## Seminar information archive

Seminar information archive ～05/23｜Today's seminar 05/24 | Future seminars 05/25～

#### Algebraic Geometry Seminar

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

Positivity in varieties of maximal Albanese dimension (ENGLISH)

**Jungkai Chen**(National Taiwan University)Positivity in varieties of maximal Albanese dimension (ENGLISH)

[ Abstract ]

Given a variety of maximal Albanese dimension, it is known that the holomorphic Euler characteristic is non-negative. It is an interesting question to characterize varieties with vanishing Euler characteristic.

In our previous work (jointly with Debarre and Jiang), we prove that Ein-Lazarsgfeld's example is essentially the only variety of maximal Albanese and Kodaira dimension with vanishing Euler characteristic in dimension three. In the recent joint work with Jiang, we prove a decomposition theorem for the push-forward of canonical sheaf. As a consequence, we are able to generalized our previous characterization. The purpose of this talk is give a survey of these two works.

Given a variety of maximal Albanese dimension, it is known that the holomorphic Euler characteristic is non-negative. It is an interesting question to characterize varieties with vanishing Euler characteristic.

In our previous work (jointly with Debarre and Jiang), we prove that Ein-Lazarsgfeld's example is essentially the only variety of maximal Albanese and Kodaira dimension with vanishing Euler characteristic in dimension three. In the recent joint work with Jiang, we prove a decomposition theorem for the push-forward of canonical sheaf. As a consequence, we are able to generalized our previous characterization. The purpose of this talk is give a survey of these two works.

#### FMSP Lectures

12:30-13:30 Room #128 (Graduate School of Math. Sci. Bldg.)

Unitary representations of reductive Lie groups III

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

**Peter Trapa**(University of Utah)Unitary representations of reductive Lie groups III

[ Abstract ]

Let G be a real reductive group. I will describe an algorithm to determine the unitary dual of G. More precisely, I will describe an algorithm to determine if an irreducible (g,K) module (specified in the Langlands classification) is unitary in the sense that it admits a positive definite invariant Hermitian form.

This is joint work with Jeffrey Adams, Marc van Leeuwen, and David Vogan.

[ Reference URL ]Let G be a real reductive group. I will describe an algorithm to determine the unitary dual of G. More precisely, I will describe an algorithm to determine if an irreducible (g,K) module (specified in the Langlands classification) is unitary in the sense that it admits a positive definite invariant Hermitian form.

This is joint work with Jeffrey Adams, Marc van Leeuwen, and David Vogan.

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

#### FMSP Lectures

9:30-10:30 Room #122 (Graduate School of Math. Sci. Bldg.)

The Geometry of Nontempered Characters

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

**Benjamin Harris**(Oklahoma State University)The Geometry of Nontempered Characters

[ Abstract ]

In this talk, we will survey the results of Rossmann and Schmid-Vilonen on geometric formulas for nontempered characters of reductive groups, and we will mention an old result of Barbasch-Vogan on the special case A_q(lambda). We will discuss what nontempered character formulas would be necessary to generalize the main formula of the second talk, and we will make conjectures.

[ Reference URL ]In this talk, we will survey the results of Rossmann and Schmid-Vilonen on geometric formulas for nontempered characters of reductive groups, and we will mention an old result of Barbasch-Vogan on the special case A_q(lambda). We will discuss what nontempered character formulas would be necessary to generalize the main formula of the second talk, and we will make conjectures.

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

#### FMSP Lectures

13:00-17:00 Room #126 (Graduate School of Math. Sci. Bldg.)

Limit order books I

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

**Frédéric Abergel**(École Centrale Paris)Limit order books I

[ Abstract ]

In this series of lectures, I will present results pertaining to the empirical properties, mathematical modeling and analysis of limit order books, an object that is now central in modern financial markets. Part of the lectures will be devoted to a survey of the quantitative finance and financial mathematics literature on the subject. I will also present some rather recent results related to the long time behaviour and stationarity of the limit order book.

[ Reference URL ]In this series of lectures, I will present results pertaining to the empirical properties, mathematical modeling and analysis of limit order books, an object that is now central in modern financial markets. Part of the lectures will be devoted to a survey of the quantitative finance and financial mathematics literature on the subject. I will also present some rather recent results related to the long time behaviour and stationarity of the limit order book.

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

#### FMSP Lectures

11:00-12:00 Room #122 (Graduate School of Math. Sci. Bldg.)

Local Theta lifting of generalized Whittaker models.

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

**Raul Gomez**(Cornell University)Local Theta lifting of generalized Whittaker models.

[ Abstract ]

In this talk, we describe the behavior of the space of generalized Whittaker models attached to a nilpotent orbit under the local theta correspondence. This description is a generalization of a result of Moeglin in the p-adic setting. This is joint work with Chengbo Zhu.

[ Reference URL ]In this talk, we describe the behavior of the space of generalized Whittaker models attached to a nilpotent orbit under the local theta correspondence. This description is a generalization of a result of Moeglin in the p-adic setting. This is joint work with Chengbo Zhu.

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

### 2015/01/25

#### FMSP Lectures

12:30-13:30 Room #126 (Graduate School of Math. Sci. Bldg.)

Unitary representations of reductive Lie groups II

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

**Peter Trapa**(University of Utah)Unitary representations of reductive Lie groups II

[ Abstract ]

Let G be a real reductive group. I will describe an algorithm to determine the unitary dual of G. More precisely, I will describe an algorithm to determine if an irreducible (g,K) module (specified in the Langlands classification) is unitary in the sense that it admits a positive definite invariant Hermitian form.

This is joint work with Jeffrey Adams, Marc van Leeuwen, and David Vogan.

[ Reference URL ]Let G be a real reductive group. I will describe an algorithm to determine the unitary dual of G. More precisely, I will describe an algorithm to determine if an irreducible (g,K) module (specified in the Langlands classification) is unitary in the sense that it admits a positive definite invariant Hermitian form.

This is joint work with Jeffrey Adams, Marc van Leeuwen, and David Vogan.

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

#### FMSP Lectures

11:00-12:00 Room #126 (Graduate School of Math. Sci. Bldg.)

The Geometry of Harmonic Analysis

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

**Benjamin Harris**(Oklahoma State University)The Geometry of Harmonic Analysis

[ Abstract ]

In this talk, we will present recent joint work with Tobias Weich. When G is a real, reductive algebraic group and X is a homogeneous space for G with an invariant measure, we will completely describe the regular, semisimple asymptotics of the support of the Plancherel measure for L^2(X). We will give concrete examples of this theorem, describing what can and cannot be deduced from this result.

[ Reference URL ]In this talk, we will present recent joint work with Tobias Weich. When G is a real, reductive algebraic group and X is a homogeneous space for G with an invariant measure, we will completely describe the regular, semisimple asymptotics of the support of the Plancherel measure for L^2(X). We will give concrete examples of this theorem, describing what can and cannot be deduced from this result.

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

#### FMSP Lectures

9:30-10:30 Room #126 (Graduate School of Math. Sci. Bldg.)

Generalized and degenerate Whittaker models associated to nilpotent orbits.

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

**Raul Gomez**(Cornell University)Generalized and degenerate Whittaker models associated to nilpotent orbits.

[ Abstract ]

In this talk, we examine the relation between the different spaces of Whittaker models that can be attached to a nilpotent orbit. We will also explore their relation to other nilpotent invariants (like the wave front set) and show some examples and applications.

This is joint work with Dmitry Gourevitch and Siddhartha Sahi.

[ Reference URL ]In this talk, we examine the relation between the different spaces of Whittaker models that can be attached to a nilpotent orbit. We will also explore their relation to other nilpotent invariants (like the wave front set) and show some examples and applications.

This is joint work with Dmitry Gourevitch and Siddhartha Sahi.

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

### 2015/01/24

#### FMSP Lectures

13:00-14:00 Room #126 (Graduate School of Math. Sci. Bldg.)

Unitary representations of reductive Lie groups I

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

**Peter Trapa**(University of Utah)Unitary representations of reductive Lie groups I

[ Abstract ]

Let G be a real reductive group. I will describe an algorithm to determine the unitary dual of G. More precisely, I will describe an algorithm to determine if an irreducible (g,K) module (specified in the Langlands classification) is unitary in the sense that it admits a positive definite invariant Hermitian form.

This is joint work with Jeffrey Adams, Marc van Leeuwen, and David Vogan.

[ Reference URL ]Let G be a real reductive group. I will describe an algorithm to determine the unitary dual of G. More precisely, I will describe an algorithm to determine if an irreducible (g,K) module (specified in the Langlands classification) is unitary in the sense that it admits a positive definite invariant Hermitian form.

This is joint work with Jeffrey Adams, Marc van Leeuwen, and David Vogan.

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

#### FMSP Lectures

16:30-17:30 Room #126 (Graduate School of Math. Sci. Bldg.)

The Geometry of Tempered Characters

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

**Benjamin Harris**(Oklahoma State University)The Geometry of Tempered Characters

[ Abstract ]

In this introductory talk, we will briefly recall parts of Harish-Chandra's theory of characters for reductive groups and the geometric formula of Rossmann and Duflo for tempered characters of reductive groups. Examples will be given in the case G=SL(2,R).

[ Reference URL ]In this introductory talk, we will briefly recall parts of Harish-Chandra's theory of characters for reductive groups and the geometric formula of Rossmann and Duflo for tempered characters of reductive groups. Examples will be given in the case G=SL(2,R).

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

#### FMSP Lectures

14:30-15:30 Room #126 (Graduate School of Math. Sci. Bldg.)

The Tor and Ext functors for smooth representations of real algebraic groups.

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

**Raul Gomez**(Cornell University)The Tor and Ext functors for smooth representations of real algebraic groups.

[ Abstract ]

Inspired by the recent work of Dipendra Prasad in the $p$-adic setting, we define the Tor and Ext functors for an appropriate category of smooth representations of a real algebraic group $G$, and give some applications. This is joint work with Birgit Speh.

[ Reference URL ]Inspired by the recent work of Dipendra Prasad in the $p$-adic setting, we define the Tor and Ext functors for an appropriate category of smooth representations of a real algebraic group $G$, and give some applications. This is joint work with Birgit Speh.

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

### 2015/01/23

#### Colloquium

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

Grothendieck and algebraic geometry

**Luc Illusie**(Université de Paris-Sud)Grothendieck and algebraic geometry

[ Abstract ]

Between 1957 and 1970 Grothendieck deeply and durably transformed algebraic geometry. I will discuss some of his revolutionary contributions.

Between 1957 and 1970 Grothendieck deeply and durably transformed algebraic geometry. I will discuss some of his revolutionary contributions.

### 2015/01/22

#### Infinite Analysis Seminar Tokyo

13:00-16:30 Room #002 (Graduate School of Math. Sci. Bldg.)

A construction of dynamical Yang-Baxter map with dynamical brace (JAPANESE)

Construction of Hopf algebroids by means of dynamical Yang-Baxter maps (JAPANESE)

**Diogo Kendy Matsumoto**(Faculty of Science and Engineerring, Waseda University) 13:00-14:30A construction of dynamical Yang-Baxter map with dynamical brace (JAPANESE)

[ Abstract ]

Brace is an algebraic system defined as a generalization of the radical ring. The radical ring means a ring $(R,+,¥cdot)$, which has a group structure with respect to $a*b:=ab+a+b$. By using brace, Rump constructs the non-degenerate Yang-Baxter map with unitary condition.

In this talk I will propose the dynamical brace, which is a generalization of the brace, and give a way to construct the dynamical Yang-Baxter map by using the dynamical brace. A dynamical Yang-Baxter map is a set-theoretical solution of the dynamical Yang-Baxter equation. Moreover, I will discuss algebraic and combinatorial properties of the dynamical brace.

Brace is an algebraic system defined as a generalization of the radical ring. The radical ring means a ring $(R,+,¥cdot)$, which has a group structure with respect to $a*b:=ab+a+b$. By using brace, Rump constructs the non-degenerate Yang-Baxter map with unitary condition.

In this talk I will propose the dynamical brace, which is a generalization of the brace, and give a way to construct the dynamical Yang-Baxter map by using the dynamical brace. A dynamical Yang-Baxter map is a set-theoretical solution of the dynamical Yang-Baxter equation. Moreover, I will discuss algebraic and combinatorial properties of the dynamical brace.

**Youichi Shibukawa**(Department of Mathematics, Hokkaido University) 15:00-16:30Construction of Hopf algebroids by means of dynamical Yang-Baxter maps (JAPANESE)

[ Abstract ]

A generalization of the Hopf algebra is a Hopf algebroid. Felder and Etingof-Varchenko constructed Hopf algebroids from the dynamical R-matrices, solutions to the quantum dynamical Yang-Baxter equation (QDYBE for short). This QDYBE was generalized, and several solutions called dynamical Yang-Baxter maps to this generalized equation were constructed. The purpose of this talk is to introduce construction of Hopf algebroids by means of dynamical Yang-Baxter maps. If time permits, I will explain that the tensor category of finite-dimensional L-operators associated with the suitable dynamical Yang-Baxter map is rigid. This tensor category is isomorphic to that consisting of finite-dimensional (dynamical) representations of the corresponding Hopf algebroid.

A generalization of the Hopf algebra is a Hopf algebroid. Felder and Etingof-Varchenko constructed Hopf algebroids from the dynamical R-matrices, solutions to the quantum dynamical Yang-Baxter equation (QDYBE for short). This QDYBE was generalized, and several solutions called dynamical Yang-Baxter maps to this generalized equation were constructed. The purpose of this talk is to introduce construction of Hopf algebroids by means of dynamical Yang-Baxter maps. If time permits, I will explain that the tensor category of finite-dimensional L-operators associated with the suitable dynamical Yang-Baxter map is rigid. This tensor category is isomorphic to that consisting of finite-dimensional (dynamical) representations of the corresponding Hopf algebroid.

#### Applied Analysis

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

On the large time behaviour of the multi-dimensional Fisher-KPP equation with compactly supported initial data

(ENGLISH)

**Arnaud Ducrot**(University of Bordeaux)On the large time behaviour of the multi-dimensional Fisher-KPP equation with compactly supported initial data

(ENGLISH)

[ Abstract ]

In this talk we discuss the asymptotic behaviour of a multi-dimensional Fisher-KPP equation posed in an asymptotically homogeneous medium and supplemented together with a compactly supported initial datum. We derive precise estimates for the location of the front before proving the convergence of the solutions towards travelling front. In particular we show that the location of the front drastically depends on the rate at which the medium become homogeneous at infinity. Fast rate of convergence only changes the location by some constant while lower rate of convergence induces further logarithmic delay.

In this talk we discuss the asymptotic behaviour of a multi-dimensional Fisher-KPP equation posed in an asymptotically homogeneous medium and supplemented together with a compactly supported initial datum. We derive precise estimates for the location of the front before proving the convergence of the solutions towards travelling front. In particular we show that the location of the front drastically depends on the rate at which the medium become homogeneous at infinity. Fast rate of convergence only changes the location by some constant while lower rate of convergence induces further logarithmic delay.

### 2015/01/21

#### Number Theory Seminar

18:00-19:00 Room #056 (Graduate School of Math. Sci. Bldg.)

Spreading-out of rigid-analytic families and observations on p-adic Hodge theory (English)

**Ofer Gabber**(CNRS, IHES)Spreading-out of rigid-analytic families and observations on p-adic Hodge theory (English)

[ Abstract ]

(Joint work with Brian Conrad.) Let $K$ be a complete rank 1 valued field with ring of integers $O_K$, $A$ an adic noetherian ring and $f:A\to O_K$ an adic morphism. If $g:X\to Y$ is a proper flat morphism between rigid analytic spaces over $K$ then locally on $Y$ a flat formal model of $g$ spreads out to a proper flat morphism between formal schemes topologically of finite type over $A$. As an application one can prove that for proper smooth $g$ and $K$ of characteristic 0, the Hodge to de Rham spectral sequence for $g$ degenerates and the $R^q g_* \Omega^p_{X/Y}$ are locally free.

(Joint work with Brian Conrad.) Let $K$ be a complete rank 1 valued field with ring of integers $O_K$, $A$ an adic noetherian ring and $f:A\to O_K$ an adic morphism. If $g:X\to Y$ is a proper flat morphism between rigid analytic spaces over $K$ then locally on $Y$ a flat formal model of $g$ spreads out to a proper flat morphism between formal schemes topologically of finite type over $A$. As an application one can prove that for proper smooth $g$ and $K$ of characteristic 0, the Hodge to de Rham spectral sequence for $g$ degenerates and the $R^q g_* \Omega^p_{X/Y}$ are locally free.

#### Classical Analysis

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

Remarks on the number of accessory parameters (JAPANESE)

**Shingo Kamimoto**(Kyoto University)Remarks on the number of accessory parameters (JAPANESE)

### 2015/01/20

#### PDE Real Analysis Seminar

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

Maximum Principle and generalized principal eigenvalue for degenerate elliptic operators (English)

**Italo Capuzzo Dolcetta**(Università degli Studi di Roma "La Sapienza")Maximum Principle and generalized principal eigenvalue for degenerate elliptic operators (English)

[ Abstract ]

In my presentation I will report on a joint paper with H. Berestycki, A. Porretta and L. Rossi to appear shortly on JMPA.

We characterize the validity of the Maximum Principle in bounded domains for fully nonlinear degenerate elliptic operators in terms of the sign of a suitably defined generalized principal eigenvalue. Here, the maximum principle refers to the property of non-positivity of viscosity subsolutions of the Dirichlet problem.

The new notion of generalized principal eigenvalue that we introduce here allows us to deal with arbitrary type of degeneracy of the elliptic operators.

We further discuss the relations between this notion and other natural generalizations of the classical notion of principal eigenvalue, some of which have been previously introduced for particular classes of operators.

In my presentation I will report on a joint paper with H. Berestycki, A. Porretta and L. Rossi to appear shortly on JMPA.

We characterize the validity of the Maximum Principle in bounded domains for fully nonlinear degenerate elliptic operators in terms of the sign of a suitably defined generalized principal eigenvalue. Here, the maximum principle refers to the property of non-positivity of viscosity subsolutions of the Dirichlet problem.

The new notion of generalized principal eigenvalue that we introduce here allows us to deal with arbitrary type of degeneracy of the elliptic operators.

We further discuss the relations between this notion and other natural generalizations of the classical notion of principal eigenvalue, some of which have been previously introduced for particular classes of operators.

#### Tuesday Seminar on Topology

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

On Lagrangian caps and their applications (JAPANESE)

**Toru Yoshiyasu**(The University of Tokyo)On Lagrangian caps and their applications (JAPANESE)

[ Abstract ]

In 2013, Y. Eliashberg and E. Murphy established the $h$-principle for

exact Lagrangian embeddings with a concave Legendrian boundary. In this

talk, I will explain a modification of their $h$-principle and show

applications to Lagrangian submanifolds in the complex projective spaces.

In 2013, Y. Eliashberg and E. Murphy established the $h$-principle for

exact Lagrangian embeddings with a concave Legendrian boundary. In this

talk, I will explain a modification of their $h$-principle and show

applications to Lagrangian submanifolds in the complex projective spaces.

### 2015/01/19

#### Seminar on Geometric Complex Analysis

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

Hyperbolic span and pseudoconvexity (Japanese)

**Hiroshi Yamaguchi**(Shia University, Prof. emeritus)Hyperbolic span and pseudoconvexity (Japanese)

[ Abstract ]

We show that the hyperbolic span for open torus (which is introduced by M. Shiba in 1993) has the intimate relation with the pseudoconvexity.

We show that the hyperbolic span for open torus (which is introduced by M. Shiba in 1993) has the intimate relation with the pseudoconvexity.

#### Numerical Analysis Seminar

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

Between error and residual in numerical computations (日本語)

**Yoshitaka Watanabe**(Kyushu University)Between error and residual in numerical computations (日本語)

#### Algebraic Geometry Seminar

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

Crepant resolutions of Slodowy slice in nilpotent orbit closure in sl_N(C) (JAPANESE)

**Ryo Yamagishi**(Kyoto University)Crepant resolutions of Slodowy slice in nilpotent orbit closure in sl_N(C) (JAPANESE)

[ Abstract ]

Nilpotent orbit closures and their intersections with Slodowy slices are typical examples of symplectic varieties. It is known that every crepant resolution of a nilpotent orbit closure is obtained as a Springer resolution. In this talk, we show that every crepant resolution of a Slodowy slice in nilpotent orbit closure in sl_N(C) is obtained as the restriction of a Springer resolution and explain how to count the number of crepant resolutions. The proof of the main results is based on the fact that Slodowy slices can be described as quiver varieties.

Nilpotent orbit closures and their intersections with Slodowy slices are typical examples of symplectic varieties. It is known that every crepant resolution of a nilpotent orbit closure is obtained as a Springer resolution. In this talk, we show that every crepant resolution of a Slodowy slice in nilpotent orbit closure in sl_N(C) is obtained as the restriction of a Springer resolution and explain how to count the number of crepant resolutions. The proof of the main results is based on the fact that Slodowy slices can be described as quiver varieties.

### 2015/01/16

#### Geometry Colloquium

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

Degeneration and curves on K3 surfaces (Japanese)

**Takeo Nishinou**(Rikkyo University)Degeneration and curves on K3 surfaces (Japanese)

[ Abstract ]

There is a well-known conjecture which states that all projective K3 surfaces contain infinitely many rational curves. By calculating obstructions in deformation theory through degeneration, we give a new approach to this problem. In particular, we show that there is a Zariski open subset in the moduli space of quartic K3 surfaces whose members fulfil the conjecture.

There is a well-known conjecture which states that all projective K3 surfaces contain infinitely many rational curves. By calculating obstructions in deformation theory through degeneration, we give a new approach to this problem. In particular, we show that there is a Zariski open subset in the moduli space of quartic K3 surfaces whose members fulfil the conjecture.

#### Seminar on Probability and Statistics

14:00-15:30 Room #052 (Graduate School of Math. Sci. Bldg.)

A stable particle filter in high-dimensions

http://www.sigmath.es.osaka-u.ac.jp/~kamatani/statseminar/2014/06.html

**Ajay Jasra**(National University of Singapore)A stable particle filter in high-dimensions

[ Abstract ]

We consider the numerical approximation of the filtering problem in high dimensions, that is, when the hidden state lies in $\mathbb{R}^d$ with $d$ large. For low dimensional problems, one of the most popular numerical procedures for consistent inference is the class of approximations termed as particle filters or sequential Monte Carlo methods. However, in high dimensions, standard particle filters (e.g. the bootstrap particle filter) can have a cost that is exponential in $d$ for the algorithm to be stable in an appropriate sense. We develop a new particle filter, called the space-time particle filter, for a specific family of state-space models in discrete time. This new class of particle filters provide consistent Monte Carlo estimates for any fixed $d$, as do standard particle filters. Moreover, under a simple i.i.d. model structure, we show that in order to achieve some stability properties this new filter has cost $\mathcal{O}(nNd^2)$, where $n$ is the time parameter and $N$ is the number of Monte Carlo samples, that are fixed and independent of $d$. Similar results hold, under a more general structure than the i.i.d. one. Here we show that, under additional assumptions and with the same cost, the asymptotic variance of the relative estimate of the normalizing constant grows at most linearly in time and independently of the dimension. Our theoretical results are supported by numerical simulations. The results suggest that it is possible to tackle some high dimensional filtering problems using the space-time particle filter that standard particle filters cannot.

This is joint work with: Alex Beskos (UCL), Dan Crisan (Imperial), Kengo Kamatani (Osaka) and Yan Zhou (NUS).

[ Reference URL ]We consider the numerical approximation of the filtering problem in high dimensions, that is, when the hidden state lies in $\mathbb{R}^d$ with $d$ large. For low dimensional problems, one of the most popular numerical procedures for consistent inference is the class of approximations termed as particle filters or sequential Monte Carlo methods. However, in high dimensions, standard particle filters (e.g. the bootstrap particle filter) can have a cost that is exponential in $d$ for the algorithm to be stable in an appropriate sense. We develop a new particle filter, called the space-time particle filter, for a specific family of state-space models in discrete time. This new class of particle filters provide consistent Monte Carlo estimates for any fixed $d$, as do standard particle filters. Moreover, under a simple i.i.d. model structure, we show that in order to achieve some stability properties this new filter has cost $\mathcal{O}(nNd^2)$, where $n$ is the time parameter and $N$ is the number of Monte Carlo samples, that are fixed and independent of $d$. Similar results hold, under a more general structure than the i.i.d. one. Here we show that, under additional assumptions and with the same cost, the asymptotic variance of the relative estimate of the normalizing constant grows at most linearly in time and independently of the dimension. Our theoretical results are supported by numerical simulations. The results suggest that it is possible to tackle some high dimensional filtering problems using the space-time particle filter that standard particle filters cannot.

This is joint work with: Alex Beskos (UCL), Dan Crisan (Imperial), Kengo Kamatani (Osaka) and Yan Zhou (NUS).

http://www.sigmath.es.osaka-u.ac.jp/~kamatani/statseminar/2014/06.html

### 2015/01/15

#### Infinite Analysis Seminar Tokyo

15:00-18:30 Room #002 (Graduate School of Math. Sci. Bldg.)

On Gram matrices of the Shapovalov form of a basic representation of a

quantum affine group (ENGLISH)

Continuous and Infinitesimal Hecke algebras (ENGLISH)

**Shunsuke Tsuchioka**(Graduate School of Mathematical Sciences, the University of Tokyo) 15:00-16:30On Gram matrices of the Shapovalov form of a basic representation of a

quantum affine group (ENGLISH)

[ Abstract ]

We consider Gram matrices of the Shapovalov form of a basic

representation

of a quantum affine group. We present a conjecture predicting the

invariant

factors of these matrices and proving that it gives the correct

invariants

when one specializes or localizes the ring $\mathbb{Z}[v,v^{-1}]$ in

certain ways.

This generalizes Evseev's theorem which settled affirmatively

the K\"{u}lshammer-Olsson-Robinson conjecture that predicts

the generalized Cartan invariants of the symmetric groups.

This is a joint work with Anton Evseev.

We consider Gram matrices of the Shapovalov form of a basic

representation

of a quantum affine group. We present a conjecture predicting the

invariant

factors of these matrices and proving that it gives the correct

invariants

when one specializes or localizes the ring $\mathbb{Z}[v,v^{-1}]$ in

certain ways.

This generalizes Evseev's theorem which settled affirmatively

the K\"{u}lshammer-Olsson-Robinson conjecture that predicts

the generalized Cartan invariants of the symmetric groups.

This is a joint work with Anton Evseev.

**Alexander Tsymbaliuk**(SCGP (Simons Center for Geometry and Physics)) 17:00-18:30Continuous and Infinitesimal Hecke algebras (ENGLISH)

[ Abstract ]

In the late 80's V. Drinfeld introduced the notion of the

degenerate affine Hecke algebras. The particular class of those, called

symplectic reflection algebras, has been rediscovered 15 years later by

[Etingof and Ginzburg]. The theory of those algebras (which include also

the rational Cherednik algebras) has attracted a lot of attention in the

last 15 years.

In this talk we will discuss their continuous and infinitesimal versions,

introduced by [Etingof, Gan, and Ginzburg]. Our key result relates those

classical algebras to the simplest 1-block finite W-algebras.

In the late 80's V. Drinfeld introduced the notion of the

degenerate affine Hecke algebras. The particular class of those, called

symplectic reflection algebras, has been rediscovered 15 years later by

[Etingof and Ginzburg]. The theory of those algebras (which include also

the rational Cherednik algebras) has attracted a lot of attention in the

last 15 years.

In this talk we will discuss their continuous and infinitesimal versions,

introduced by [Etingof, Gan, and Ginzburg]. Our key result relates those

classical algebras to the simplest 1-block finite W-algebras.

### 2015/01/14

#### Number Theory Seminar

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

Iterate extensions and relative Lubin-Tate groups

**Laurent Berger**(ENS de Lyon)Iterate extensions and relative Lubin-Tate groups

[ Abstract ]

Let K be a p-adic field, let P(T) be a polynomial with coefficients in K, and let {$u_n$} be a sequence such that $P(u_{n+1}) = u_n$ for all n and $u_0$ belongs to K. The extension of K generated by the $u_n$ is called an iterate extension. I will discuss these extensions, show that under certain favorable conditions there is a theory of Coleman power series, and explain the relationship with relative Lubin-Tate groups.

Let K be a p-adic field, let P(T) be a polynomial with coefficients in K, and let {$u_n$} be a sequence such that $P(u_{n+1}) = u_n$ for all n and $u_0$ belongs to K. The extension of K generated by the $u_n$ is called an iterate extension. I will discuss these extensions, show that under certain favorable conditions there is a theory of Coleman power series, and explain the relationship with relative Lubin-Tate groups.

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