## Seminar information archive

Seminar information archive ～12/08｜Today's seminar 12/09 | Future seminars 12/10～

### 2008/12/18

#### Operator Algebra Seminars

16:30-18:00 Room #128 (Graduate School of Math. Sci. Bldg.)

Some geometric and probabilistic properties of the free quantum group $A_o(n)$

**Benoit Collins**(東大数理/Ottawa 大学)Some geometric and probabilistic properties of the free quantum group $A_o(n)$

### 2008/12/17

#### Seminar on Probability and Statistics

13:40-14:50 Room #002 (Graduate School of Math. Sci. Bldg.)

Goodness of fit tests for ergodic diffusions by discrete sampling schemes

https://www.ms.u-tokyo.ac.jp/~kengok/statseminar/2008/09.html

**Ilia Negri**(University of Bergamo, Italy)Goodness of fit tests for ergodic diffusions by discrete sampling schemes

[ Abstract ]

We consider a nonparametric goodness of fit test problem for the drift coefficient of one-dimensional ergodic diffusions, where the diffusion coefficient is a nuisance function which is estimated in some sense. Using a theory for the continuous observation case, we construct two kinds of tests based on different types of discrete observations, namely, the data observed discretely in time or in space. We prove that the limit distribution of our tests is the supremum of the standard Brownian motion, and thus our tests are asymptotically distribution free. We also show that our tests are consistent under any fixed alternatives.

joint with Yoichi Nishiyama (Inst. Statist. Math.)

[ Reference URL ]We consider a nonparametric goodness of fit test problem for the drift coefficient of one-dimensional ergodic diffusions, where the diffusion coefficient is a nuisance function which is estimated in some sense. Using a theory for the continuous observation case, we construct two kinds of tests based on different types of discrete observations, namely, the data observed discretely in time or in space. We prove that the limit distribution of our tests is the supremum of the standard Brownian motion, and thus our tests are asymptotically distribution free. We also show that our tests are consistent under any fixed alternatives.

joint with Yoichi Nishiyama (Inst. Statist. Math.)

https://www.ms.u-tokyo.ac.jp/~kengok/statseminar/2008/09.html

#### Seminar on Probability and Statistics

15:00-16:10 Room #002 (Graduate School of Math. Sci. Bldg.)

Divergences Test Statistics for Discretely Observed Diffusion Processes

https://www.ms.u-tokyo.ac.jp/~kengok/statseminar/2008/10.html

**Stefano Maria Iacus**(Universita degli Studi di Milano, Italy)Divergences Test Statistics for Discretely Observed Diffusion Processes

[ Abstract ]

In this paper we propose the use of $\\phi$-divergences as test statistics to verify simple hypotheses about a one-dimensional parametric diffusion process dXt = b(Xt, theta)dt + sigma(Xt, theta) dWt, from discrete observations at times ti = i*Dn, i=0, 1, ..., n, under the asymptotic scheme Dn - 0, n*Dn - +oo and n*Dn^2 - 0. The class of phi-divergences is wide and includes several special members like Kullback-Leibler, Renyi, power and alpha-divergences. We derive the asymptotic distribution of the test statistics based on phi- divergences. The limiting law takes different forms depending on the regularity of phi. These convergence differ from the classical results for independent and identically distributed random variables. Numerical analysis is used to show the small sample properties of the test statistics in terms of estimated level and power of the test.

joint work with A. De Gregorio

[ Reference URL ]In this paper we propose the use of $\\phi$-divergences as test statistics to verify simple hypotheses about a one-dimensional parametric diffusion process dXt = b(Xt, theta)dt + sigma(Xt, theta) dWt, from discrete observations at times ti = i*Dn, i=0, 1, ..., n, under the asymptotic scheme Dn - 0, n*Dn - +oo and n*Dn^2 - 0. The class of phi-divergences is wide and includes several special members like Kullback-Leibler, Renyi, power and alpha-divergences. We derive the asymptotic distribution of the test statistics based on phi- divergences. The limiting law takes different forms depending on the regularity of phi. These convergence differ from the classical results for independent and identically distributed random variables. Numerical analysis is used to show the small sample properties of the test statistics in terms of estimated level and power of the test.

joint work with A. De Gregorio

https://www.ms.u-tokyo.ac.jp/~kengok/statseminar/2008/10.html

#### Seminar on Probability and Statistics

16:20-17:30 Room #002 (Graduate School of Math. Sci. Bldg.)

Stein estimation of Poisson process intensities

https://www.ms.u-tokyo.ac.jp/~kengok/statseminar/2008/11.html

**Nicolas Privault**(City University of Hong Kong)Stein estimation of Poisson process intensities

[ Abstract ]

In this talk we will construct superefficient estimators of Stein type for the intensity parameter lambda > 0 of a Poisson process, using integration by parts and superharmonic functionals on the Poisson space.

[ Reference URL ]In this talk we will construct superefficient estimators of Stein type for the intensity parameter lambda > 0 of a Poisson process, using integration by parts and superharmonic functionals on the Poisson space.

https://www.ms.u-tokyo.ac.jp/~kengok/statseminar/2008/11.html

### 2008/12/12

#### Lecture Series on Mathematical Sciences in Soceity

16:20-17:50 Room #128 (Graduate School of Math. Sci. Bldg.)

ガスタービン翼の伝熱について

**吉野 伸**(東京電力)ガスタービン翼の伝熱について

### 2008/12/11

#### Operator Algebra Seminars

16:30-18:00 Room #128 (Graduate School of Math. Sci. Bldg.)

Certain aperiodic automorphisms of unital simple projectionless $C^*$-algebras

**佐藤康彦**(北大理)Certain aperiodic automorphisms of unital simple projectionless $C^*$-algebras

### 2008/12/10

#### Geometry Seminar

14:45-18:00 Room #122 (Graduate School of Math. Sci. Bldg.)

Acyclic polarizations and localization of Riemann-Roch numbers

The topology of symplectic and hyperkahler quotients

**吉田 尚彦**(明治大学大学院理工学研究科) 14:45-16:15Acyclic polarizations and localization of Riemann-Roch numbers

[ Abstract ]

前量子化可能な閉シンプレクティック多様体が(特異)Lagrange ファイバー空間の構造を持つ場合,Riemann-Roch 数が Bohr-Sommerfeld ファイバーの個数と一致することがトーリック多様体,ユニタリー群の Gelfand-Cetlin 系や Riemann 面上の平坦 SU(2) 束のモジュライなどの例で,双方を別々に計算し比較することにより,確かめられている.本講演では,spin^c Dirac 作用素の指数に対する Witten 流の局所化を用いることによって,Riemann-Roch 数が非特異 Bohr-Sommerfeld ファイバー及び特異ファイバーに局所化することを示す.(古田幹雄氏(東大数理),藤田玄氏(学習院大学)との共同研究.論文:arXiv:0804.3258)

前量子化可能な閉シンプレクティック多様体が(特異)Lagrange ファイバー空間の構造を持つ場合,Riemann-Roch 数が Bohr-Sommerfeld ファイバーの個数と一致することがトーリック多様体,ユニタリー群の Gelfand-Cetlin 系や Riemann 面上の平坦 SU(2) 束のモジュライなどの例で,双方を別々に計算し比較することにより,確かめられている.本講演では,spin^c Dirac 作用素の指数に対する Witten 流の局所化を用いることによって,Riemann-Roch 数が非特異 Bohr-Sommerfeld ファイバー及び特異ファイバーに局所化することを示す.(古田幹雄氏(東大数理),藤田玄氏(学習院大学)との共同研究.論文:arXiv:0804.3258)

**Megumi Harada**(McMaster University) 16:30-18:00The topology of symplectic and hyperkahler quotients

[ Abstract ]

Symplectic geometry lies at the crossroads of many exciting areas of research due to its relationship to geometric representation theory, combinatorics, and algebraic geometry, among others. As often happens in mathematics, the presence of symmetry in these geometric structures -- in this context, a Hamiltonian G-action for a Lie group G, i.e. an action with an associated moment map -- turns out to be crucial in the computation of topological invariants, such as the Betti numbers, the cohomology ring, or the K-theory, of symplectic manifolds which arise as Hamiltonian quotients. In the first part of the talk, I will give a bird's-eye, motivating overview of this subject, and in particular will introduce one of the main technical tools of the field, which is the Morse theory associated to the moment map. In the second part, I will give a more detailed account of recent joint work with Graeme Wilkin, which deals with Nakajima quiver varieties, a special case of hyperkahler Hamiltonian quotients. In particular, we develop a Morse theory for the hyperkahler moment map analogous to the case of the moduli space of Higgs bundles. In particular, we show that the Harder-Narasimhan stratification of spaces of representations of quivers coincide with the Morse-theoretic stratification associated to the norm-square of the real moment map. Our approach also provides insight into the topology of specific examples of small-rank quiver varieties, including hyperpolygon spaces and some ADHM quivers.

Symplectic geometry lies at the crossroads of many exciting areas of research due to its relationship to geometric representation theory, combinatorics, and algebraic geometry, among others. As often happens in mathematics, the presence of symmetry in these geometric structures -- in this context, a Hamiltonian G-action for a Lie group G, i.e. an action with an associated moment map -- turns out to be crucial in the computation of topological invariants, such as the Betti numbers, the cohomology ring, or the K-theory, of symplectic manifolds which arise as Hamiltonian quotients. In the first part of the talk, I will give a bird's-eye, motivating overview of this subject, and in particular will introduce one of the main technical tools of the field, which is the Morse theory associated to the moment map. In the second part, I will give a more detailed account of recent joint work with Graeme Wilkin, which deals with Nakajima quiver varieties, a special case of hyperkahler Hamiltonian quotients. In particular, we develop a Morse theory for the hyperkahler moment map analogous to the case of the moduli space of Higgs bundles. In particular, we show that the Harder-Narasimhan stratification of spaces of representations of quivers coincide with the Morse-theoretic stratification associated to the norm-square of the real moment map. Our approach also provides insight into the topology of specific examples of small-rank quiver varieties, including hyperpolygon spaces and some ADHM quivers.

### 2008/12/09

#### Tuesday Seminar on Topology

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

Tits alternative in $Diff^1(S^1)$

**Bertrand Deroin**(CNRS, Orsay, Universit\'e Paris-Sud 11)Tits alternative in $Diff^1(S^1)$

[ Abstract ]

The following form of Tits alternative for subgroups of

homeomorphisms of the circle has been proved by Margulis: or the group

preserve a probability measure on the circle, or it contains a free

subgroup on two generators. We will prove that if the group acts by diffeomorphisms of

class $C^1$ and does not preserve a probability measure on the circle, then

in fact it contains a subgroup topologically conjugated to a Schottky group.

This is a joint work with V. Kleptsyn and A. Navas.

The following form of Tits alternative for subgroups of

homeomorphisms of the circle has been proved by Margulis: or the group

preserve a probability measure on the circle, or it contains a free

subgroup on two generators. We will prove that if the group acts by diffeomorphisms of

class $C^1$ and does not preserve a probability measure on the circle, then

in fact it contains a subgroup topologically conjugated to a Schottky group.

This is a joint work with V. Kleptsyn and A. Navas.

### 2008/12/08

#### Seminar on Geometric Complex Analysis

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

Critically finite holomorphic maps on projective spaces

**上田 哲生**(京大理)Critically finite holomorphic maps on projective spaces

### 2008/12/05

#### Lecture Series on Mathematical Sciences in Soceity

16:20-17:50 Room #128 (Graduate School of Math. Sci. Bldg.)

金融リスク管理と数理Ⅰ(基礎編)

**池森 俊文**(みずほ第一フィナンシャルテクノロジー)金融リスク管理と数理Ⅰ(基礎編)

### 2008/12/04

#### Operator Algebra Seminars

16:30-18:00 Room #128 (Graduate School of Math. Sci. Bldg.)

カッツ環の作用の分類

**戸松玲治**(東大数理)カッツ環の作用の分類

#### Lie Groups and Representation Theory

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

Realization of quanternionic discrete series as spaces of H-holomorphic

functions

[ Reference URL ]

https://www.ms.u-tokyo.ac.jp/~toshi/seminar/ut-seminar.html

**Genkai Zhang**(Chalmers and Gothenburg University)Realization of quanternionic discrete series as spaces of H-holomorphic

functions

[ Reference URL ]

https://www.ms.u-tokyo.ac.jp/~toshi/seminar/ut-seminar.html

### 2008/12/03

#### Mathematical Finance

17:30-19:00 Room #122 (Graduate School of Math. Sci. Bldg.)

The structure of dynamic utility functions in a Brownian Filtration

**Freddy Delaben**(ETH)The structure of dynamic utility functions in a Brownian Filtration

[ Abstract ]

The penalty function for monetary dynamic utility functions

has a special form. They can be seen as potentials. In the Brownian Filtration Rao's theorem permits to give a complete description.

The penalty function for monetary dynamic utility functions

has a special form. They can be seen as potentials. In the Brownian Filtration Rao's theorem permits to give a complete description.

#### Number Theory Seminar

16:30-17:30 Room #056 (Graduate School of Math. Sci. Bldg.)

Mean-periodicity and analytic properties of zeta-functions

**鈴木正俊**(東京大学大学院数理科学研究科)Mean-periodicity and analytic properties of zeta-functions

[ Abstract ]

Mean-periodicityというのは周期性の概念のひとつの一般化である。最近、I. Fesenko, G. Ricottaとの共同研究により、数論的スキームのゼータ関数を含むある複素関数のクラスと、mean-periodicityとの関連性が新しく見出された。

これはHecke-Weilによる, 解析接続と関数等式を持つDirichlet級数と保型形式との対応の一つの拡張ともみなせる. この背景には, I. Fesenkoの高次元アデール上のゼータ積分の理論があり、数論的スキームのHasseゼータ関数の解析接続を高次元アデール上の調和解析から導こうというプログラムの一環となっている。

この講演ではそのような背景にも若干触れた上、ゼータ関数の解析的性質とmean-periodicityの関連、特に解析接続と関数等式との関連について解説する。

Mean-periodicityというのは周期性の概念のひとつの一般化である。最近、I. Fesenko, G. Ricottaとの共同研究により、数論的スキームのゼータ関数を含むある複素関数のクラスと、mean-periodicityとの関連性が新しく見出された。

これはHecke-Weilによる, 解析接続と関数等式を持つDirichlet級数と保型形式との対応の一つの拡張ともみなせる. この背景には, I. Fesenkoの高次元アデール上のゼータ積分の理論があり、数論的スキームのHasseゼータ関数の解析接続を高次元アデール上の調和解析から導こうというプログラムの一環となっている。

この講演ではそのような背景にも若干触れた上、ゼータ関数の解析的性質とmean-periodicityの関連、特に解析接続と関数等式との関連について解説する。

### 2008/12/02

#### Lie Groups and Representation Theory

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

消滅と剛性

https://www.ms.u-tokyo.ac.jp/~toshi/seminar/ut-seminar.html

**金井雅彦**(名古屋大学)消滅と剛性

[ Abstract ]

The aim of my talk is to reveal an unforeseen link between the classical vanishing theorems of Matsushima and Weil, on the one hand, and rigidity of the Weyl chamber flow, a dynamical system arising from a higher-rank noncompact Lie group, on the other.

The connection is established via "transverse extension theorems": Roughly speaking, they claim that a tangential 1- form of the orbit foliation of the Weyl chamber flow that is tangentially closed (and satisfies a certain mild additional condition) can be extended to a closed 1- form on the whole space in a canonical manner. In particular, infinitesimal rigidity of the orbit foliation of the Weyl chamber flow is proved as an application.

[ Reference URL ]The aim of my talk is to reveal an unforeseen link between the classical vanishing theorems of Matsushima and Weil, on the one hand, and rigidity of the Weyl chamber flow, a dynamical system arising from a higher-rank noncompact Lie group, on the other.

The connection is established via "transverse extension theorems": Roughly speaking, they claim that a tangential 1- form of the orbit foliation of the Weyl chamber flow that is tangentially closed (and satisfies a certain mild additional condition) can be extended to a closed 1- form on the whole space in a canonical manner. In particular, infinitesimal rigidity of the orbit foliation of the Weyl chamber flow is proved as an application.

https://www.ms.u-tokyo.ac.jp/~toshi/seminar/ut-seminar.html

#### Tuesday Seminar on Topology

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

Vanishing and Rigidity

**金井 雅彦**(名古屋大学多元数理科学研究科)Vanishing and Rigidity

[ Abstract ]

The aim of my talk is to reveal an unforeseen link between

the classical vanishing theorems of Matsushima and Weil, on the one hand,

andrigidity of the Weyl chamber flow, a dynamical system arising from a higher-rank

noncompact Lie group, on the other. The connection is established via

"transverse extension theorems": Roughly speaking, they claim that a tangential 1- form of the

orbit foliation of the Weyl chamber flow that is tangentially closed

(and satisfies a certain mild additional condition) can be extended to a closed 1- form on the

whole space in a canonical manner. In particular, infinitesimal

rigidity of the orbit foliation of the Weyl chamber flow is proved as an application.

The aim of my talk is to reveal an unforeseen link between

the classical vanishing theorems of Matsushima and Weil, on the one hand,

andrigidity of the Weyl chamber flow, a dynamical system arising from a higher-rank

noncompact Lie group, on the other. The connection is established via

"transverse extension theorems": Roughly speaking, they claim that a tangential 1- form of the

orbit foliation of the Weyl chamber flow that is tangentially closed

(and satisfies a certain mild additional condition) can be extended to a closed 1- form on the

whole space in a canonical manner. In particular, infinitesimal

rigidity of the orbit foliation of the Weyl chamber flow is proved as an application.

### 2008/12/01

#### Seminar on Geometric Complex Analysis

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

双曲的Gauss写像の値分布

**川上 裕**(九大数理/大阪市立大)双曲的Gauss写像の値分布

#### Kavli IPMU Komaba Seminar

17:00-18:30 Room #002 (Graduate School of Math. Sci. Bldg.)

A pair of non-birational but derived equivalent Calabi-Yau

manifolds from non-Abelian gauge theories

**Kentaro Hori**(University of Toronto / IPMU)A pair of non-birational but derived equivalent Calabi-Yau

manifolds from non-Abelian gauge theories

[ Abstract ]

We construct a family of (2,2) supersymmetric gauge theories

in 2-dimensions that flows to a family of (2,2) superconformal fields theories with \\hat{c}=3. The family has two limits and three singular points. The two limits correspond to two Calabi-Yau manifolds which are not birationally equivalent. The two are, however, derived equivalent

by general principle of supersymmetric quantum field theory.

We construct a family of (2,2) supersymmetric gauge theories

in 2-dimensions that flows to a family of (2,2) superconformal fields theories with \\hat{c}=3. The family has two limits and three singular points. The two limits correspond to two Calabi-Yau manifolds which are not birationally equivalent. The two are, however, derived equivalent

by general principle of supersymmetric quantum field theory.

### 2008/11/28

#### Lecture Series on Mathematical Sciences in Soceity

16:20-17:50 Room #128 (Graduate School of Math. Sci. Bldg.)

製鐵プロセスの数学Ⅱ

**中川 淳一**(新日本製鐵先端技術研究所)製鐵プロセスの数学Ⅱ

#### Colloquium

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

曲線の錐体と因子の錐体

[開催日にご注意下さい]

**川又雄二郎**(東京大学大学院数理科学研究科)曲線の錐体と因子の錐体

[ Abstract ]

代数多様体上に載っている曲線と因子の交点数を使うと、互いに双対な有限次元実ベクトル空間内の、互いに双対な閉凸錐体 -- 曲線の錐体と因子の錐体が定義される。極小モデル理論では、曲線の錐体の端射線から収縮写像が構成されるが、双有理同値な代数多様体をたくさん同時に考えるためには、因子の錐体のほうが便利である。標準環の有限生成定理は、因子の錐体の集まりの間の壁越えの様子を詳しく調べることによって証明された。一般の代数多様体に対する極小モデルの存在は未解決問題であるが、そのためには因子の錐体についてのより深い理解が必要と思われる。この講演ではそのあたりの事情を解説する。

[ Reference URL ]代数多様体上に載っている曲線と因子の交点数を使うと、互いに双対な有限次元実ベクトル空間内の、互いに双対な閉凸錐体 -- 曲線の錐体と因子の錐体が定義される。極小モデル理論では、曲線の錐体の端射線から収縮写像が構成されるが、双有理同値な代数多様体をたくさん同時に考えるためには、因子の錐体のほうが便利である。標準環の有限生成定理は、因子の錐体の集まりの間の壁越えの様子を詳しく調べることによって証明された。一般の代数多様体に対する極小モデルの存在は未解決問題であるが、そのためには因子の錐体についてのより深い理解が必要と思われる。この講演ではそのあたりの事情を解説する。

[開催日にご注意下さい]

#### GCOE lecture series

14:40-16:10 Room #002 (Graduate School of Math. Sci. Bldg.)

Circle-valued Morse theory

, Lecture 2

**Andrei Pajitnov**(Univ. de Nantes)Circle-valued Morse theory

, Lecture 2

### 2008/11/26

#### Algebraic Geometry Seminar

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

Diagonal subschemes and vector bundles

**Piotr Pragacz**

(Banach Institute)Diagonal subschemes and vector bundles

#### GCOE lecture series

14:40-16:10 Room #002 (Graduate School of Math. Sci. Bldg.)

Circle-valued Morse theory, Lecture 1

**Andrei Pajitnov**(Univ. de Nantes)Circle-valued Morse theory, Lecture 1

[ Abstract ]

Morse theory of circle-valued functions, initiated by S. P. Novikov in 1980-1982 is now a rapidly developing domain with applications and connections to many other fields of geometry and topology such as dynamical systems, Lagrangian intersections,

knots and links in three-dimensional sphere.

We will start with the basics of the theory, discuss the construction of the Novikov complex, relations with the dynamical zeta functions, and the knot theory. We will conclude with a list of the open problems of the theory.

Morse theory of circle-valued functions, initiated by S. P. Novikov in 1980-1982 is now a rapidly developing domain with applications and connections to many other fields of geometry and topology such as dynamical systems, Lagrangian intersections,

knots and links in three-dimensional sphere.

We will start with the basics of the theory, discuss the construction of the Novikov complex, relations with the dynamical zeta functions, and the knot theory. We will conclude with a list of the open problems of the theory.

#### Number Theory Seminar

16:30-17:30 Room #056 (Graduate School of Math. Sci. Bldg.)

Lang's Observation in Diophantine Problems

**平田典子**(日本大学理工学部)Lang's Observation in Diophantine Problems

[ Abstract ]

In 1964, Serge Lang suggested the following problem, which reads now as follows:

Let $E$ be an elliptic curve defined over a number field $K$, and $\\varphi$ be a rational function on $E$. Then, for every point $P\\in E(K)$ where $\\varphi$ does not vanish at $P$, the logarithms of a norm of $\\varphi(P)$ is at worst linear in the logarithms of the Neron-Tate height of the point $P$.

We give a simultaneous Diophantine approximation for linear forms in elliptic logarithms which actually implies this conjecture. We also present Lang's observations in Diophantine problems.

In 1964, Serge Lang suggested the following problem, which reads now as follows:

Let $E$ be an elliptic curve defined over a number field $K$, and $\\varphi$ be a rational function on $E$. Then, for every point $P\\in E(K)$ where $\\varphi$ does not vanish at $P$, the logarithms of a norm of $\\varphi(P)$ is at worst linear in the logarithms of the Neron-Tate height of the point $P$.

We give a simultaneous Diophantine approximation for linear forms in elliptic logarithms which actually implies this conjecture. We also present Lang's observations in Diophantine problems.

### 2008/11/25

#### Tuesday Seminar of Analysis

16:30-18:00 Room #128 (Graduate School of Math. Sci. Bldg.)

Heat kernels for subelliptic operators

**Ovidiu Calin**(Eastern Michigan University)Heat kernels for subelliptic operators

[ Abstract ]

Subelliptic operators are differential operators with missing

directions. Their behavior is very different than the behavior or

elliptic operators. Among the most well known subelliptic operators

are the Grusin operator, the Heisenberg operator, and the Kolmogorov

operator. There are several methods of finding the heat kernels of

subelliptic operators. The heat kernels of subelliptic operators are

usually represented in integral form, but in the case of the

Kolmogorov operator we shall show that the heat kernel is of function

type. We shall spend some time on other subelliptic operators too.

Subelliptic operators are differential operators with missing

directions. Their behavior is very different than the behavior or

elliptic operators. Among the most well known subelliptic operators

are the Grusin operator, the Heisenberg operator, and the Kolmogorov

operator. There are several methods of finding the heat kernels of

subelliptic operators. The heat kernels of subelliptic operators are

usually represented in integral form, but in the case of the

Kolmogorov operator we shall show that the heat kernel is of function

type. We shall spend some time on other subelliptic operators too.

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