## 講演会

過去の記録 ～10/22｜次回の予定｜今後の予定 10/23～

**過去の記録**

### 2018年09月21日(金)

17:00-18:00 数理科学研究科棟(駒場) 056号室

On the geometry of some p-adic period domains (ENGLISH)

**Laurent Fargues 氏**(CNRS, Institut Mathématique de Jussieu)On the geometry of some p-adic period domains (ENGLISH)

[ 講演概要 ]

p-adic period spaces have been introduced by Rapoport and Zink as a generalization of Drinfeld upper half spaces and Lubin-Tate spaces. Those are open subsets of a rigid analytic p-adic flag manifold. An approximation of this open subset is the so called weakly admissible locus obtained by removing a profinite set of closed Schubert varieties. I will explain a recent theorem characterizing when the period space coincides with the weakly admissible locus. As an application we can compute the p-adic period space of K3 surfaces with supersingular reduction. The talk will be mainly introductory, presenting the objects showing up in this theorem. This is joint work with Miaofen Chen and Xu Shen.

p-adic period spaces have been introduced by Rapoport and Zink as a generalization of Drinfeld upper half spaces and Lubin-Tate spaces. Those are open subsets of a rigid analytic p-adic flag manifold. An approximation of this open subset is the so called weakly admissible locus obtained by removing a profinite set of closed Schubert varieties. I will explain a recent theorem characterizing when the period space coincides with the weakly admissible locus. As an application we can compute the p-adic period space of K3 surfaces with supersingular reduction. The talk will be mainly introductory, presenting the objects showing up in this theorem. This is joint work with Miaofen Chen and Xu Shen.

### 2018年09月18日(火)

17:00-18:00 数理科学研究科棟(駒場) 056号室

Topological epsilon-factors (ENGLISH)

**Alexander Beilinson 氏**(University of Chicago)Topological epsilon-factors (ENGLISH)

[ 講演概要 ]

I will explain (following mostly my old article arXiv:0610055) how the Kashiwara-Shapira Morse theory construction of the characteristic cycle of a constructible R-sheaf can be refined to yield the cycle with coefficients in the K-theory spectrum K(R). The construction can be viewed as a topological analog of the arithmetic theory of epsilon-factors.

I will explain (following mostly my old article arXiv:0610055) how the Kashiwara-Shapira Morse theory construction of the characteristic cycle of a constructible R-sheaf can be refined to yield the cycle with coefficients in the K-theory spectrum K(R). The construction can be viewed as a topological analog of the arithmetic theory of epsilon-factors.

### 2018年07月10日(火)

15:00-16:00 数理科学研究科棟(駒場) 128号室

Canceled!! (諸事情により,講演は取りやめとなりました.)

On the moduli space of flat symplectic surface bundles

Canceled!! (諸事情により,講演は取りやめとなりました.)

**Sam Nariman 氏**(Northwestern University)On the moduli space of flat symplectic surface bundles

[ 講演概要 ]

There are at least three different approaches to construct characteristic invariants of flat symplectic bundles. Reznikov generalized Chern-Weil theory for finite dimension Lie groups to the infinite dimensional group of symplectomorphisms. He constructed nontrivial invariants of symplectic bundles whose fibers are diffeomorphic to complex projective spaces. Kontsevich used formal symplectic geometry to build interesting classes that are not yet known to be nontrivial. Also for surface bundles whose holonomy groups preserve the symplectic form, Kotschick and Morita used the flux homomorphism to construct many nontrivial stable classes.

In this talk, we introduce infinite loop spaces whose cohomolgy groups describe the stable characteristic invariants of symplectic flat surface bundles. As an application, we give a homotopy theoretic description of

Kotschick and Morita's classes and prove a result about codimension 2 foliations that implies the nontriviality of KM classes.

There are at least three different approaches to construct characteristic invariants of flat symplectic bundles. Reznikov generalized Chern-Weil theory for finite dimension Lie groups to the infinite dimensional group of symplectomorphisms. He constructed nontrivial invariants of symplectic bundles whose fibers are diffeomorphic to complex projective spaces. Kontsevich used formal symplectic geometry to build interesting classes that are not yet known to be nontrivial. Also for surface bundles whose holonomy groups preserve the symplectic form, Kotschick and Morita used the flux homomorphism to construct many nontrivial stable classes.

In this talk, we introduce infinite loop spaces whose cohomolgy groups describe the stable characteristic invariants of symplectic flat surface bundles. As an application, we give a homotopy theoretic description of

Kotschick and Morita's classes and prove a result about codimension 2 foliations that implies the nontriviality of KM classes.

### 2018年06月22日(金)

16:00-17:00 数理科学研究科棟(駒場) 128号室

Fibrations of R^3 by oriented lines

**Michael Harrison 氏**(Lehigh University)Fibrations of R^3 by oriented lines

[ 講演概要 ]

Is it possible to cover 3-dimensional space by a collection of lines, such that no two lines intersect and no two lines are parallel? More precisely, does there exist a fibration of R^3 by pairwise skew lines? We give some examples and provide a complete topological classification of such objects, by exhibiting a deformation retract from the space of skew fibrations of R^3 to its subspace of Hopf fibrations. As a corollary of the proof we obtain Gluck and Warner's classification of great circle fibrations of S^3. We continue with some recent results regarding contact structures on R^3 which are naturally induced by skew fibrations. Finally, we discuss fibrations of R^3 which may contain parallel fibers, and discuss when such objects induce contact structures.

Is it possible to cover 3-dimensional space by a collection of lines, such that no two lines intersect and no two lines are parallel? More precisely, does there exist a fibration of R^3 by pairwise skew lines? We give some examples and provide a complete topological classification of such objects, by exhibiting a deformation retract from the space of skew fibrations of R^3 to its subspace of Hopf fibrations. As a corollary of the proof we obtain Gluck and Warner's classification of great circle fibrations of S^3. We continue with some recent results regarding contact structures on R^3 which are naturally induced by skew fibrations. Finally, we discuss fibrations of R^3 which may contain parallel fibers, and discuss when such objects induce contact structures.

### 2018年06月12日(火)

15:00-16:00 数理科学研究科棟(駒場) 056号室

Cellular E_2-algebras and the unstable homology of mapping class groups

**Alexander Kupers 氏**(Harvard University)Cellular E_2-algebras and the unstable homology of mapping class groups

[ 講演概要 ]

We discuss joint work with Soren Galatius and Oscar Randal-Williams on the application of higher-algebraic techniques to classical questions about the homology of mapping class groups. This uses a new "multiplicative" approach to homological stability -- in contrast to the "additive" one due to Quillen -- which has the advantage of providing information outside of the stable range.

We discuss joint work with Soren Galatius and Oscar Randal-Williams on the application of higher-algebraic techniques to classical questions about the homology of mapping class groups. This uses a new "multiplicative" approach to homological stability -- in contrast to the "additive" one due to Quillen -- which has the advantage of providing information outside of the stable range.

### 2018年05月11日(金)

13:00-14:00 数理科学研究科棟(駒場) 123号室

The Langlands-Kottwitz method for deformation spaces of Hodge type

**Alex Youcis 氏**(University of California, Berkeley)The Langlands-Kottwitz method for deformation spaces of Hodge type

[ 講演概要 ]

Cohomology of global Shimura varieties is an object of universal importance in the Langlands program. Given a Shimura datum (G,X) and a (sufficiently nice) representation ¥xi of G, one obtains an l-adic sheaf F_{¥xi,l} on Sh(G,X) with a G(A_f)-structure. Thus, in the standard way, the cohomology group H^*(Sh(G,X),F_¥xi) has an admissible action of Gal(¥overline{E}/E) ¥times G(A_f), where E=E(G,X) is the reflex field of (G,X). Extending work of Kottwitz, Scholze, and others we discuss a method for computing the traces of this action, more specifically of an element ¥tau ¥times g where ¥tau ¥in W_{E_¥mathfrak{p}} for some prime ¥mathfrak{p} of E dividing p and g ¥in G(A_f^p) ¥times G(Z_p), in terms of a weighted point count on the Shimura variety's special fiber, as well as the traces of various local Shimura varieties over E_¥mathfrak{p}, at least in the case when (G,X) is a abelian-type Shimura datum unramified at p.

Cohomology of global Shimura varieties is an object of universal importance in the Langlands program. Given a Shimura datum (G,X) and a (sufficiently nice) representation ¥xi of G, one obtains an l-adic sheaf F_{¥xi,l} on Sh(G,X) with a G(A_f)-structure. Thus, in the standard way, the cohomology group H^*(Sh(G,X),F_¥xi) has an admissible action of Gal(¥overline{E}/E) ¥times G(A_f), where E=E(G,X) is the reflex field of (G,X). Extending work of Kottwitz, Scholze, and others we discuss a method for computing the traces of this action, more specifically of an element ¥tau ¥times g where ¥tau ¥in W_{E_¥mathfrak{p}} for some prime ¥mathfrak{p} of E dividing p and g ¥in G(A_f^p) ¥times G(Z_p), in terms of a weighted point count on the Shimura variety's special fiber, as well as the traces of various local Shimura varieties over E_¥mathfrak{p}, at least in the case when (G,X) is a abelian-type Shimura datum unramified at p.

### 2018年05月10日(木)

11:00-12:00 数理科学研究科棟(駒場) 123号室

The Cohomology of Rapoport-Zink Spaces of EL-Type

**Alexander Bertoloni Meli 氏**(University of California, Berkeley)The Cohomology of Rapoport-Zink Spaces of EL-Type

[ 講演概要 ]

I will discuss Rapoport-Zink spaces of EL-type and how to explicitly compute a certain variant of their cohomology in terms of the local Langlands correspondence for general linear groups. I will then show how this computation can be used to resolve certain cases of a conjecture of Harris.

I will discuss Rapoport-Zink spaces of EL-type and how to explicitly compute a certain variant of their cohomology in terms of the local Langlands correspondence for general linear groups. I will then show how this computation can be used to resolve certain cases of a conjecture of Harris.

### 2018年05月08日(火)

13:00-14:00 数理科学研究科棟(駒場) 122号室

Langlands-Rapoport for the Modular Curve

**Sander Mack-Crane 氏**(University of California, Berkeley)Langlands-Rapoport for the Modular Curve

[ 講演概要 ]

We discuss a concrete version of the Langlands-Rapoport conjecture in the case of the modular curve, and use this case to illuminate some of the more abstract features of the Langlands-Rapoport conjecture for general (abelian type) Shimura varieties.

We discuss a concrete version of the Langlands-Rapoport conjecture in the case of the modular curve, and use this case to illuminate some of the more abstract features of the Langlands-Rapoport conjecture for general (abelian type) Shimura varieties.

### 2018年03月13日(火)

10:00-11:00 数理科学研究科棟(駒場) 126号室

GSpにおけるDeligne-Lusztig多様体とaffine Deligne-Lusztig多様体との比較

**高松 哲平 氏**(東大数理)GSpにおけるDeligne-Lusztig多様体とaffine Deligne-Lusztig多様体との比較

[ 講演概要 ]

Deligne-Lusztig理論とは、有限体上の簡約代数群の有理点の表現を、Deligne-Lusztig多様体と呼ばれる代数多様体のエタールコホモロジーに実現する理論であった。Lusztigは、この理論の非Archimedes的局所体K上での類似の存在を予想した。しかし、Deligne-Lusztig多様体の局所体上の直接の類似物は、アプリオリにはscheme構造を持たないという問題がある。

他方で、局所体K上の簡約代数群に対して、affine Deligne-Lusztig多様体とよばれるDeligne-Lusztig多様体の岩堀類似物が存在し、それらはKの剰余体の閉包上のscheme構造を持つことが知られている。

本講演では、Chan-IvanovのGLの場合での研究の方針にならい、GSpの場合に、Deligne-Lusztig多様体及びaffine Deligne-Lusztig多様体のσ線形代数的記述を示し、その応用としてDeligne-Lusztig多様体にpro-scheme構造を入れられることを説明する。

Deligne-Lusztig理論とは、有限体上の簡約代数群の有理点の表現を、Deligne-Lusztig多様体と呼ばれる代数多様体のエタールコホモロジーに実現する理論であった。Lusztigは、この理論の非Archimedes的局所体K上での類似の存在を予想した。しかし、Deligne-Lusztig多様体の局所体上の直接の類似物は、アプリオリにはscheme構造を持たないという問題がある。

他方で、局所体K上の簡約代数群に対して、affine Deligne-Lusztig多様体とよばれるDeligne-Lusztig多様体の岩堀類似物が存在し、それらはKの剰余体の閉包上のscheme構造を持つことが知られている。

本講演では、Chan-IvanovのGLの場合での研究の方針にならい、GSpの場合に、Deligne-Lusztig多様体及びaffine Deligne-Lusztig多様体のσ線形代数的記述を示し、その応用としてDeligne-Lusztig多様体にpro-scheme構造を入れられることを説明する。

### 2018年03月09日(金)

13:30-14:30 数理科学研究科棟(駒場) 056号室

Sliced nearby cycles and duality, after W. Zheng (ENGLISH)

**Luc Illusie 氏**(パリ南大学名誉教授)Sliced nearby cycles and duality, after W. Zheng (ENGLISH)

[ 講演概要 ]

In the early 1980's Gabber proved duality for nearby cycles and, by a different method, Beilinson proved duality for vanishing cycles in the strictly local case (up to a twist of the inertia action on the tame part). Recently W. Zheng found a simple proof of a result, conjectured by Deligne, which implies them both, and extended it over finite dimensional excellent bases. I will explain the main ideas of his work, which relies on new developments, due to him, of Deligne's theory of fibered and oriented products.

In the early 1980's Gabber proved duality for nearby cycles and, by a different method, Beilinson proved duality for vanishing cycles in the strictly local case (up to a twist of the inertia action on the tame part). Recently W. Zheng found a simple proof of a result, conjectured by Deligne, which implies them both, and extended it over finite dimensional excellent bases. I will explain the main ideas of his work, which relies on new developments, due to him, of Deligne's theory of fibered and oriented products.

### 2017年10月25日(水)

11:00-12:00 数理科学研究科棟(駒場) 128号室

On Faltings' main comparison theorem in p-adic Hodge theory : the relative case (ENGLISH)

**Ahmed Abbes 氏**(CNRS/IHES)On Faltings' main comparison theorem in p-adic Hodge theory : the relative case (ENGLISH)

[ 講演概要 ]

In the appendix of his 2002 Asterisque article, Faltings roughly sketched a proof of a relative version of his main comparison theorem in p-adic Hodge theory. I will briefly review the absolute case and then explain some of the key new inputs for the proof of the relative case (joint work with Michel Gros).

In the appendix of his 2002 Asterisque article, Faltings roughly sketched a proof of a relative version of his main comparison theorem in p-adic Hodge theory. I will briefly review the absolute case and then explain some of the key new inputs for the proof of the relative case (joint work with Michel Gros).

### 2017年10月17日(火)

17:00-18:00 数理科学研究科棟(駒場) 128号室

Integer partitions and hook length formulas (ENGLISH)

www-irma.u-strasbg.fr/~guoniu/

**Guoniu Han 氏**(Université de Strasbourg/CNRS)Integer partitions and hook length formulas (ENGLISH)

[ 講演概要 ]

Integer partitions were first studied by Euler.

The Ferrers diagram of an integer partition is a very useful tool for

visualizing partitions. A Ferrers diagram is turned into a Young tableau

by filling each cell with a unique integer satisfying some conditions.

The number of Young tableaux is given by the famous hook length formula,

discovered by Frame-Robinson-Thrall.

In this talk, we introduce the hook length expansion technique and

explain how to find old and new hook length formulas for integer

partitions. In particular, we derive an expansion formula for the

powers of the Euler Product in terms of hook lengths, which is also

discovered by Nekrasov-Okounkov and Westburg. We obtain an extension

by adding two more parameters. It appears to be a discrete

interpolation between the Macdonald identities and the generating

function for t-cores. Several other summations involving hook length,

in particular, the Okada-Panova formula, will also be discussed.

[ 講演参考URL ]Integer partitions were first studied by Euler.

The Ferrers diagram of an integer partition is a very useful tool for

visualizing partitions. A Ferrers diagram is turned into a Young tableau

by filling each cell with a unique integer satisfying some conditions.

The number of Young tableaux is given by the famous hook length formula,

discovered by Frame-Robinson-Thrall.

In this talk, we introduce the hook length expansion technique and

explain how to find old and new hook length formulas for integer

partitions. In particular, we derive an expansion formula for the

powers of the Euler Product in terms of hook lengths, which is also

discovered by Nekrasov-Okounkov and Westburg. We obtain an extension

by adding two more parameters. It appears to be a discrete

interpolation between the Macdonald identities and the generating

function for t-cores. Several other summations involving hook length,

in particular, the Okada-Panova formula, will also be discussed.

www-irma.u-strasbg.fr/~guoniu/

### 2017年10月11日(水)

11:00-12:00 数理科学研究科棟(駒場) 128号室

On Faltings' main comparison theorem in p-adic Hodge theory : the relative case (ENGLISH)

**Ahmed Abbes 氏**(CNRS/IHES)On Faltings' main comparison theorem in p-adic Hodge theory : the relative case (ENGLISH)

[ 講演概要 ]

In the appendix of his 2002 Asterisque article, Faltings roughly sketched a proof of a relative version of his main comparison theorem in p-adic Hodge theory. I will briefly review the absolute case and then explain some of the key new inputs for the proof of the relative case (joint work with Michel Gros).

In the appendix of his 2002 Asterisque article, Faltings roughly sketched a proof of a relative version of his main comparison theorem in p-adic Hodge theory. I will briefly review the absolute case and then explain some of the key new inputs for the proof of the relative case (joint work with Michel Gros).

### 2017年09月11日(月)

15:30-16:30 数理科学研究科棟(駒場) 002号室

3D field theories with Chern-Simons term for large N in the Weyl gauge

(ENGLISH)

**Jean Zinn-Justin 氏**(CEA Saclay)3D field theories with Chern-Simons term for large N in the Weyl gauge

(ENGLISH)

[ 講演概要 ]

ADS/CFT correspondance has led to a number of conjectures concerning, conformal invariant, U(N) symmetric 3D field theories with Chern-Simons term for N large. An example is boson-fermion duality. This has prompted a number of calculations to shed extra light on the ADS/CFT correspondance.

We study here the example of gauge invariant fermion matter coupled to a Chern-Simons term. In contrast with previous calculations, which employ the light-cone gauge, we use the more conventional temporal gauge. We calculate several gauge invariant correlation functions. We consider general massive matter and determine the conditions for conformal invariance. We compare massless results with previous calculations, providing a check of gauge independence.

We examine also the possibility of spontaneous breaking of scale invariance and show that this requires the addition of an auxiliary scalar field.

Our method is based on field integral and steepest descent. The saddle point equations involve non-local fields and take the form of a set of integral equations that we solve exactly.

ADS/CFT correspondance has led to a number of conjectures concerning, conformal invariant, U(N) symmetric 3D field theories with Chern-Simons term for N large. An example is boson-fermion duality. This has prompted a number of calculations to shed extra light on the ADS/CFT correspondance.

We study here the example of gauge invariant fermion matter coupled to a Chern-Simons term. In contrast with previous calculations, which employ the light-cone gauge, we use the more conventional temporal gauge. We calculate several gauge invariant correlation functions. We consider general massive matter and determine the conditions for conformal invariance. We compare massless results with previous calculations, providing a check of gauge independence.

We examine also the possibility of spontaneous breaking of scale invariance and show that this requires the addition of an auxiliary scalar field.

Our method is based on field integral and steepest descent. The saddle point equations involve non-local fields and take the form of a set of integral equations that we solve exactly.

### 2017年05月23日(火)

17:00-18:00 数理科学研究科棟(駒場) 126号室

講演後の質疑応答の状況によっては、終了時間が多少遅れるかもしれません。

Enumeration of fully commutative elements in classical Coxeter groups (English)

http://math.univ-lyon1.fr/homes-www/jouhet/

講演後の質疑応答の状況によっては、終了時間が多少遅れるかもしれません。

**Frédéric Jouhet 氏**(Université Claude Bernard Lyon 1 / Institut Camille Jordan)Enumeration of fully commutative elements in classical Coxeter groups (English)

[ 講演概要 ]

An element of a Coxeter group W is fully commutative if any two of its reduced decompositions are related by a series of transpositions of adjacent commuting generators. They index naturally a basis of the (generalized) Temperley-Lieb algebra associated with W. In this talk, focusing on the (affine) type A, I will describe how to

enumerate these elements according to their Coxeter length, in all classical finite and affine Coxeter groups. The methods, which generalize previous work of Stembridge,

involve many combinatorial objects, such as heaps, walks, or parallelogram

polyominoes. This talk is based on joint works with R. Biagioli, M. Bousquet-Mélou and

P. Nadeau.

[ 講演参考URL ]An element of a Coxeter group W is fully commutative if any two of its reduced decompositions are related by a series of transpositions of adjacent commuting generators. They index naturally a basis of the (generalized) Temperley-Lieb algebra associated with W. In this talk, focusing on the (affine) type A, I will describe how to

enumerate these elements according to their Coxeter length, in all classical finite and affine Coxeter groups. The methods, which generalize previous work of Stembridge,

involve many combinatorial objects, such as heaps, walks, or parallelogram

polyominoes. This talk is based on joint works with R. Biagioli, M. Bousquet-Mélou and

P. Nadeau.

http://math.univ-lyon1.fr/homes-www/jouhet/

### 2015年07月28日(火)

17:00-18:00 数理科学研究科棟(駒場) 056号室

Convergence of some horocyclic deformations to the Gardiner-Masur

boundary of Teichmueller space. (ENGLISH)

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

boundary of Teichmueller space. (ENGLISH)

[ 講演概要 ]

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.

### 2015年05月21日(木)

16:00-17:00 数理科学研究科棟(駒場) 056号室

Tea:15:30～16:00 コモンルーム

The Shape of Data

(ENGLISH)

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

Tea:15:30～16:00 コモンルーム

**Gunnar Carlsson 氏**(Stanford University, Ayasdi INC)The Shape of Data

(ENGLISH)

[ 講演概要 ]

There is a tremendous amount of attention being paid to the notion of

"Big Data". In many situations, however, the problem is not so much the

size of the data but rather its complexity. This observation shows that

it is now important to find methods for representing complex data in a

compressed and understandable fashion. Representing data by shapes

turns out to be useful in many situations, and therefore topology, the

mathematical sub discipline which studies shape, becomes quite

relevant. There is now a collection of methods based on topology for

analyzing complex data, and in this talk we will discuss these methods,

with numerous examples.

[ 講演参考URL ]There is a tremendous amount of attention being paid to the notion of

"Big Data". In many situations, however, the problem is not so much the

size of the data but rather its complexity. This observation shows that

it is now important to find methods for representing complex data in a

compressed and understandable fashion. Representing data by shapes

turns out to be useful in many situations, and therefore topology, the

mathematical sub discipline which studies shape, becomes quite

relevant. There is now a collection of methods based on topology for

analyzing complex data, and in this talk we will discuss these methods,

with numerous examples.

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

### 2014年12月03日(水)

16:30-17:30 数理科学研究科棟(駒場) 128号室

New isoperimetric inequalities with densities, part II: Detailed proofs and related works (ENGLISH)

**Xavier Cabre 氏**(ICREA and UPC, Barcelona)New isoperimetric inequalities with densities, part II: Detailed proofs and related works (ENGLISH)

[ 講演概要 ]

This is a sequel to the Tuesday Analysis Seminar on December 2 by the same speaker.

In joint works with X. Ros-Oton and J. Serra, the study of the regularity of stable solutions to reaction-diffusion problems has led us to certain Sobolev and isoperimetric inequalities with weights. We will present our results in these new isoperimetric inequalities with the best constant, that we establish via the ABP method.

More precisely, we obtain a new family of sharp isoperimetric inequalities with weights (or densities) in open convex cones of R^n. Our results apply to all nonnegative homogeneous weights satisfying a concavity condition in the cone. Surprisingly, even that our weights are not radially symmetric, Euclidean balls centered at the origin (intersected with the cone) minimize the weighted isoperimetric quotient. As a particular case of our results, we provide with new proofs of classical results such as the Wulff inequality and the isoperimetric inequality in convex cones of Lions and Pacella. Furthermore, we also study the anisotropic isoperimetric problem for the same class of weights and we prove that the Wulff shape always minimizes the anisotropic weighted perimeter under the weighted volume constraint.

This is a sequel to the Tuesday Analysis Seminar on December 2 by the same speaker.

In joint works with X. Ros-Oton and J. Serra, the study of the regularity of stable solutions to reaction-diffusion problems has led us to certain Sobolev and isoperimetric inequalities with weights. We will present our results in these new isoperimetric inequalities with the best constant, that we establish via the ABP method.

More precisely, we obtain a new family of sharp isoperimetric inequalities with weights (or densities) in open convex cones of R^n. Our results apply to all nonnegative homogeneous weights satisfying a concavity condition in the cone. Surprisingly, even that our weights are not radially symmetric, Euclidean balls centered at the origin (intersected with the cone) minimize the weighted isoperimetric quotient. As a particular case of our results, we provide with new proofs of classical results such as the Wulff inequality and the isoperimetric inequality in convex cones of Lions and Pacella. Furthermore, we also study the anisotropic isoperimetric problem for the same class of weights and we prove that the Wulff shape always minimizes the anisotropic weighted perimeter under the weighted volume constraint.

### 2014年11月26日(水)

16:00-17:00 数理科学研究科棟(駒場) 123号室

Intertwinings, wave equations and beta ensembles (ENGLISH)

**Mykhaylo Shkolnikov 氏**(Princeton University)Intertwinings, wave equations and beta ensembles (ENGLISH)

[ 講演概要 ]

We will discuss a general theory of intertwined diffusion processes of any dimension. Intertwined processes arise in many different contexts in probability theory, most notably in the study of random matrices, random polymers and path decompositions of Brownian motion. Recently, they turned out to be also closely related to wave equations and more general hyperbolic partial differential equations. The talk will be devoted to this recent development, as well as an algebraic perspective on intertwinings which, in particular, gives rise to a novel intertwining in beta random matrix theory. Based on joint works with Vadim Gorin and Soumik Pal.

We will discuss a general theory of intertwined diffusion processes of any dimension. Intertwined processes arise in many different contexts in probability theory, most notably in the study of random matrices, random polymers and path decompositions of Brownian motion. Recently, they turned out to be also closely related to wave equations and more general hyperbolic partial differential equations. The talk will be devoted to this recent development, as well as an algebraic perspective on intertwinings which, in particular, gives rise to a novel intertwining in beta random matrix theory. Based on joint works with Vadim Gorin and Soumik Pal.

### 2014年09月04日(木)

12:10-13:00 数理科学研究科棟(駒場) 470号室

X-ray imaging of moving objects (ENGLISH)

**Samuli Siltanen 氏**(University of Helsinki, Finland)X-ray imaging of moving objects (ENGLISH)

### 2014年06月10日(火)

14:40-16:10 数理科学研究科棟(駒場) 056号室

Bipartite knots (ENGLISH)

**Sergei Duzhin 氏**(Steklov Institute of Mathematics)Bipartite knots (ENGLISH)

[ 講演概要 ]

We give a solution to a part of Problem 1.60 in Kirby's list of open

problems in topology thus proving a conjecture raised in 1987 by

J.Przytycki. A knot is said to be bipartite if it has a "matched" diagram,

that is, a plane diagram that has an even number of crossings which can be

split into pairs that look like a simple braid on two strands with two

crossings. The conjecture was that there exist knots that do not have such

diagrams. I will prove this fact using higher Alexander ideals.

This talk is based on a joint work with my student M.Shkolnikov

We give a solution to a part of Problem 1.60 in Kirby's list of open

problems in topology thus proving a conjecture raised in 1987 by

J.Przytycki. A knot is said to be bipartite if it has a "matched" diagram,

that is, a plane diagram that has an even number of crossings which can be

split into pairs that look like a simple braid on two strands with two

crossings. The conjecture was that there exist knots that do not have such

diagrams. I will prove this fact using higher Alexander ideals.

This talk is based on a joint work with my student M.Shkolnikov

### 2014年05月15日(木)

16:30-17:30 数理科学研究科棟(駒場) 050号室

Synthetic theory of Ricci curvature

― When Monge, Riemann and Boltzmann meet ― (ENGLISH)

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

**Cédric Villani 氏**(Université de Lyon, Institut Henri Poincaré)Synthetic theory of Ricci curvature

― When Monge, Riemann and Boltzmann meet ― (ENGLISH)

[ 講演概要 ]

Optimal transport theory, non-Euclidean geometry and statistical physics met fifteen years ago with the discovery that Ricci curvature can be studied quantitatively thanks to entropy and

Monge-Kantorovich transport.

This unexpected encounter was very fruitful, leading to progress in each of these fields.

[ 講演参考URL ]Optimal transport theory, non-Euclidean geometry and statistical physics met fifteen years ago with the discovery that Ricci curvature can be studied quantitatively thanks to entropy and

Monge-Kantorovich transport.

This unexpected encounter was very fruitful, leading to progress in each of these fields.

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

### 2014年03月13日(木)

10:15-11:45 数理科学研究科棟(駒場) 470号室

Almost sure triviality of the $C^1$-centralizer of random circle diffeomorphisms with periodic points (ENGLISH)

**Michele Triestino 氏**(Ecole Normale Superieure de Lyon)Almost sure triviality of the $C^1$-centralizer of random circle diffeomorphisms with periodic points (ENGLISH)

[ 講演概要 ]

By the end of the 80s, Malliavin and Shavgulidze introduced a measure on the space of C^1 circle diffeomorphisms which carries many interesting features. Perhaps the most interesting aspect is that it can be considered as an analog of the Haar measure for the group Diff^1_+(S^1).

The nature of this measure has been mostly investigated in connection to representation theory.

For people working in dynamical systems, the MS measure offers a way to quantify dynamical phenomena: for example, which is the probability that a random diffeomorphism is irrational? Even if this question have occupied my mind for a long time, it remains still unanswered, as many other interesting ones. However, it is possible to understand precisely what are the typical features of a diffeomorphism with periodic points.

By the end of the 80s, Malliavin and Shavgulidze introduced a measure on the space of C^1 circle diffeomorphisms which carries many interesting features. Perhaps the most interesting aspect is that it can be considered as an analog of the Haar measure for the group Diff^1_+(S^1).

The nature of this measure has been mostly investigated in connection to representation theory.

For people working in dynamical systems, the MS measure offers a way to quantify dynamical phenomena: for example, which is the probability that a random diffeomorphism is irrational? Even if this question have occupied my mind for a long time, it remains still unanswered, as many other interesting ones. However, it is possible to understand precisely what are the typical features of a diffeomorphism with periodic points.

### 2014年03月12日(水)

10:15-11:45 数理科学研究科棟(駒場) 470号室

Invariant distributions for circle diffeomorphisms of

irrational rotation number and low regularity (ENGLISH)

**Michele Triestino 氏**(Ecole Normale Superieure de Lyon)Invariant distributions for circle diffeomorphisms of

irrational rotation number and low regularity (ENGLISH)

[ 講演概要 ]

The main inspiration of this joint work with Andrés Navas is the beautiful result of Ávila and Kocsard: if f is a C^\\infty circle diffeomorphism of irrational rotation number, then the unique invariant probability measure is also the unique (up to rescaling) invariant distribution.

Using conceptual geometric arguments (Hahn-Banach...), we investigate the uniqueness of invariant distributions for C^1 circle diffeomorphisms of irrational rotation number, with particular attention to sharp regularity.

We prove that If the diffeomorphism is C^{1+bv}, then there is a unique invariant distribution of order 1. On the other side, examples by Douady and Yoccoz, and by Kodama and Matsumoto exhibit differentiable dynamical systems for which the uniqueness does not hold.

The main inspiration of this joint work with Andrés Navas is the beautiful result of Ávila and Kocsard: if f is a C^\\infty circle diffeomorphism of irrational rotation number, then the unique invariant probability measure is also the unique (up to rescaling) invariant distribution.

Using conceptual geometric arguments (Hahn-Banach...), we investigate the uniqueness of invariant distributions for C^1 circle diffeomorphisms of irrational rotation number, with particular attention to sharp regularity.

We prove that If the diffeomorphism is C^{1+bv}, then there is a unique invariant distribution of order 1. On the other side, examples by Douady and Yoccoz, and by Kodama and Matsumoto exhibit differentiable dynamical systems for which the uniqueness does not hold.

### 2014年03月10日(月)

15:15-16:45 数理科学研究科棟(駒場) 123号室

Energy fluctuations in the disordered harmonic chain (ENGLISH)

**Marielle Simon 氏**(ENS Lyon, UMPA)Energy fluctuations in the disordered harmonic chain (ENGLISH)

[ 講演概要 ]

We study the energy diffusion in the disordered harmonic chain of oscillators: the usual Hamiltonian dynamics is provided with random masses and perturbed by a degenerate energy conserving noise. After rescaling space and time diffusively, we prove that energy fluctuations evolve following an infinite dimensional linear stochastic differential equation driven by the linearized heat equation. We also give variational expressions for the thermal diffusivity and an equivalent definition through the Green-Kubo formula. Since the model is non gradient, and the perturbation is very degenerate, the standard Varadhan's approach is reviewed under new perspectives.

We study the energy diffusion in the disordered harmonic chain of oscillators: the usual Hamiltonian dynamics is provided with random masses and perturbed by a degenerate energy conserving noise. After rescaling space and time diffusively, we prove that energy fluctuations evolve following an infinite dimensional linear stochastic differential equation driven by the linearized heat equation. We also give variational expressions for the thermal diffusivity and an equivalent definition through the Green-Kubo formula. Since the model is non gradient, and the perturbation is very degenerate, the standard Varadhan's approach is reviewed under new perspectives.