## Seminar information archive

Seminar information archive ～11/01｜Today's seminar 11/02 | Future seminars 11/03～

#### Tuesday Seminar on Topology

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

Topological Blow-up (JAPANESE)

**Taro Yoshino**(The University of Tokyo)Topological Blow-up (JAPANESE)

[ Abstract ]

Suppose that a Lie group $G$ acts on a manifold

$M$. The quotient space $X:=G\\backslash M$ is locally compact,

but not Hausdorff in general. Our aim is to understand

such a non-Hausdorff space $X$.

The space $X$ has the crack $S$. Roughly speaking, $S$ is

the causal subset of non-Hausdorffness of $X$, and especially

$X\\setminus S$ is Hausdorff.

We introduce the concept of `topological blow-up' as a `repair'

of the crack. The `repaired' space $\\tilde{X}$ is

locally compact and Hausdorff space containing $X\\setminus S$

as its open subset. Moreover, the original space $X$ can be

recovered from the pair of $(\\tilde{X}, S)$.

Suppose that a Lie group $G$ acts on a manifold

$M$. The quotient space $X:=G\\backslash M$ is locally compact,

but not Hausdorff in general. Our aim is to understand

such a non-Hausdorff space $X$.

The space $X$ has the crack $S$. Roughly speaking, $S$ is

the causal subset of non-Hausdorffness of $X$, and especially

$X\\setminus S$ is Hausdorff.

We introduce the concept of `topological blow-up' as a `repair'

of the crack. The `repaired' space $\\tilde{X}$ is

locally compact and Hausdorff space containing $X\\setminus S$

as its open subset. Moreover, the original space $X$ can be

recovered from the pair of $(\\tilde{X}, S)$.

#### Tuesday Seminar of Analysis

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

On the system of fifth-order differential equations which describes surfaces containing six continuous families of circles (JAPANESE)

**Kiyoomi KATAOKA**(Graduate School of Mathematical Sciences, the University of Tokyo)On the system of fifth-order differential equations which describes surfaces containing six continuous families of circles (JAPANESE)

### 2011/04/25

#### Algebraic Geometry Seminar

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

Mirror symmetry and projective geometry of Reye congruences (JAPANESE)

**Hiromichi Takagi**(University of Tokyo)Mirror symmetry and projective geometry of Reye congruences (JAPANESE)

[ Abstract ]

This is a joint work with Shinobu Hosono.

It is well-known that the projective dual of the second Veronese variety v_2(P^n) is the symmetric determinantal hypersurface H. However, in the context of homological projective duality after Kuznetsov, it is natural to consider that the Chow^2 P^n and H are dual (note that Chow^2 P^n is the secant variety of v_2(P^n)).

Though we did not yet formulate what this duality exactly means in full generality, we show some results in this context for the values n¥leq 4.

For example, let n=4. We consider Chow^2 P^4 in P(S^2 V) and H in P(S^2 V^*), where V is the vector space such that P^4 =P(V). Take a general 4-plane P in

P(S^2 V^*) and let P' be the orthogonal space to P in P(S^2 V). Then X:=Chow^2 P^4 ¥cap P' is a smooth Calabi-Yau 3-fold, and there exists a natural double cover Y -> H¥cap P with a smooth Calabi-Yau 3-fold Y. It is easy to check

that X and Y are not birational each other.

Our main result asserts the derived equivalence of X and Y. This derived equivalence is given by the Fourier Mukai functor D(X)-> D(Y) whose kernel is the ideal sheaf in X×Y of a flat family of curves on Y parameterized by X.

Curves on Y in this family have degree 5 and arithmetic genus 3, and these have a nice interpretation by a BPS number of Y. The proof of the derived equivalence is slightly involved so I explain a similar result in the case where n=3. In this case, we obtain a fully faithful functor from D(X)-> D(Y), where X is a so called the Reye congruence Enriques surface and Y is the 'big resolution' of the Artin-Mumford quartic double solid.

This is a joint work with Shinobu Hosono.

It is well-known that the projective dual of the second Veronese variety v_2(P^n) is the symmetric determinantal hypersurface H. However, in the context of homological projective duality after Kuznetsov, it is natural to consider that the Chow^2 P^n and H are dual (note that Chow^2 P^n is the secant variety of v_2(P^n)).

Though we did not yet formulate what this duality exactly means in full generality, we show some results in this context for the values n¥leq 4.

For example, let n=4. We consider Chow^2 P^4 in P(S^2 V) and H in P(S^2 V^*), where V is the vector space such that P^4 =P(V). Take a general 4-plane P in

P(S^2 V^*) and let P' be the orthogonal space to P in P(S^2 V). Then X:=Chow^2 P^4 ¥cap P' is a smooth Calabi-Yau 3-fold, and there exists a natural double cover Y -> H¥cap P with a smooth Calabi-Yau 3-fold Y. It is easy to check

that X and Y are not birational each other.

Our main result asserts the derived equivalence of X and Y. This derived equivalence is given by the Fourier Mukai functor D(X)-> D(Y) whose kernel is the ideal sheaf in X×Y of a flat family of curves on Y parameterized by X.

Curves on Y in this family have degree 5 and arithmetic genus 3, and these have a nice interpretation by a BPS number of Y. The proof of the derived equivalence is slightly involved so I explain a similar result in the case where n=3. In this case, we obtain a fully faithful functor from D(X)-> D(Y), where X is a so called the Reye congruence Enriques surface and Y is the 'big resolution' of the Artin-Mumford quartic double solid.

#### Seminar on Geometric Complex Analysis

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

On the classification of CR mappings between generalized pseudoellipsoids (JAPANESE)

**Atsushi Hayashimoto**(Nagano National College of Technology)On the classification of CR mappings between generalized pseudoellipsoids (JAPANESE)

### 2011/04/20

#### PDE Real Analysis Seminar

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

Stochastic power law fluids (JAPANESE)

http://www.math.kyoto-u.ac.jp/~nobuo/

**Yoshida, Nobuo**(Department of Mathematics, Kyoto University)Stochastic power law fluids (JAPANESE)

[ Abstract ]

This talk is based in part on a joint work with Yutaka Terasawa.

We consider a SPDE (stochastic partial differential equation) which describes the velocity field of a viscous, incompressible non-Newtonian fluid subject to a random force.

Here, the extra stress tensor of the fluid is given by a polynomial of degree $p-1$ of the rate of strain tensor, while the colored noise is considered as a random force.

We first investigate the existence and the uniqueness of weak solutions to this SPDE.

We next turn to the special case: $p \\in [1 + {d \\over 2},{2d\\overd-2})$,

where $d$ is the dimension of the space. We prove there that the Galerkin scheme approximates the velocity field in a strong sense. As a consequence, we establish the energy equality for the velocity field.

[ Reference URL ]This talk is based in part on a joint work with Yutaka Terasawa.

We consider a SPDE (stochastic partial differential equation) which describes the velocity field of a viscous, incompressible non-Newtonian fluid subject to a random force.

Here, the extra stress tensor of the fluid is given by a polynomial of degree $p-1$ of the rate of strain tensor, while the colored noise is considered as a random force.

We first investigate the existence and the uniqueness of weak solutions to this SPDE.

We next turn to the special case: $p \\in [1 + {d \\over 2},{2d\\overd-2})$,

where $d$ is the dimension of the space. We prove there that the Galerkin scheme approximates the velocity field in a strong sense. As a consequence, we establish the energy equality for the velocity field.

http://www.math.kyoto-u.ac.jp/~nobuo/

### 2011/04/18

#### Algebraic Geometry Seminar

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

Ideal-adic semi-continuity problem for minimal log discrepancies (JAPANESE)

**Masayuki Kawakita**(Research Institute for Mathematical Sciences, Kyoto University)Ideal-adic semi-continuity problem for minimal log discrepancies (JAPANESE)

[ Abstract ]

De Fernex, Ein and Mustaţă, after Kollár, proved the ideal-adic semi-continuity of log canonicity to obtain Shokurov's ACC conjecture for log canonical thresholds on l.c.i. varieties. I discuss its generalisation to minimal log discrepancies, proposed by Mustaţă.

De Fernex, Ein and Mustaţă, after Kollár, proved the ideal-adic semi-continuity of log canonicity to obtain Shokurov's ACC conjecture for log canonical thresholds on l.c.i. varieties. I discuss its generalisation to minimal log discrepancies, proposed by Mustaţă.

#### Seminar on Geometric Complex Analysis

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

Asymptotic cohomology vanishing and a converse of the Andreotti-Grauert theorem on surface (JAPANESE)

**Shinichi Matsumura**(University of Tokyo)Asymptotic cohomology vanishing and a converse of the Andreotti-Grauert theorem on surface (JAPANESE)

### 2011/04/14

#### Operator Algebra Seminars

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

Amenable actions and crossed products of $C^*$-algebras (JAPANESE)

**Masayoshi Matsumura**(Univ. Tokyo)Amenable actions and crossed products of $C^*$-algebras (JAPANESE)

#### Applied Analysis

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

Homoclinic and heteroclinic orbits for a semilinear parabolic equation (ENGLISH)

**Marek FILA**(Comenius University (Slovakia))Homoclinic and heteroclinic orbits for a semilinear parabolic equation (ENGLISH)

[ Abstract ]

We study the existence of connecting orbits for the Fujita equation

u_t=\\Delta u+u^p

with a critical or supercritical exponent $p$. For certain ranges of the exponent we prove the existence of heteroclinic connections from positive steady states to zero and the existence of a homoclinic orbit with respect to zero. This is a joint work with Eiji Yanagida.

We study the existence of connecting orbits for the Fujita equation

u_t=\\Delta u+u^p

with a critical or supercritical exponent $p$. For certain ranges of the exponent we prove the existence of heteroclinic connections from positive steady states to zero and the existence of a homoclinic orbit with respect to zero. This is a joint work with Eiji Yanagida.

### 2011/04/13

#### Functional Analysis Seminar

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

Spectral theory for functions of self-adjoint operators (ENGLISH)

**Alexander Pushnitski**(King's College, London)Spectral theory for functions of self-adjoint operators (ENGLISH)

[ Abstract ]

Let A, B be self-adjoint operators such that the standard assumptions of smooth scattering theory for the pair A, B are satisfied. The spectral theory of the operators of the type f(A)-f(B) will be discussed, with a particular attention to the case of discontinuous functions f. It turns out that the spectrum of f(A)-f(B) can often be explicitly described in terms of the spectrum of the scattering matrix for the pair A,B. This is joint work with D.Yafaev.

Let A, B be self-adjoint operators such that the standard assumptions of smooth scattering theory for the pair A, B are satisfied. The spectral theory of the operators of the type f(A)-f(B) will be discussed, with a particular attention to the case of discontinuous functions f. It turns out that the spectrum of f(A)-f(B) can often be explicitly described in terms of the spectrum of the scattering matrix for the pair A,B. This is joint work with D.Yafaev.

### 2011/04/12

#### Tuesday Seminar on Topology

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

On diffeomorphisms over non-orientable surfaces embedded in the 4-sphere (JAPANESE)

**Susumu Hirose**(Tokyo University of Science)On diffeomorphisms over non-orientable surfaces embedded in the 4-sphere (JAPANESE)

[ Abstract ]

For a closed orientable surface standardly embedded in the 4-sphere,

it was known that a diffeomorphism over this surface is extendable to

the 4-sphere if and only if this diffeomorphism preserves

the Rokhlin quadratic form of this surafce.

In this talk, we will explain an approach to the same kind of problem for

closed non-orientable surfaces.

For a closed orientable surface standardly embedded in the 4-sphere,

it was known that a diffeomorphism over this surface is extendable to

the 4-sphere if and only if this diffeomorphism preserves

the Rokhlin quadratic form of this surafce.

In this talk, we will explain an approach to the same kind of problem for

closed non-orientable surfaces.

### 2011/04/11

#### Seminar on Geometric Complex Analysis

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

Algebraic analysis of resolvents and an exact algorithm for computing Spectral decomposition matrices (JAPANESE)

**Shinichi Tajima**(University of Tsukuba)Algebraic analysis of resolvents and an exact algorithm for computing Spectral decomposition matrices (JAPANESE)

### 2011/03/31

#### Lectures

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

Dynamical localization for unitary Anderson models (JAPANESE)

**Alain Joye**(Univ. Grenoble)Dynamical localization for unitary Anderson models (JAPANESE)

#### Lectures

14:30-15:30 Room #118 (Graduate School of Math. Sci. Bldg.)

Stable limits for biased random walks on random trees (JAPANESE)

**Gerard Ben Arous**(Courant Institute, New York Univ.)Stable limits for biased random walks on random trees (JAPANESE)

[ Abstract ]

It is well know that transport in random media can be hampered by dead-end regions and that the velocity can even vanish for strong drifts. We study this phenomenon in great detail for random trees. That is, we study the behavior of biased random walks on supercritical random trees with leaves, in the sub-ballistic regime. When the drift is strong enough it is well known that trapping in the dead-ends of the tree, causes the velocity to vanish. We study the behavior of the walk in this regime, and in particular find the exponents for the mean displacement and the time to reach a given large distance. We also establish a scaling limit result in the case where the drift are random and a non-lattice condition is satisfied. (Joint work with Alexander Fribergh, Alan Hammond, Nina Gantert)

It is well know that transport in random media can be hampered by dead-end regions and that the velocity can even vanish for strong drifts. We study this phenomenon in great detail for random trees. That is, we study the behavior of biased random walks on supercritical random trees with leaves, in the sub-ballistic regime. When the drift is strong enough it is well known that trapping in the dead-ends of the tree, causes the velocity to vanish. We study the behavior of the walk in this regime, and in particular find the exponents for the mean displacement and the time to reach a given large distance. We also establish a scaling limit result in the case where the drift are random and a non-lattice condition is satisfied. (Joint work with Alexander Fribergh, Alan Hammond, Nina Gantert)

### 2011/03/22

#### Lectures

14:00-15:00 Room #128 (Graduate School of Math. Sci. Bldg.)

Potts models and Bethe states on sparse random graphs (JAPANESE)

**Amir Dembo**(Stanford Univ.)Potts models and Bethe states on sparse random graphs (JAPANESE)

[ Abstract ]

Theoretical models of disordered materials lead to challenging mathematical problems with applications to random combinatorial problems and coding theory. The underlying mathematical structure is that of many discrete variables that are strongly interacting according to a mean field model determined by a random sparse graph. Focusing on ferromagnetic Potts measures on random finite graphs that converge locally to trees we validate the `cavity' prediction for the limiting free energy per spin and show that local marginals are approximated well by the belief propagation algorithm. This is a concrete example of the more general approximation by Bethe measures, namely, the local convergence of the Boltzmann distribution on the original graph to the Boltzmann distribution on an appropriate infinite random tree (this talk is based on a joint work with Andrea Montanari and Nike Sun).

Theoretical models of disordered materials lead to challenging mathematical problems with applications to random combinatorial problems and coding theory. The underlying mathematical structure is that of many discrete variables that are strongly interacting according to a mean field model determined by a random sparse graph. Focusing on ferromagnetic Potts measures on random finite graphs that converge locally to trees we validate the `cavity' prediction for the limiting free energy per spin and show that local marginals are approximated well by the belief propagation algorithm. This is a concrete example of the more general approximation by Bethe measures, namely, the local convergence of the Boltzmann distribution on the original graph to the Boltzmann distribution on an appropriate infinite random tree (this talk is based on a joint work with Andrea Montanari and Nike Sun).

### 2011/03/08

#### GCOE Seminars

15:00-16:30 Room #128 (Graduate School of Math. Sci. Bldg.)

Diagonal singularities of the scattering matrix and the inverse problem at a fixed energy (ENGLISH)

**Dimitri Yafaev**(Univ. Rennes 1)Diagonal singularities of the scattering matrix and the inverse problem at a fixed energy (ENGLISH)

### 2011/03/04

#### GCOE Seminars

17:00-18:00 Room #370 (Graduate School of Math. Sci. Bldg.)

Inverse boundary value problem by measuring Dirichlet data and Neumann data on disjoint sets (ENGLISH)

**Oleg Emanouilov**(Colorado State University)Inverse boundary value problem by measuring Dirichlet data and Neumann data on disjoint sets (ENGLISH)

[ Abstract ]

We discuss the inverse boundary value problem of determining the conductivity in two dimensions from the pair of all input Dirichlet data supported on an open subset S1 and all the corresponding Neumann data measured on an open subset S2.

We prove the global uniqueness under some additional geometric condition, in the case where the intersection of S_1 and S_2 has no interior points, and we prove also the uniqueness for a similar inverse problem for the stationary Schr"odinger equation.

The key of the proof isthe construction of appropriate complex geometrical optics solutions using Carleman estimates with a singular weight.

We discuss the inverse boundary value problem of determining the conductivity in two dimensions from the pair of all input Dirichlet data supported on an open subset S1 and all the corresponding Neumann data measured on an open subset S2.

We prove the global uniqueness under some additional geometric condition, in the case where the intersection of S_1 and S_2 has no interior points, and we prove also the uniqueness for a similar inverse problem for the stationary Schr"odinger equation.

The key of the proof isthe construction of appropriate complex geometrical optics solutions using Carleman estimates with a singular weight.

### 2011/03/03

#### Lectures

13:30-14:30 Room #270 (Graduate School of Math. Sci. Bldg.)

Energy Diffusion: hydrodynamic, weak coupling, kinetic limits (ENGLISH)

**Stefano Olla**(Univ. Paris Dauphine)Energy Diffusion: hydrodynamic, weak coupling, kinetic limits (ENGLISH)

[ Abstract ]

I will review recent results about weak coupling and kinetic limits for the energy diffusive evolution in hamiltonian systems perturbed by energy-conservating noise. Two universality classes of diffusion are obtained: Ginzburg-Landau dynamics that arise from weak coupling limit of anharmonic oscillators, and exclusion type processes that arise from kinetic limit (rarefied collisions) of interacting billiards. Works in collaboration with Carlangelo Liverani (weak coupling) and Francois Huveneers (kinetic limits).

I will review recent results about weak coupling and kinetic limits for the energy diffusive evolution in hamiltonian systems perturbed by energy-conservating noise. Two universality classes of diffusion are obtained: Ginzburg-Landau dynamics that arise from weak coupling limit of anharmonic oscillators, and exclusion type processes that arise from kinetic limit (rarefied collisions) of interacting billiards. Works in collaboration with Carlangelo Liverani (weak coupling) and Francois Huveneers (kinetic limits).

#### Lectures

14:45-15:45 Room #270 (Graduate School of Math. Sci. Bldg.)

Singularity and absolute continuity of Palm measures of Ginibre random fields

(ENGLISH)

**Hirofumi Osada**(Kyushu Univ.)Singularity and absolute continuity of Palm measures of Ginibre random fields

(ENGLISH)

[ Abstract ]

The Ginibre random point field is a probability measure on the configuration space over the complex plane $\\mathbb{C}$, which is translation and rotation invariant. Intuitively, the interaction potential of this random point field is the two dimensional Coulomb potential with $\\beta = 2 $. This fact is justified by the integration by parts formula.

Since the two dimensional Coulomb potential is quite strong at infinity, the property of the Ginibre random point field is different from that of Gibbs measure with Ruelle class potentials. As an instance, we prove that the Palm measure of the Ginibre random point field is singular to the original Ginibre random point field. Moreover, all Palm measures conditioned at $x \\in \\mathbb{C}$ are mutually absolutely continuous.

The Ginibre random point field is a probability measure on the configuration space over the complex plane $\\mathbb{C}$, which is translation and rotation invariant. Intuitively, the interaction potential of this random point field is the two dimensional Coulomb potential with $\\beta = 2 $. This fact is justified by the integration by parts formula.

Since the two dimensional Coulomb potential is quite strong at infinity, the property of the Ginibre random point field is different from that of Gibbs measure with Ruelle class potentials. As an instance, we prove that the Palm measure of the Ginibre random point field is singular to the original Ginibre random point field. Moreover, all Palm measures conditioned at $x \\in \\mathbb{C}$ are mutually absolutely continuous.

#### Lectures

16:00-16:30 Room #270 (Graduate School of Math. Sci. Bldg.)

A proof of the Brascamp-Lieb inequality based on Skorokhod embedding (ENGLISH)

**Yuu Hariya**(Tohoku Univ.)A proof of the Brascamp-Lieb inequality based on Skorokhod embedding (ENGLISH)

[ Abstract ]

In this talk, we provide a probabilistic approach to the Brascamp-Lieb inequality based on Skorokhod embedding. An extension of the inequality to non-convex potentials will also be discussed.

In this talk, we provide a probabilistic approach to the Brascamp-Lieb inequality based on Skorokhod embedding. An extension of the inequality to non-convex potentials will also be discussed.

### 2011/02/28

#### Lectures

17:00-18:00 Room #470 (Graduate School of Math. Sci. Bldg.)

Extremum Seeking Control: history and recent developments (ENGLISH)

**Ying Tan**(The University of Melbourne)Extremum Seeking Control: history and recent developments (ENGLISH)

[ Abstract ]

A control system which is to determine and maintain the extremum value of a function is called extremum seeking control. The first extremum seeking control application appeared in 1922, in which the extremum seeking control was applied to electric railways. The first rigorous local stability analysis for an ESC scheme was recently proved in 2000 and later extended to semi-global stability analysis 2006.. This has spurred a renewed interest in this research area, leading to numerous practical implementations of the scheme. This talk will first revisit the history of extremum seeking control. It is followed by an explanation how the extremum seeking works. Finally, it will focus on the latest unifying framework that combines arbitrary continuous optimization algorithms with an estimator for derivatives of the unknown reference-to-output steady state map that contains an extremum.

A control system which is to determine and maintain the extremum value of a function is called extremum seeking control. The first extremum seeking control application appeared in 1922, in which the extremum seeking control was applied to electric railways. The first rigorous local stability analysis for an ESC scheme was recently proved in 2000 and later extended to semi-global stability analysis 2006.. This has spurred a renewed interest in this research area, leading to numerous practical implementations of the scheme. This talk will first revisit the history of extremum seeking control. It is followed by an explanation how the extremum seeking works. Finally, it will focus on the latest unifying framework that combines arbitrary continuous optimization algorithms with an estimator for derivatives of the unknown reference-to-output steady state map that contains an extremum.

### 2011/02/24

#### Applied Analysis

16:00-18:10 Room #002 (Graduate School of Math. Sci. Bldg.)

Travelling waves for a size and space structured model in population dynamics: Point to sustained oscillating solution connections (ENGLISH)

Some open problems in PDE control (ENGLISH)

**Arnaud Ducrot**(University of Bordeaux 2) 16:00-17:00Travelling waves for a size and space structured model in population dynamics: Point to sustained oscillating solution connections (ENGLISH)

[ Abstract ]

This work is devoted to the study of travelling wave solutions for some size structured model in population dynamics. The population under consideration is also spatially structured and has a nonlocal spatial reproduction. This phenomenon may model the invasion of plants within some empty landscape. Since the corresponding unspatially structured size structured models may induce oscillating dynamics due to Hopf bifurcations, the aim of this work is to prove the existence of point to sustained oscillating solution travelling waves for the spatially structured problem. From a biological viewpoint, such solutions represent the spatial invasion of some species with spatio-temporal patterns at the place where the population is established.

This work is devoted to the study of travelling wave solutions for some size structured model in population dynamics. The population under consideration is also spatially structured and has a nonlocal spatial reproduction. This phenomenon may model the invasion of plants within some empty landscape. Since the corresponding unspatially structured size structured models may induce oscillating dynamics due to Hopf bifurcations, the aim of this work is to prove the existence of point to sustained oscillating solution travelling waves for the spatially structured problem. From a biological viewpoint, such solutions represent the spatial invasion of some species with spatio-temporal patterns at the place where the population is established.

**Enrique Zuazua**(Basque Center for Applied Mathematics) 17:10-18:10Some open problems in PDE control (ENGLISH)

[ Abstract ]

The field of PDE control has experienced a great progress in the last decades, developing new theories and tools that have also influenced other disciplines as Inverse Problem and Optimal Design Theories and Numerical Analysis. PDE control arises in most applications ranging from classical problems in fluid mechanics or structural engineering to modern molecular design experiments.

From a mathematical viewpoint the problems arising in this field are extremely challenging since the existing theory of existence and uniqueness of solutions and the corresponding numerical schemes is insufficient when addressing realistic control problems. Indeed, an efficient controller requires of an in depth understanding of how solutions depend on the various parameters of the problem (shape of the domain, time of control, coefficients of the equation, location

of the controller, nonlinearity in the equation,...)

In this lecture we shall briefly discuss some important advances and some challenging open problems. All of them shear some features. In particular they are simple to state and very likely hard to solve. We shall discuss in particular:

1.- Semilinear wave equations and their control properties.

2.- Microlocal optimal design of wave processes

3.- Sharp observability estimates for heat processes.

4.- Robustness on the control of finite-dimensional systems.

5.- Unique continuation for discrete elliptic models

6.- Control of Kolmogorov equations and other hypoelliptic models.

The field of PDE control has experienced a great progress in the last decades, developing new theories and tools that have also influenced other disciplines as Inverse Problem and Optimal Design Theories and Numerical Analysis. PDE control arises in most applications ranging from classical problems in fluid mechanics or structural engineering to modern molecular design experiments.

From a mathematical viewpoint the problems arising in this field are extremely challenging since the existing theory of existence and uniqueness of solutions and the corresponding numerical schemes is insufficient when addressing realistic control problems. Indeed, an efficient controller requires of an in depth understanding of how solutions depend on the various parameters of the problem (shape of the domain, time of control, coefficients of the equation, location

of the controller, nonlinearity in the equation,...)

In this lecture we shall briefly discuss some important advances and some challenging open problems. All of them shear some features. In particular they are simple to state and very likely hard to solve. We shall discuss in particular:

1.- Semilinear wave equations and their control properties.

2.- Microlocal optimal design of wave processes

3.- Sharp observability estimates for heat processes.

4.- Robustness on the control of finite-dimensional systems.

5.- Unique continuation for discrete elliptic models

6.- Control of Kolmogorov equations and other hypoelliptic models.

### 2011/02/23

#### Functional Analysis Seminar

14:00-18:00 Room #118 (Graduate School of Math. Sci. Bldg.)

The semiclassical limit of eigenfunctions of the Schroedinger equation and the Bohr-Sommerfeld quantization condition, revisited (ENGLISH)

Uniform localization (ENGLISH)

Global solutions to the eikonal equation (ENGLISH)

Applications of microlocal analysis to quantum field theory on curved space-times (ENGLISH)

**Dimitri Yafaev**(Univ. Rennes 1) 14:00-14:45The semiclassical limit of eigenfunctions of the Schroedinger equation and the Bohr-Sommerfeld quantization condition, revisited (ENGLISH)

**David Damanik**(Rice University) 15:00-15:45Uniform localization (ENGLISH)

**Erik Skibsted**(Aarhus University) 16:15-17:00Global solutions to the eikonal equation (ENGLISH)

**Christian Gerard**(Univ. Paris Sud 11) 17:15-18:00Applications of microlocal analysis to quantum field theory on curved space-times (ENGLISH)

### 2011/02/18

#### Operator Algebra Seminars

10:30-12:00 Room #122 (Graduate School of Math. Sci. Bldg.)

Dirac families and 1-cocycles (ENGLISH)

**Pedram Hekmati**(Univ. Adelaide)Dirac families and 1-cocycles (ENGLISH)

[ Abstract ]

Families of Dirac type operators, transforming covariantly under the projective action of the loop group $LG$, determine a class in twisted K-theory on compact Lie groups $G$. The loop group is the gauge group of a principal $G$-bundle over the circle and an interesting problem is to try to generalise the circle to a higher dimensional compact manifold. This is far from obvious and some of the difficulties can be modelled in a slightly simpler setting, by replacing $LG$ and gauge connections by objects which have only small differentiability in the Sobolev sense. In this talk, I will provide some background to this problem and explain how 1-cocycles naturally appear in this construction.

Families of Dirac type operators, transforming covariantly under the projective action of the loop group $LG$, determine a class in twisted K-theory on compact Lie groups $G$. The loop group is the gauge group of a principal $G$-bundle over the circle and an interesting problem is to try to generalise the circle to a higher dimensional compact manifold. This is far from obvious and some of the difficulties can be modelled in a slightly simpler setting, by replacing $LG$ and gauge connections by objects which have only small differentiability in the Sobolev sense. In this talk, I will provide some background to this problem and explain how 1-cocycles naturally appear in this construction.

#### Classical Analysis

11:00-15:45 Room #126 (Graduate School of Math. Sci. Bldg.)

Connection problem on the Hahn-Exton $q$-Bessel functions (ENGLISH)

Rigidity index and middle convolution of $q$-difference equations (Joint work with H. Sakai)

(ENGLISH)

Arithmetic theory of $q$-difference equations and applications (Joint work with C. Hardouin)

(ENGLISH)

**T. Morita**(Osaka University) 11:00-12:00Connection problem on the Hahn-Exton $q$-Bessel functions (ENGLISH)

**M. Yamaguchi**(University of Tokyo) 13:30-14:30Rigidity index and middle convolution of $q$-difference equations (Joint work with H. Sakai)

(ENGLISH)

**L. Di Vizio**(Universite Paris 7) 14:45-15:45Arithmetic theory of $q$-difference equations and applications (Joint work with C. Hardouin)

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