## GCOE lecture series

Seminar information archive ～12/05｜Next seminar｜Future seminars 12/06～

**Seminar information archive**

### 2013/01/08

16:30-17:30 Room #118 (Graduate School of Math. Sci. Bldg.)

Probing factions and Direct sampling method (ENGLISH)

**Kazufumi Ito**(North Carolina State Univ.)Probing factions and Direct sampling method (ENGLISH)

[ Abstract ]

We develop probing functions for inverse medium problems arising in Helmholtz and conductivity and Schrodinger equations. direct sampling methods for determining inhomogeneous inclusions using a limited number of Cauchy data.

We develop probing functions for inverse medium problems arising in Helmholtz and conductivity and Schrodinger equations. direct sampling methods for determining inhomogeneous inclusions using a limited number of Cauchy data.

### 2013/01/07

16:30-17:30 Room #118 (Graduate School of Math. Sci. Bldg.)

Traffic flow modeling and Hamilton Jacobi equation (ENGLISH)

**Kazufumi Ito**(North Carolina State Univ.)Traffic flow modeling and Hamilton Jacobi equation (ENGLISH)

[ Abstract ]

A traffic model based on Hamilton Jacobi equations are developed. The model incorporates the sag and source conditions of traffic flows and predict and classify the traffic congestion.

A traffic model based on Hamilton Jacobi equations are developed. The model incorporates the sag and source conditions of traffic flows and predict and classify the traffic congestion.

### 2012/12/21

15:00-16:00 Room #126 (Graduate School of Math. Sci. Bldg.)

Evolution equation approach to Fractional Differential Equations (ENGLISH)

**Kazufumi Ito**(North Carolina State Univ.)Evolution equation approach to Fractional Differential Equations (ENGLISH)

[ Abstract ]

A class of fractional differential equations is formulated as an evolution equation on the memory space with non-local boundary condition. Based on such a formulation the mathematical theory of evolution equations is applied to concrete examples of nonlinear fractional PDEs.

A class of fractional differential equations is formulated as an evolution equation on the memory space with non-local boundary condition. Based on such a formulation the mathematical theory of evolution equations is applied to concrete examples of nonlinear fractional PDEs.

### 2012/11/29

10:00-12:10 Room #123 (Graduate School of Math. Sci. Bldg.)

Sobolev maps with values into the circle (ENGLISH)

**Haim Brezis**(Rutgers University / Technion)Sobolev maps with values into the circle (ENGLISH)

[ Abstract ]

Sobolev functions with values into R are very well understood and play an immense role in many branches of Mathematics. By contrast, the theory of Sobolev maps with values into the unit circle is still under construction. Such maps occur e.g. in the asymptotic analysis of the Ginzburg-Landau model. The reason one is interested in Sobolev maps, rather than smooth maps is to allow singularities such as x/|x| in 2D or line singularities 3D which appear in physical problems. Our focus in these lectures is not the Ginzburg-Landau equation per se, but rather the intrinsic study of the function space W^{1,p} of maps from a smooth domain in R^N taking their values into the unit circle. Such classes of maps have an amazingly rich structure. Geometrical and Topological effects are already noticeable in this simple framework, since S^1 has nontrivial topology. Moreover the fact that the target space is the circle (as opposed to higher-dimensional manifolds) offers the option to introduce a lifting. We'll see that "optimal liftings" are in one-to-one correspondence with minimal connections (resp. minimal surfaces) spanned by the topological singularities of u.

I will also discuss the question of uniqueness of lifting . A key ingredient in some of the proofs is a formula (due to myself, Bourgain and Mironescu) which provides an original way of approximating Sobolev norms (or the total variation) by nonlocal functionals. Nonconvex versions of these functionals raise very challenging questions recently tackled together with H.-M. Nguyen. Comparable functionals also occur in Image Processing and suggest exciting interactions with this field.

Sobolev functions with values into R are very well understood and play an immense role in many branches of Mathematics. By contrast, the theory of Sobolev maps with values into the unit circle is still under construction. Such maps occur e.g. in the asymptotic analysis of the Ginzburg-Landau model. The reason one is interested in Sobolev maps, rather than smooth maps is to allow singularities such as x/|x| in 2D or line singularities 3D which appear in physical problems. Our focus in these lectures is not the Ginzburg-Landau equation per se, but rather the intrinsic study of the function space W^{1,p} of maps from a smooth domain in R^N taking their values into the unit circle. Such classes of maps have an amazingly rich structure. Geometrical and Topological effects are already noticeable in this simple framework, since S^1 has nontrivial topology. Moreover the fact that the target space is the circle (as opposed to higher-dimensional manifolds) offers the option to introduce a lifting. We'll see that "optimal liftings" are in one-to-one correspondence with minimal connections (resp. minimal surfaces) spanned by the topological singularities of u.

I will also discuss the question of uniqueness of lifting . A key ingredient in some of the proofs is a formula (due to myself, Bourgain and Mironescu) which provides an original way of approximating Sobolev norms (or the total variation) by nonlocal functionals. Nonconvex versions of these functionals raise very challenging questions recently tackled together with H.-M. Nguyen. Comparable functionals also occur in Image Processing and suggest exciting interactions with this field.

### 2012/11/28

13:30-14:30 Room #123 (Graduate School of Math. Sci. Bldg.)

How Poincare became my hero (ENGLISH)

**Haim Brezis**(Rutgers University / Technion)How Poincare became my hero (ENGLISH)

[ Abstract ]

I recently discovered little-known texts of Poincare which include fundamental results on PDEs together with prophetic insights into their future impact on various branches of modern mathematics.

I recently discovered little-known texts of Poincare which include fundamental results on PDEs together with prophetic insights into their future impact on various branches of modern mathematics.

### 2012/11/28

14:50-17:30 Room #123 (Graduate School of Math. Sci. Bldg.)

Can you hear the degree of a map from the circle into itself? An intriguing story which is not yet finished (ENGLISH)

**Haim Brezis**(Rutgers University / Technion)Can you hear the degree of a map from the circle into itself? An intriguing story which is not yet finished (ENGLISH)

[ Abstract ]

A few years ago - following a suggestion by I. M. Gelfand - I discovered an intriguing connection between the topological degree of a map from the circle into itself and its Fourier coefficients. This relation is easily justified when the map is smooth. However, the situation turns out to be much more delicate if one assumes only continuity, or even Holder continuity. I will present recent developments and open problems. The initial motivation for this direction of research came from the analysis of the Ginzburg-Landau model.

A few years ago - following a suggestion by I. M. Gelfand - I discovered an intriguing connection between the topological degree of a map from the circle into itself and its Fourier coefficients. This relation is easily justified when the map is smooth. However, the situation turns out to be much more delicate if one assumes only continuity, or even Holder continuity. I will present recent developments and open problems. The initial motivation for this direction of research came from the analysis of the Ginzburg-Landau model.

### 2012/07/25

16:30-18:00 Room #118 (Graduate School of Math. Sci. Bldg.)

A survey of recent results on the classification of C*-algebras (ENGLISH)

**George Elliott**(University of Toronto)A survey of recent results on the classification of C*-algebras (ENGLISH)

### 2012/07/17

16:30-18:00 Room #118 (Graduate School of Math. Sci. Bldg.)

An introduction to C*-algebra classification theory (ENGLISH)

**George Elliott**(University of Toronto)An introduction to C*-algebra classification theory (ENGLISH)

### 2012/06/08

14:00-15:30 Room #118 (Graduate School of Math. Sci. Bldg.)

Derived categories and cohomological invariants II (ENGLISH)

**Mihnea Popa**(University of Illinois at Chicago)Derived categories and cohomological invariants II (ENGLISH)

[ Abstract ]

(Abstract for both Parts I and II)

I will discuss results on the derived invariance of various cohomological quantities, like the Hodge numbers, a twisted version of Hochschild cohomology, the Picard variety, and cohomological support loci. I will include a small discussion of current work on orbifolds if time permits.

(Abstract for both Parts I and II)

I will discuss results on the derived invariance of various cohomological quantities, like the Hodge numbers, a twisted version of Hochschild cohomology, the Picard variety, and cohomological support loci. I will include a small discussion of current work on orbifolds if time permits.

### 2012/06/05

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

Random walk on reductive groups II (ENGLISH)

**Yves Benoist**(CNRS, Orsay)Random walk on reductive groups II (ENGLISH)

[ Abstract ]

The asymptotic behavior of the sum of real numbers chosen independantly with same probability law is controled by many classical theorems: Law of Large Numbers, Central Limit Theorem, Law of Iterated Logarithm, Local Limit Theorem, Large deviation Principle, 0-1 Law,... In these introductory talks I will recall these classical results and explain their analogs for products of matrices chosen independantly with same probability law, when the action of the support of the law is semisimple. We will see that the dynamics of the corresponding action on the flag variety is a crucial tool for studying these non-commutative random walks.

The asymptotic behavior of the sum of real numbers chosen independantly with same probability law is controled by many classical theorems: Law of Large Numbers, Central Limit Theorem, Law of Iterated Logarithm, Local Limit Theorem, Large deviation Principle, 0-1 Law,... In these introductory talks I will recall these classical results and explain their analogs for products of matrices chosen independantly with same probability law, when the action of the support of the law is semisimple. We will see that the dynamics of the corresponding action on the flag variety is a crucial tool for studying these non-commutative random walks.

### 2012/06/01

14:00-15:30 Room #118 (Graduate School of Math. Sci. Bldg.)

Derived categories and cohomological invariants I (ENGLISH)

**Mihnea Popa**(University of Illinois at Chicago)Derived categories and cohomological invariants I (ENGLISH)

[ Abstract ]

(Abstract for both Parts I and II)

I will discuss results on the derived invariance of various cohomological quantities, like the Hodge numbers, a twisted version of Hochschild cohomology, the Picard variety, and cohomological support loci. I will include a small discussion of current work on orbifolds if time permits.

(Abstract for both Parts I and II)

I will discuss results on the derived invariance of various cohomological quantities, like the Hodge numbers, a twisted version of Hochschild cohomology, the Picard variety, and cohomological support loci. I will include a small discussion of current work on orbifolds if time permits.

### 2012/05/29

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

Random walk on reductive groups. (ENGLISH)

**Yves Benoist**(CNRS, Orsay)Random walk on reductive groups. (ENGLISH)

[ Abstract ]

The asymptotic behavior of the sum of real numbers chosen independantly with same probability law is controled by many classical theorems: Law of Large Numbers, Central Limit Theorem, Law of Iterated Logarithm, Local Limit Theorem, Large deviation Principle, 0-1 Law,... In these introductory talks I will recall these classical results and explain their analogs for products of matrices chosen independantly with same probability law, when the action of the support of the law is semisimple. We will see that the dynamics of the corresponding action on the flag variety is a crucial tool for studying these non-commutative random walks.

The asymptotic behavior of the sum of real numbers chosen independantly with same probability law is controled by many classical theorems: Law of Large Numbers, Central Limit Theorem, Law of Iterated Logarithm, Local Limit Theorem, Large deviation Principle, 0-1 Law,... In these introductory talks I will recall these classical results and explain their analogs for products of matrices chosen independantly with same probability law, when the action of the support of the law is semisimple. We will see that the dynamics of the corresponding action on the flag variety is a crucial tool for studying these non-commutative random walks.

### 2012/05/25

14:00-15:30 Room #118 (Graduate School of Math. Sci. Bldg.)

Generic vanishing theory and connections with derived categories (ENGLISH)

**Mihnea Popa**(University of Illinois at Chicago)Generic vanishing theory and connections with derived categories (ENGLISH)

[ Abstract ]

I will give a basic introduction to the main results regarding the cohomology of deformations of the canonical bundle, and explain a connection with certain t-structures on the derived categories of Picard varieties. (This will also serve as an introduction for the talk at AG seminar on 5/28, 15:30-17:00.)

I will give a basic introduction to the main results regarding the cohomology of deformations of the canonical bundle, and explain a connection with certain t-structures on the derived categories of Picard varieties. (This will also serve as an introduction for the talk at AG seminar on 5/28, 15:30-17:00.)

### 2012/01/23

13:30-15:00 Room #122 (Graduate School of Math. Sci. Bldg.)

TBA(中止になりました) (ENGLISH)

**Mihai Paun**(Institut Élie Cartan and KIAS)TBA(中止になりました) (ENGLISH)

### 2012/01/18

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

TBA(中止になりました) (ENGLISH)

**Mihai Paun**(Institut Élie Cartan and KIAS)TBA(中止になりました) (ENGLISH)

### 2012/01/16

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

TBA(中止になりました) (ENGLISH)

**Mihai Paun**(Institut Élie Cartan and KIAS)TBA(中止になりました) (ENGLISH)

### 2011/11/17

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

Recovery of weakly coupled system from partial Cauchy data (ENGLISH)

**Oleg Emanouilov**(Colorado State University)Recovery of weakly coupled system from partial Cauchy data (ENGLISH)

[ Abstract ]

We consider the inverse problem for recovery of coefficients of weakly coupled system of elliptic equations in a bounded 2D domain.

We consider the inverse problem for recovery of coefficients of weakly coupled system of elliptic equations in a bounded 2D domain.

### 2011/11/14

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

Recovery of weakly coupled system from partial Cauchy data (ENGLISH)

**Oleg Emanouilov**(Colorado State University)Recovery of weakly coupled system from partial Cauchy data (ENGLISH)

[ Abstract ]

We consider the inverse problem for recovery of coefficients of weakly coupled system of elliptic equations in a bounded 2D domain.

We consider the inverse problem for recovery of coefficients of weakly coupled system of elliptic equations in a bounded 2D domain.

### 2011/11/10

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

Inverse boundary value problem for Schroedinger equation in two dimensions (ENGLISH)

**Oleg Emanouilov**(Colorado State University)Inverse boundary value problem for Schroedinger equation in two dimensions (ENGLISH)

[ Abstract ]

We relax the regularity condition on potentials of Schroedinger equations in uniqueness results on the inverse boundary value problem recently proved in A.Bukhgeim (2008) and O. Imanuvilov, G.Uhlmann and M. Yamamoto (2010).

We relax the regularity condition on potentials of Schroedinger equations in uniqueness results on the inverse boundary value problem recently proved in A.Bukhgeim (2008) and O. Imanuvilov, G.Uhlmann and M. Yamamoto (2010).

### 2011/11/07

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

Inverse boundary value problem for Schroedinger equation in two dimensions (ENGLISH)

**Oleg Emanouilov**(Colorado State University)Inverse boundary value problem for Schroedinger equation in two dimensions (ENGLISH)

[ Abstract ]

We relax the regularity condition on potentials of Schroedinger equations in uniqueness results on the inverse boundary value problem recently proved in A.Bukhgeim (2008) and O. Imanuvilov, G.Uhlmann and M. Yamamoto (2010).

We relax the regularity condition on potentials of Schroedinger equations in uniqueness results on the inverse boundary value problem recently proved in A.Bukhgeim (2008) and O. Imanuvilov, G.Uhlmann and M. Yamamoto (2010).

### 2010/11/08

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

Invariant differential operators on the sphere (ENGLISH)

https://www.ms.u-tokyo.ac.jp/~toshi/seminar/ut-seminar2010.html#20101102eastwood

**Michael Eastwood**(Australian National University)Invariant differential operators on the sphere (ENGLISH)

[ Abstract ]

The circle is acted upon by the rotation group SO(2) and there are plenty of differential operators invariant under this action. But the circle is also acted upon by SL(2,R) and this larger symmetry group cuts down the list of invariant differential operators to something smaller but more interesting! I shall explain what happens and how this phenomenon generalises to spheres. These constructions are part of a general theory but have numerous unexpected applications, for example in suggesting a new stable finite-element scheme in linearised elasticity (due to Arnold, Falk, and Winther).

[ Reference URL ]The circle is acted upon by the rotation group SO(2) and there are plenty of differential operators invariant under this action. But the circle is also acted upon by SL(2,R) and this larger symmetry group cuts down the list of invariant differential operators to something smaller but more interesting! I shall explain what happens and how this phenomenon generalises to spheres. These constructions are part of a general theory but have numerous unexpected applications, for example in suggesting a new stable finite-element scheme in linearised elasticity (due to Arnold, Falk, and Winther).

https://www.ms.u-tokyo.ac.jp/~toshi/seminar/ut-seminar2010.html#20101102eastwood

### 2010/11/05

16:30-18:00 Room #123 (Graduate School of Math. Sci. Bldg.)

How to recognise the geodesics of a metric connection (ENGLISH)

https://www.ms.u-tokyo.ac.jp/~toshi/seminar/ut-seminar2010.html#20101102eastwood

**Michael Eastwood**(Australian National University)How to recognise the geodesics of a metric connection (ENGLISH)

[ Abstract ]

The geodesics on a Riemannian manifold form a distinguished family of curves, one in every direction through every point. Sometimes two metrics can provide the same family of curves: the Euclidean metric and the round metric on the hemisphere have this property. It is also possible that a family of curves does not arise from a metric at all. Following a classical procedure due to Roger Liouville, I shall explain how to tell these cases apart on a surface. This is joint work with Robert Bryant and Maciej Dunajski.

[ Reference URL ]The geodesics on a Riemannian manifold form a distinguished family of curves, one in every direction through every point. Sometimes two metrics can provide the same family of curves: the Euclidean metric and the round metric on the hemisphere have this property. It is also possible that a family of curves does not arise from a metric at all. Following a classical procedure due to Roger Liouville, I shall explain how to tell these cases apart on a surface. This is joint work with Robert Bryant and Maciej Dunajski.

https://www.ms.u-tokyo.ac.jp/~toshi/seminar/ut-seminar2010.html#20101102eastwood

### 2010/06/26

13:50-14:50 Room #056 (Graduate School of Math. Sci. Bldg.)

On a subfactor generalization of Wall's conjecture (ENGLISH)

**Feng Xu**(UC Riverside)On a subfactor generalization of Wall's conjecture (ENGLISH)

### 2010/06/17

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

On a relative version of Wall's conjecture (ENGLISH)

**Feng Xu**(UC Riverside)On a relative version of Wall's conjecture (ENGLISH)

### 2010/06/03

16:30-18:00 Room #470 (Graduate School of Math. Sci. Bldg.)

Introduction to the cohomology of locally symmetric spaces 2

(ENGLISH)

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

**Birgit Speh**(Cornel University)Introduction to the cohomology of locally symmetric spaces 2

(ENGLISH)

[ Abstract ]

I will give an introduction to the cohomology of noncompact locally symmetric spaces $X_\\Gamma =K \\backslash G / \\Gamma $.

If $X_\\Gamma $ is cocompact this cohomology can be expressed as the $(\\bg,K) $-cohomology of automorphic representations. I will explain how representation theory and automorphic forms can be used to study the cohomology in this case.

[ Reference URL ]I will give an introduction to the cohomology of noncompact locally symmetric spaces $X_\\Gamma =K \\backslash G / \\Gamma $.

If $X_\\Gamma $ is cocompact this cohomology can be expressed as the $(\\bg,K) $-cohomology of automorphic representations. I will explain how representation theory and automorphic forms can be used to study the cohomology in this case.

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