmueller space equipped with the so-called Teichmueller metric. In this talk, we will consider the image by this embedding of a particular horocycle whose points will be called an horocyclic deformation. To be more precise, we will be interested in the closure of this subset in the Gardiner-Masur compactification. As the embedding of the disc does not admit a continuous extension to boundaries, we cannot say that the boundary of the set of horocyclic deformations consists of one point.

However, according to Miyachi's results, we will see that it is the case if the given foliation is either a simple closed curve or a uniquely ergodic foliation.

#### Lie Groups and Representation Theory

17:00-18:30 Room #122 (Graduate School of Math. Sci. Bldg.)

On g,K-modules over arbitrary fields and applications to special values of L-functions

**Fabian Januszewski**(Karlsruhe Institute of Technology)On g,K-modules over arbitrary fields and applications to special values of L-functions

[ Abstract ]

I will introduce g,K-modules over arbitrary fields of characteristic 0 and discuss their fundamental properties and constructions, including Zuckerman functors. This may be applied to produce models of certain standard modules over number fields, which has applications to special values of automorphic L-functions, and also furnishes the space of regular algebraic cusp forms of GL(n) with a natural global Q-structure.

I will introduce g,K-modules over arbitrary fields of characteristic 0 and discuss their fundamental properties and constructions, including Zuckerman functors. This may be applied to produce models of certain standard modules over number fields, which has applications to special values of automorphic L-functions, and also furnishes the space of regular algebraic cusp forms of GL(n) with a natural global Q-structure.

### 2015/07/27

#### Tokyo Probability Seminar

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

Convergence of Brownian motions on RCD*(K,N) spaces

**Kohei Suzuki**(Graduate School of Science, Kyoto University)Convergence of Brownian motions on RCD*(K,N) spaces

### 2015/07/24

#### Operator Algebra Seminars

16:45-18:15 Room #122 (Graduate School of Math. Sci. Bldg.)

Applications of Boltzmann's S=k log W in algebra and analysis

**Mikael Pichot**(McGill Univ.)Applications of Boltzmann's S=k log W in algebra and analysis

#### Geometry Colloquium

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

High-dimensional metric-measure limit of Stiefel manifolds (Japanese)

**Asuka Takatsu**(Tokyo Metropolitan University)High-dimensional metric-measure limit of Stiefel manifolds (Japanese)

[ Abstract ]

A metric measure space is the triple of a complete separable metric space with a Borel measure on this space. Gromov defined a concept of convergence of metric measure spaces by the convergence of the sets of 1-Lipschitz functions on the spaces. We study and specify the high-dimensional limit of Stiefel manifolds in the sense of this convergence; the limit is the infinite-dimensional Gaussian space, which is drastically different from the manifolds. This is a joint work with Takashi SHIOYA (Tohoku univ).

A metric measure space is the triple of a complete separable metric space with a Borel measure on this space. Gromov defined a concept of convergence of metric measure spaces by the convergence of the sets of 1-Lipschitz functions on the spaces. We study and specify the high-dimensional limit of Stiefel manifolds in the sense of this convergence; the limit is the infinite-dimensional Gaussian space, which is drastically different from the manifolds. This is a joint work with Takashi SHIOYA (Tohoku univ).

#### FMSP Lectures

16:00-19:00 Room #268 (Graduate School of Math. Sci. Bldg.)

Solvability and approximate solution of a coefficient inverse problem for the kinetic equation (ENGLISH)

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

**Fikret Golgeleyen**(Bulent Ecevit University)Solvability and approximate solution of a coefficient inverse problem for the kinetic equation (ENGLISH)

[ Abstract ]

The existence, uniqueness and stability of the solution of a coefficient inverse problem for the kinetic equation are proven.

The approximate solution of the problem in one-dimensional case is investigated by using two different techniques: finite difference approximation (FDA) and symbolic computation approach (SCA).

A comparison among the exact solution of the problem, the numerical solution obtained from FDA and the approximate analytical solution obtained from SCA is presented.

[ Reference URL ]The existence, uniqueness and stability of the solution of a coefficient inverse problem for the kinetic equation are proven.

The approximate solution of the problem in one-dimensional case is investigated by using two different techniques: finite difference approximation (FDA) and symbolic computation approach (SCA).

A comparison among the exact solution of the problem, the numerical solution obtained from FDA and the approximate analytical solution obtained from SCA is presented.

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

### 2015/07/23

#### Number Theory Seminar

13:00-16:30 Room #056 (Graduate School of Math. Sci. Bldg.)

Representation of squares by cubic forms - Estimates for the appearing exponential sums (English)

Ramification and nearby cycles for $\ell$-adic sheaves on relative curves (English)

Explicit computation of the number of dormant opers and duality (Japanese)

**Lasse Grimmelt**(University of Göttingen/Waseda University) 13:00-14:00Representation of squares by cubic forms - Estimates for the appearing exponential sums (English)

**Haoyu Hu**(University of Tokyo) 14:15-15:15Ramification and nearby cycles for $\ell$-adic sheaves on relative curves (English)

[ Abstract ]

I will present a new approach for a formula of Deligne and Kato that computes the dimension of the nearby cycle complex of an $\ell$-adic sheaf on a smooth relative curve over a strictly henselian trait such that $p$ is not one of its uniformizer. Deligne considered the case where the sheaf has no vertical ramification and Kato extended the formula to the general case. My approach is based on ramification theory of Abbes and Saito. It computes the nearby cycle complex in terms of the refined Swan conductor. In fact, I compare Abbes-Saito's refined Swan conductor with Kato's Swan conductor with differential values, which is the key ingredient in Kato's formula; the case of rank one sheaves is due to Abbes and Saito. My approach provides also a new independent proof of Deligne-Kato's formula.

I will present a new approach for a formula of Deligne and Kato that computes the dimension of the nearby cycle complex of an $\ell$-adic sheaf on a smooth relative curve over a strictly henselian trait such that $p$ is not one of its uniformizer. Deligne considered the case where the sheaf has no vertical ramification and Kato extended the formula to the general case. My approach is based on ramification theory of Abbes and Saito. It computes the nearby cycle complex in terms of the refined Swan conductor. In fact, I compare Abbes-Saito's refined Swan conductor with Kato's Swan conductor with differential values, which is the key ingredient in Kato's formula; the case of rank one sheaves is due to Abbes and Saito. My approach provides also a new independent proof of Deligne-Kato's formula.

**Yasuhiro Wakabayashi**(University of Tokyo) 15:30-16:30Explicit computation of the number of dormant opers and duality (Japanese)

### 2015/07/22

#### Operator Algebra Seminars

16:45-18:15 Room #122 (Graduate School of Math. Sci. Bldg.)

Recent progress in the classification of amenable $C^*$-algebras

**George Elliott**(Univ. Toronto)Recent progress in the classification of amenable $C^*$-algebras

#### FMSP Lectures

16:45-18:15 Room #122 (Graduate School of Math. Sci. Bldg.)

Recent progress in the classification of amenable C*-algebras (ENGLISH)

[ Reference URL ]

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

**George Elliott**(Univ. Toronto)Recent progress in the classification of amenable C*-algebras (ENGLISH)

[ Reference URL ]

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

### 2015/07/21

#### Tuesday Seminar on Topology

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

Ribbon concordance and 0-surgeries along knots (JAPANESE)

**Keiji Tagami**(Tokyo Institute of Technology)Ribbon concordance and 0-surgeries along knots (JAPANESE)

[ Abstract ]

Akbulut and Kirby conjectured that two knots with

the same 0-surgery are concordant. Recently, Yasui

gave a counterexample of this conjecture.

In this talk, we introduce a technique to construct

non-ribbon concordant knots with the same 0-surgery.

Moreover, we give a potential counterexample of the

slice-ribbon conjecture. This is a joint work with

Tetsuya Abe (Osaka City University, OCAMI).

Akbulut and Kirby conjectured that two knots with

the same 0-surgery are concordant. Recently, Yasui

gave a counterexample of this conjecture.

In this talk, we introduce a technique to construct

non-ribbon concordant knots with the same 0-surgery.

Moreover, we give a potential counterexample of the

slice-ribbon conjecture. This is a joint work with

Tetsuya Abe (Osaka City University, OCAMI).

#### Tuesday Seminar of Analysis

16:30-18:00 Room #126 (Graduate School of Math. Sci. Bldg.)

Born-Oppenheimer approximation for an atom in constant magnetic fields (Japanese)

**Sohei Ashida**(Department of Mathematics, Kyoto University)Born-Oppenheimer approximation for an atom in constant magnetic fields (Japanese)

[ Abstract ]

We obtain a reduction scheme for the study of the quantum evolution of an atom in constant magnetic fields using the method developed by Martinez, Nenciu and Sordoni based on the construction of almost invariant subspace. Martinez and Sordoni also dealt with such a case but their reduced Hamiltonian includes the vector potential terms. Using the center of mass coordinates and constructing the almost invariant subspace different from theirs, we obtain the reduced Hamiltonian which does not include the vector potential terms. Using the reduced evolution we also obtain the asymptotic expantion of the evolution for a specific localized initial data, which verifies the straight motion of an atom in constatnt magnetic fields.

We obtain a reduction scheme for the study of the quantum evolution of an atom in constant magnetic fields using the method developed by Martinez, Nenciu and Sordoni based on the construction of almost invariant subspace. Martinez and Sordoni also dealt with such a case but their reduced Hamiltonian includes the vector potential terms. Using the center of mass coordinates and constructing the almost invariant subspace different from theirs, we obtain the reduced Hamiltonian which does not include the vector potential terms. Using the reduced evolution we also obtain the asymptotic expantion of the evolution for a specific localized initial data, which verifies the straight motion of an atom in constatnt magnetic fields.

#### Lie Groups and Representation Theory

17:00-18:30 Room #122 (Graduate School of Math. Sci. Bldg.)

GEOMETRIC STRUCTURE IN SMOOTH DUAL

**Paul Baum**(Penn State University)GEOMETRIC STRUCTURE IN SMOOTH DUAL

[ Abstract ]

Let G be a connected split reductive p-adic group. Examples are GL(n, F) , SL(n, F) , SO(n, F) , Sp(2n, F) , PGL(n, F) where n can be any positive integer and F can be any finite extension of the field Q_p of p-adic numbers. The smooth (or admissible) dual of G is the set of equivalence classes of smooth irreducible representations of G. This talk will first review the theory of the Bernstein center. According to this theory, the smooth dual of G is the disjoint union of subsets known as the Bernstein components. The talk will then explain the ABPS (Aubert-Baum-Plymen-Solleveld) conjecture which states that each Bernstein component is a complex affine variety. Each of these complex affine varieties is explicitly identified as the extended quotient associated to the given Bernstein component.

The ABPS conjecture has been proved for GL(n, F), SO(n, F), and Sp(2n, F).

Let G be a connected split reductive p-adic group. Examples are GL(n, F) , SL(n, F) , SO(n, F) , Sp(2n, F) , PGL(n, F) where n can be any positive integer and F can be any finite extension of the field Q_p of p-adic numbers. The smooth (or admissible) dual of G is the set of equivalence classes of smooth irreducible representations of G. This talk will first review the theory of the Bernstein center. According to this theory, the smooth dual of G is the disjoint union of subsets known as the Bernstein components. The talk will then explain the ABPS (Aubert-Baum-Plymen-Solleveld) conjecture which states that each Bernstein component is a complex affine variety. Each of these complex affine varieties is explicitly identified as the extended quotient associated to the given Bernstein component.

The ABPS conjecture has been proved for GL(n, F), SO(n, F), and Sp(2n, F).

#### Lie Groups and Representation Theory

15:30-16:30 Room #122 (Graduate School of Math. Sci. Bldg.)

(Japanese)

**Toshiaki Hattori**(Tokyo Institute of Technology)(Japanese)

### 2015/07/17

#### Geometry Colloquium

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

Realization of tropical curves in complex tori (Japanese)

**Takeo Nishinou**(Rikkyo University)Realization of tropical curves in complex tori (Japanese)

[ Abstract ]

Tropical curves are combinatorial object satisfying certain harmonicity condition. They reflect properties of holomorphic curves, and rather precise correspondence is known between tropical curves in real affine spaces and holomorphic curves in toric varieties. In this talk we extend this correspondence to the periodic case. Namely, we give a correspondence between periodic plane tropical curves and holomorphic curves in complex tori. This is a joint work with Tony Yue Yu.

Tropical curves are combinatorial object satisfying certain harmonicity condition. They reflect properties of holomorphic curves, and rather precise correspondence is known between tropical curves in real affine spaces and holomorphic curves in toric varieties. In this talk we extend this correspondence to the periodic case. Namely, we give a correspondence between periodic plane tropical curves and holomorphic curves in complex tori. This is a joint work with Tony Yue Yu.

#### Infinite Analysis Seminar Tokyo

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

Classifying simple modules at admissible levels through symmetric polynomials (ENGLISH)

**Simon Wood**(The Australian National University)Classifying simple modules at admissible levels through symmetric polynomials (ENGLISH)

[ Abstract ]

From infinite dimensional Lie algebras such as the Virasoro

algebra or affine Lie (super)algebras one can construct universal

vertex operator algebras. These vertex operator algebras are simple at

generic central charges or levels and only contain proper ideals at so

called admissible levels. The simple quotient vertex operator algebras

at these admissible levels are called minimal model algebras. In this

talk I will present free field realisations of the universal vertex

operator algebras and show how they allow one to elegantly classify

the simple modules over the simple quotient vertex operator algebras

by using a deep connection to symmetric polynomials.

From infinite dimensional Lie algebras such as the Virasoro

algebra or affine Lie (super)algebras one can construct universal

vertex operator algebras. These vertex operator algebras are simple at

generic central charges or levels and only contain proper ideals at so

called admissible levels. The simple quotient vertex operator algebras

at these admissible levels are called minimal model algebras. In this

talk I will present free field realisations of the universal vertex

operator algebras and show how they allow one to elegantly classify

the simple modules over the simple quotient vertex operator algebras

by using a deep connection to symmetric polynomials.

### 2015/07/16

#### Applied Analysis

16:00-17:30 Room #128 (Graduate School of Math. Sci. Bldg.)

(Japanese)

**Yoshihiro Tonegawa**(Tokyo Institute of Technology)(Japanese)

### 2015/07/14

#### Tuesday Seminar on Topology

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

How homoclinic orbits explain some algebraic relations holding in Novikov rings. (ENGLISH)

**Carlos Moraga Ferrandiz**(The University of Tokyo, JSPS)How homoclinic orbits explain some algebraic relations holding in Novikov rings. (ENGLISH)

[ Abstract ]

Given u, a de-Rham cohomology class of degree 1 of a closed manifold M, we consider the space F_u of (closed) Morse 1-forms in this class. In Morse theory, it is important to equip each α in F_u with a descending pseudo-gradient X. The case u=0 yields usual Morse theory, while u ≠ 0 yields Morse-Novikov theory, which is devoted to the understanding of the space of equipped 1-forms (α,X) with α in F_u.

Here, X is a descending pseudo-gradient, which is said to be adapted to α.

The morphism π1(M) → R induced by u (given by the integral of any α in F_u over a loop of M) determines a set of u-negative loops.

We show that for every u-negative g in π1(M), there exists a co-dimension 1 C∞-stratum Sg of F_u which is naturally co-oriented. The stratum Sg is made of elements (α, X) such that X has exactly one homoclinic orbit L whose homotopy class is g.

The goal of this talk is to show that there exists a co-dimension 1 C∞-stratum Sg (0) of Sg which lies in the closure of Sg^2. This result explains geometrically an easy algebraic relation holding in the Novikov ring associated with u.

We will mention how this study generalizes to produce some non-evident symmetric formulas holding in the Novikov ring.

Given u, a de-Rham cohomology class of degree 1 of a closed manifold M, we consider the space F_u of (closed) Morse 1-forms in this class. In Morse theory, it is important to equip each α in F_u with a descending pseudo-gradient X. The case u=0 yields usual Morse theory, while u ≠ 0 yields Morse-Novikov theory, which is devoted to the understanding of the space of equipped 1-forms (α,X) with α in F_u.

Here, X is a descending pseudo-gradient, which is said to be adapted to α.

The morphism π1(M) → R induced by u (given by the integral of any α in F_u over a loop of M) determines a set of u-negative loops.

We show that for every u-negative g in π1(M), there exists a co-dimension 1 C∞-stratum Sg of F_u which is naturally co-oriented. The stratum Sg is made of elements (α, X) such that X has exactly one homoclinic orbit L whose homotopy class is g.

The goal of this talk is to show that there exists a co-dimension 1 C∞-stratum Sg (0) of Sg which lies in the closure of Sg^2. This result explains geometrically an easy algebraic relation holding in the Novikov ring associated with u.

We will mention how this study generalizes to produce some non-evident symmetric formulas holding in the Novikov ring.

#### PDE Real Analysis Seminar

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

Viscosity solutions of Hamilton-Jacobi equations from a dynamical viewpoint (English)

**Lin Wang**(Tsinghua University)Viscosity solutions of Hamilton-Jacobi equations from a dynamical viewpoint (English)

[ Abstract ]

By establishing an implicit variational principle for contact Hamiltonian systems, we detect some properties of viscosity solutions of Hamilton-Jacobi equations of certain Hamilton-Jacobi equations depending on unknown functions, including large time behavior and regularity on certain sets. Besides, I will talk about some connections with contact geometry, thermodynamics and nonholonomic mechanics.

By establishing an implicit variational principle for contact Hamiltonian systems, we detect some properties of viscosity solutions of Hamilton-Jacobi equations of certain Hamilton-Jacobi equations depending on unknown functions, including large time behavior and regularity on certain sets. Besides, I will talk about some connections with contact geometry, thermodynamics and nonholonomic mechanics.

#### Tuesday Seminar of Analysis

16:30-18:00 Room #126 (Graduate School of Math. Sci. Bldg.)

Small-time Asymptotics for Subelliptic Heat Kernels (English)

**Li Yutian**(Department of Mathematics, Hong Kong Baptist University)Small-time Asymptotics for Subelliptic Heat Kernels (English)

[ Abstract ]

Subelliptic operators are the natural generalizations of the Laplace- Beltrami operators, and they play important roles in geometry, several complex variables, probability and physics. As in the classical spectral theory for the elliptic operators, some geometrical properties of the induced subRiemannian geometry can be extracted from the analysis of the heat kernels for subelliptic operators. In this talk we shall review the recent progress in the heat kernel asymptotics for subelliptic operators. We concentrate on the small-time asymptotics of the heat kernel on the diagonal, or equivalently, the asymptotics for the trace. Our interest is to find the exact form of the leading term, and this will lead to a Weyl’s asymptotic formula for the subelliptic operators. This is a joint work with Professor Der-Chen Chang.

Subelliptic operators are the natural generalizations of the Laplace- Beltrami operators, and they play important roles in geometry, several complex variables, probability and physics. As in the classical spectral theory for the elliptic operators, some geometrical properties of the induced subRiemannian geometry can be extracted from the analysis of the heat kernels for subelliptic operators. In this talk we shall review the recent progress in the heat kernel asymptotics for subelliptic operators. We concentrate on the small-time asymptotics of the heat kernel on the diagonal, or equivalently, the asymptotics for the trace. Our interest is to find the exact form of the leading term, and this will lead to a Weyl’s asymptotic formula for the subelliptic operators. This is a joint work with Professor Der-Chen Chang.

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