## Seminar information archive

Seminar information archive ～09/15｜Today's seminar 09/16 | Future seminars 09/17～

#### Number Theory Seminar

17:30-18:30 Room #056 (Graduate School of Math. Sci. Bldg.)

Around the Mordell-Lang conjecture in positive characteristic (ENGLISH)

**Damian Rossler**(CNRS, Universite de Toulouse)Around the Mordell-Lang conjecture in positive characteristic (ENGLISH)

[ Abstract ]

Let V be a subvariety of an abelian variety A over C and let G\\subseteq A(C) be a subgroup. The classical Mordell-Lang conjecture predicts that if V is of general type and G\\otimesQ is finite dimensional, then V\\cap G is not Zariski dense in V. This statement contains the Mordell conjecture as well as the Manin-Mumford conjecture (for curves). The positive characteristic analog of the Mordell-Lang conjecture makes sense, when A is supposed to have no subquotient, which is defined over a finite field. This positive characteristic analog was proven in 1996 by E. Hrushovski using model-theoretic methods. We shall discuss the prehistory and context of this proof. We shall also discuss the proof (due to the speaker) of the fact that in positive characteristic, the Manin-Mumford conjecture implies the Mordell-Lang conjecture (whereas this seems far from true in characteristic 0).

Let V be a subvariety of an abelian variety A over C and let G\\subseteq A(C) be a subgroup. The classical Mordell-Lang conjecture predicts that if V is of general type and G\\otimesQ is finite dimensional, then V\\cap G is not Zariski dense in V. This statement contains the Mordell conjecture as well as the Manin-Mumford conjecture (for curves). The positive characteristic analog of the Mordell-Lang conjecture makes sense, when A is supposed to have no subquotient, which is defined over a finite field. This positive characteristic analog was proven in 1996 by E. Hrushovski using model-theoretic methods. We shall discuss the prehistory and context of this proof. We shall also discuss the proof (due to the speaker) of the fact that in positive characteristic, the Manin-Mumford conjecture implies the Mordell-Lang conjecture (whereas this seems far from true in characteristic 0).

### 2012/04/10

#### Tuesday Seminar on Topology

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

On homology of symplectic derivation Lie algebras of

the free associative algebra and the free Lie algebra (JAPANESE)

**Takuya Sakasai**(The University of Tokyo)On homology of symplectic derivation Lie algebras of

the free associative algebra and the free Lie algebra (JAPANESE)

[ Abstract ]

We discuss homology of symplectic derivation Lie algebras of

the free associative algebra and the free Lie algebra

with particular stress on their abelianizations (degree 1 part).

Then, by using a theorem of Kontsevich,

we give some applications to rational cohomology of the moduli spaces of

Riemann surfaces and metric graphs.

This is a joint work with Shigeyuki Morita and Masaaki Suzuki.

We discuss homology of symplectic derivation Lie algebras of

the free associative algebra and the free Lie algebra

with particular stress on their abelianizations (degree 1 part).

Then, by using a theorem of Kontsevich,

we give some applications to rational cohomology of the moduli spaces of

Riemann surfaces and metric graphs.

This is a joint work with Shigeyuki Morita and Masaaki Suzuki.

### 2012/04/09

#### Algebraic Geometry Seminar

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

On mirror symmetry for weighted Calabi-Yau hypersurfaces (JAPANESE)

**Kazushi Ueda**(Osaka University)On mirror symmetry for weighted Calabi-Yau hypersurfaces (JAPANESE)

[ Abstract ]

In the talk, I will discuss relation between homological mirror symmetry for weighted projective spaces, their Calabi-Yau hypersurfaces, and weighted homogeneous singularities.

If the time permits, I will also discuss an application to monodromy of hypergeometric functions.

In the talk, I will discuss relation between homological mirror symmetry for weighted projective spaces, their Calabi-Yau hypersurfaces, and weighted homogeneous singularities.

If the time permits, I will also discuss an application to monodromy of hypergeometric functions.

#### Seminar on Geometric Complex Analysis

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

Effective estimate on the number of deformation types of families of canonically polarized manifolds over curves

(JAPANESE)

**Shigeharu TAKAYAMA**(University of Tokyo)Effective estimate on the number of deformation types of families of canonically polarized manifolds over curves

(JAPANESE)

### 2012/04/04

#### PDE Real Analysis Seminar

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

Multi linear formulation of differential geometry and matrix regularizations (ENGLISH)

**Jens Hoppe**(Sogang University / KTH Royal Institute of Technology)Multi linear formulation of differential geometry and matrix regularizations (ENGLISH)

[ Abstract ]

We prove that many aspects of the differential geometry of embedded Riemannian manifolds can be formulated in terms of multi linear algebraic structures on the space of smooth functions. In particular, we find algebraic expressions for Weingarten's formula, the Ricci curvature and the Codazzi-Mainardi equations.

For matrix analogues of embedded surfaces we define discrete curvatures and Euler characteristics, and a non-commutative Gauss–Bonnet theorem is shown to follow. We derive simple expressions for the discrete Gauss curvature in terms of matrices representing the embedding coordinates, and a large class of explicit examples is provided.

We prove that many aspects of the differential geometry of embedded Riemannian manifolds can be formulated in terms of multi linear algebraic structures on the space of smooth functions. In particular, we find algebraic expressions for Weingarten's formula, the Ricci curvature and the Codazzi-Mainardi equations.

For matrix analogues of embedded surfaces we define discrete curvatures and Euler characteristics, and a non-commutative Gauss–Bonnet theorem is shown to follow. We derive simple expressions for the discrete Gauss curvature in terms of matrices representing the embedding coordinates, and a large class of explicit examples is provided.

### 2012/04/03

#### Monthly Seminar on Arithmetic of Automorphic Forms

13:30-16:00 Room #123 (Graduate School of Math. Sci. Bldg.)

Explicit formula for the formal degree of the discrete series representations of GL_m(D). (JAPANESE)

Moments of the derivatives of the Riemann zeta function (JAPANESE)

**Kazutoshi Kariyama**(Onomichi city university) 13:30-14:30Explicit formula for the formal degree of the discrete series representations of GL_m(D). (JAPANESE)

**Keijyu Souno**(Math.-Sci., Tokyo Univ.) 15:00-16:00Moments of the derivatives of the Riemann zeta function (JAPANESE)

[ Abstract ]

In my talk, we consider the integral moments of the derivatives of the Riemann zeta function on the critical line. We give certain lower bounds for these moments under the assumption of the Riemann hypothesis.

In my talk, we consider the integral moments of the derivatives of the Riemann zeta function on the critical line. We give certain lower bounds for these moments under the assumption of the Riemann hypothesis.

### 2012/03/23

#### Operator Algebra Seminars

16:30-18:00 Room #117 (Graduate School of Math. Sci. Bldg.)

Higher Rank Graph $C^*$-algebras (ENGLISH)

**Alex Kumjian**(University of Nevada, Reno)Higher Rank Graph $C^*$-algebras (ENGLISH)

#### Lectures

10:30-11:30 Room #117 (Graduate School of Math. Sci. Bldg.)

Cell decomposition of homotopy Deligne-Mumford. (ENGLISH)

**R. Penner**(Aarhus/Caltech)Cell decomposition of homotopy Deligne-Mumford. (ENGLISH)

[ Abstract ]

A long-standing problem has been to extend the ideal cell decomposition of Riemann's moduli space to its compactification by stable curves. In joint work with Doug LaFountain, we have solved this problem with an explicit generalization of fatgraphs. The solution immediately provides a construction of odd-degree cycles, which are conjectured to be non-trivial, thus addressing yet another long-standing issue.

A long-standing problem has been to extend the ideal cell decomposition of Riemann's moduli space to its compactification by stable curves. In joint work with Doug LaFountain, we have solved this problem with an explicit generalization of fatgraphs. The solution immediately provides a construction of odd-degree cycles, which are conjectured to be non-trivial, thus addressing yet another long-standing issue.

### 2012/03/21

#### Lectures

10:15-12:00 Room #123 (Graduate School of Math. Sci. Bldg.)

Geochemical structure of biological macromolecules (ENGLISH)

**R. Penner**(Aarhus/Caltech)Geochemical structure of biological macromolecules (ENGLISH)

[ Abstract ]

This first of two lectures will explain the basic combinatorial and geometrical structures of both protein and RNA. It is intended to set the stage of subsequent discussions for an audience with mathematical background.

This first of two lectures will explain the basic combinatorial and geometrical structures of both protein and RNA. It is intended to set the stage of subsequent discussions for an audience with mathematical background.

#### Lectures

15:15-17:00 Room #123 (Graduate School of Math. Sci. Bldg.)

Moduli space techniques in computational biology

(ENGLISH)

**R. Penner**(Aarhus/Caltech)Moduli space techniques in computational biology

(ENGLISH)

[ Abstract ]

Basic fatgraph models of RNA and protein will be discussed, where edges are associated with base pairs in the former case and with hydrogen bonds between backbone atoms in the latter. For RNA, this leads to new methods described by context-free grammars of RNA folding prediction including certain classes of pseudo-knots. For protein, beyond these discrete invariants lie continuous ones which associate a rotation of

3-dimensional space to each hydrogen bond linking a pair of peptide units. Histograms of these rotations over the entire database of proteins exhibit a small number of "peptide unit legos" which can be used to advantage for the protein folding problem.

Basic fatgraph models of RNA and protein will be discussed, where edges are associated with base pairs in the former case and with hydrogen bonds between backbone atoms in the latter. For RNA, this leads to new methods described by context-free grammars of RNA folding prediction including certain classes of pseudo-knots. For protein, beyond these discrete invariants lie continuous ones which associate a rotation of

3-dimensional space to each hydrogen bond linking a pair of peptide units. Histograms of these rotations over the entire database of proteins exhibit a small number of "peptide unit legos" which can be used to advantage for the protein folding problem.

#### PDE Real Analysis Seminar

10:00-11:00 Room #056 (Graduate School of Math. Sci. Bldg.)

The asymptotic behaviors of the solutions of Poisson-Boltzmann type of equations (ENGLISH)

**Chiun-Chang Lee**(National Taiwan University)The asymptotic behaviors of the solutions of Poisson-Boltzmann type of equations (ENGLISH)

[ Abstract ]

Understanding the existence of electrical double layers around particles in the colloidal dispersion (system) is a crucial phenomenon of the colloid science. The Poisson-Boltzmann (PB) equation is one of the most widely used models to describe the equilibrium phenomenon of an electrical double layer in colloidal systems. This motivates us to study the asymptotic behavior for the boundary layer of the solutions of the PB equation. In this talk, we introduce the precise asymptotic formulas for the slope of the boundary layers with the exact leading order term and the second-order term. In particular, these formulas show that the mean curvature of the boundary exactly appears in the second-order term. This part is my personal work.

On the other hand, to study how the ionic concentrations and ionic valences affect the difference between the boundary and interior potentials in an electrolyte solution, we also introduce a modified PB equation - New Poisson-Boltzmann (PB_n) equation - joint works with Prof. Tai-Chia Lin and Chun Liu and YunKyong Hyon. We give a specific formula showing the influence of these crucial physical quantities on the potential difference in an electrolyte solution. This cannot be found in the PB equation.

Understanding the existence of electrical double layers around particles in the colloidal dispersion (system) is a crucial phenomenon of the colloid science. The Poisson-Boltzmann (PB) equation is one of the most widely used models to describe the equilibrium phenomenon of an electrical double layer in colloidal systems. This motivates us to study the asymptotic behavior for the boundary layer of the solutions of the PB equation. In this talk, we introduce the precise asymptotic formulas for the slope of the boundary layers with the exact leading order term and the second-order term. In particular, these formulas show that the mean curvature of the boundary exactly appears in the second-order term. This part is my personal work.

On the other hand, to study how the ionic concentrations and ionic valences affect the difference between the boundary and interior potentials in an electrolyte solution, we also introduce a modified PB equation - New Poisson-Boltzmann (PB_n) equation - joint works with Prof. Tai-Chia Lin and Chun Liu and YunKyong Hyon. We give a specific formula showing the influence of these crucial physical quantities on the potential difference in an electrolyte solution. This cannot be found in the PB equation.

### 2012/03/16

#### Colloquium

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

On Complex Fluids (ENGLISH)

**Chun LIU**(University of Tokyo / Penn State University)On Complex Fluids (ENGLISH)

[ Abstract ]

The talk is on the mathematical theories, in particular the energetic variational approaches, of anisotropic complex fluids, such as viscoelastic materials, liquid crystals and ionic fluids in proteins and bio-solutions.

Complex fluids, including mixtures and solutions, are abundant in our daily life. The complicated phenomena and properties exhibited by these materials reflects the coupling and competition between the microscopic interactions and the macroscopic dynamics. We study the underlying energetic variational structures that is common among all these multiscale-multiphysics systems.

In this talk, I will demonstrate the modeling as well as analysis and numerical issues arising from various complex fluids.

The talk is on the mathematical theories, in particular the energetic variational approaches, of anisotropic complex fluids, such as viscoelastic materials, liquid crystals and ionic fluids in proteins and bio-solutions.

Complex fluids, including mixtures and solutions, are abundant in our daily life. The complicated phenomena and properties exhibited by these materials reflects the coupling and competition between the microscopic interactions and the macroscopic dynamics. We study the underlying energetic variational structures that is common among all these multiscale-multiphysics systems.

In this talk, I will demonstrate the modeling as well as analysis and numerical issues arising from various complex fluids.

### 2012/03/14

#### PDE Real Analysis Seminar

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

Exponential convergence to equilibria for a general model in hydrodynamics (ENGLISH)

**Jürgen Saal**(Technische Universität Darmstadt)Exponential convergence to equilibria for a general model in hydrodynamics (ENGLISH)

[ Abstract ]

We present a thorough analysis of the Navier-Stokes-Nernst-Planck-Poisson equations. This system describes the dynamics of charged particles dispersed in an incompressible fluid.

In contrast to existing literature and in view of its physical relevance, we also allow for different diffusion coefficients of the charged species.

In addition, the commonly assumed electro-neutrality condition is not required by our approach.

Our aim is to present results on local and global well-posedness as well as exponential stability of equilibria. The results are obtained jointly with Dieter Bothe and Andre Fischer at the Center of Smart Interfaces at TU Darmstadt.

We present a thorough analysis of the Navier-Stokes-Nernst-Planck-Poisson equations. This system describes the dynamics of charged particles dispersed in an incompressible fluid.

In contrast to existing literature and in view of its physical relevance, we also allow for different diffusion coefficients of the charged species.

In addition, the commonly assumed electro-neutrality condition is not required by our approach.

Our aim is to present results on local and global well-posedness as well as exponential stability of equilibria. The results are obtained jointly with Dieter Bothe and Andre Fischer at the Center of Smart Interfaces at TU Darmstadt.

### 2012/03/13

#### Colloquium

15:00-16:00 Room #050 (Graduate School of Math. Sci. Bldg.)

Problems and results on Hardy's Z-function (JAPANESE)

**Aleksandar Ivic**(University of Belgrade, the Serbian Academy of Science and Arts)Problems and results on Hardy's Z-function (JAPANESE)

[ Abstract ]

The title is self-explanatory: G.H. Hardy first used the function

$Z(t)$ to show that there are infinitely many zeta-zeros on the

critical line $\\Re s = 1/2$. In recent years there is a revived

interest in this function, with many results and open problems.

The title is self-explanatory: G.H. Hardy first used the function

$Z(t)$ to show that there are infinitely many zeta-zeros on the

critical line $\\Re s = 1/2$. In recent years there is a revived

interest in this function, with many results and open problems.

#### Mathematical Biology Seminar

14:00-15:00 Room #154 (Graduate School of Math. Sci. Bldg.)

On construction of Lyapunov functions and functionals (JAPANESE)

**Tsuyoshi Kajiwara**(Okayama University)On construction of Lyapunov functions and functionals (JAPANESE)

### 2012/03/09

#### Infinite Analysis Seminar Tokyo

13:30-14:30 Room #002 (Graduate School of Math. Sci. Bldg.)

On Hall algebra of complexes (JAPANESE)

**Shintarou Yanagida**(Kobe Univ.)On Hall algebra of complexes (JAPANESE)

[ Abstract ]

The topic of my talk is the Hall algebra of complexes,

which is recently introduced by T. Bridgeland.

I will discuss its properties and relation to

auto-equivalences of derived category.

If I have enough time,

I will also discuss the relation

of this Hall algebra to the so-called Ding-Iohara algebra.

The topic of my talk is the Hall algebra of complexes,

which is recently introduced by T. Bridgeland.

I will discuss its properties and relation to

auto-equivalences of derived category.

If I have enough time,

I will also discuss the relation

of this Hall algebra to the so-called Ding-Iohara algebra.

### 2012/03/07

#### GCOE Seminars

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

Nonsmooth Optimization, Theory and Applications. (ENGLISH)

**Kazufumi Ito**(North Carolina State Univ.)Nonsmooth Optimization, Theory and Applications. (ENGLISH)

[ Abstract ]

We develop a Lagrange multiplier theory for Nonsmooth optimization, including $L^¥infty$ and $L^1$ optimizations, $¥ell^0$ (counting meric) and $L^0$ (Ekeland mertic), Binary and Mixed integer optimizations and Data mining. A multitude of important problems can be treated by our approach and numerical algorithms are developed based on the Lagrange multiplier theory.

We develop a Lagrange multiplier theory for Nonsmooth optimization, including $L^¥infty$ and $L^1$ optimizations, $¥ell^0$ (counting meric) and $L^0$ (Ekeland mertic), Binary and Mixed integer optimizations and Data mining. A multitude of important problems can be treated by our approach and numerical algorithms are developed based on the Lagrange multiplier theory.

### 2012/03/06

#### GCOE Seminars

16:00-17:00 Room #370 (Graduate School of Math. Sci. Bldg.)

On the phase field approach to shape and topology optimization (ENGLISH)

**Dietmar Hoemberg**(Weierstrass Institute, Berlin)On the phase field approach to shape and topology optimization (ENGLISH)

[ Abstract ]

Owing to different densities of the respective phases, solid-solid phase transitions often are accompanied by (often undesired) changes in workpiece size and shape. In my talk I will address the reverse question of finding an optimal phase mixture in order to accomplish a desired workpiece shape.

From mathematical point of view this corresponds to an optimal shape design problem subject to a static mechanical equilibrium problem with phase dependent stiffness tensor, in which the two phases exhibit different densities leading to different internal stresses. Our goal is to tackle this problem using a phasefield relaxation.

To this end we first briefly recall previous works regarding phasefield approaches to topology optimization (e.g. by Bourdin ¥& Chambolle, Burger ¥& Stainko and Blank, Garcke et al.).

We add a Ginzburg-Landau term to our cost functional, derive an adjoint equation for the displacement and choose a gradient flow dynamics with an articial time variable for our phasefield variable. We discuss well-posedness results for the resulting system and conclude with some numerical results.

Owing to different densities of the respective phases, solid-solid phase transitions often are accompanied by (often undesired) changes in workpiece size and shape. In my talk I will address the reverse question of finding an optimal phase mixture in order to accomplish a desired workpiece shape.

From mathematical point of view this corresponds to an optimal shape design problem subject to a static mechanical equilibrium problem with phase dependent stiffness tensor, in which the two phases exhibit different densities leading to different internal stresses. Our goal is to tackle this problem using a phasefield relaxation.

To this end we first briefly recall previous works regarding phasefield approaches to topology optimization (e.g. by Bourdin ¥& Chambolle, Burger ¥& Stainko and Blank, Garcke et al.).

We add a Ginzburg-Landau term to our cost functional, derive an adjoint equation for the displacement and choose a gradient flow dynamics with an articial time variable for our phasefield variable. We discuss well-posedness results for the resulting system and conclude with some numerical results.

#### GCOE Seminars

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

Finite element simulations of induction hardening of steel parts (ENGLISH)

**Thomas Petzold**(Weierstrass Institute, Berlin)Finite element simulations of induction hardening of steel parts (ENGLISH)

[ Abstract ]

Induction hardening is a modern method for the heat treatment of steel parts.

A well directed heating by electromagnetic waves and subsequent quenching of the workpiece increases the hardness of the surface layer.

The process is very fast and energy efficient and plays a big role in modern manufacturing facilities in many industrial application areas.

In this talk a model for induction hardening of steel parts is presented. It consist of a system of partial differential equations including Maxwell's equations and the heat equation.

The finite element method is used to perform numerical simulations in 3D.

This requires a suitable discretization of Maxwell's equations leading to so called edge-finite-elements.

We will give a short overview of edge elements and present numerical simulations of induction hardening.

We will address some of the difficulties arising when solving the large system of non-linear coupled PDEs in three space dimensions.

Induction hardening is a modern method for the heat treatment of steel parts.

A well directed heating by electromagnetic waves and subsequent quenching of the workpiece increases the hardness of the surface layer.

The process is very fast and energy efficient and plays a big role in modern manufacturing facilities in many industrial application areas.

In this talk a model for induction hardening of steel parts is presented. It consist of a system of partial differential equations including Maxwell's equations and the heat equation.

The finite element method is used to perform numerical simulations in 3D.

This requires a suitable discretization of Maxwell's equations leading to so called edge-finite-elements.

We will give a short overview of edge elements and present numerical simulations of induction hardening.

We will address some of the difficulties arising when solving the large system of non-linear coupled PDEs in three space dimensions.

### 2012/02/29

#### GCOE Seminars

16:00-17:00 Room #270 (Graduate School of Math. Sci. Bldg.)

Direct and inverse scattering of elastic waves by diffraction gratings (ENGLISH)

**Johannes Elschner**(Weierstrass Institute, Germany)Direct and inverse scattering of elastic waves by diffraction gratings (ENGLISH)

[ Abstract ]

The talk presents joint work with Guanghui Hu on the scattering of time-harmonic plane elastic waves by two-dimensional periodic structures. The first part presents existence and uniqueness results for the direct problem , using a variational approach. For the inverse problem, we discuss global uniqueness results with a minimal number of incident pressure or shear waves under the boundary conditions of the third and fourth kind. Generalizations to biperiodic elastic diffraction gratings in 3D are also mentioned. Finally we consider a reconstruction method applied to the inverse Dirichlet problem for the quasi-periodic 2D Navier equation.

The talk presents joint work with Guanghui Hu on the scattering of time-harmonic plane elastic waves by two-dimensional periodic structures. The first part presents existence and uniqueness results for the direct problem , using a variational approach. For the inverse problem, we discuss global uniqueness results with a minimal number of incident pressure or shear waves under the boundary conditions of the third and fourth kind. Generalizations to biperiodic elastic diffraction gratings in 3D are also mentioned. Finally we consider a reconstruction method applied to the inverse Dirichlet problem for the quasi-periodic 2D Navier equation.

### 2012/02/22

#### Number Theory Seminar

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

Twistor $D$-module and harmonic bundle (ENGLISH)

**Takuro Mochizuki**(Research Institute for Mathematical Sciences, Kyoto University)Twistor $D$-module and harmonic bundle (ENGLISH)

[ Abstract ]

Abstract:

We shall overview the theory of twistor $D$-modules and

harmonic bundles. I am planning to survey the following topics,

motivated by the Hard Lefschetz Theorem for semisimple holonomic

$D$-modules:

1. What is a twistor $D$-module?

2. Local structure of meromorphic flat bundles

3. Wild harmonic bundles from local and global viewpoints

(本講演は「東京パリ数論幾何セミナー」として、インターネットによる東大数理とIHESとの双方向同時中継で行います。)

Abstract:

We shall overview the theory of twistor $D$-modules and

harmonic bundles. I am planning to survey the following topics,

motivated by the Hard Lefschetz Theorem for semisimple holonomic

$D$-modules:

1. What is a twistor $D$-module?

2. Local structure of meromorphic flat bundles

3. Wild harmonic bundles from local and global viewpoints

(本講演は「東京パリ数論幾何セミナー」として、インターネットによる東大数理とIHESとの双方向同時中継で行います。)

#### GCOE Seminars

15:00-16:00 Room #270 (Graduate School of Math. Sci. Bldg.)

Justification of a Shallow Shell Model in Unilateral Contact with an Obstacle (ENGLISH)

**Bernadette Miara**(Université Paris-Est, ESIEE, France)Justification of a Shallow Shell Model in Unilateral Contact with an Obstacle (ENGLISH)

[ Abstract ]

We consider a three-dimensional elastic shell in unilateral contact with a plane. This lecture aims at justifying the asymptotic limit of the set of equilibrium equations of the structure when the thickness of the shell goes to zero. More precisely, we start with the 3D Signorini problem (with finite thickness) and obtain at the limit an obstacle 2D problem. This problem has already been studied [4] in the Cartesian framework on the basis of the bi-lateral problem [3]. The interest and the difficulty of the approach in the curvilinear framework (more appropriate to handle general shells) is due to the coupling between the tangential and transverse covariant components of the elastic field in the expression of the nonpenetrability conditions.

The procedure is the same as the one used in the asymptotic analysis of 3D bilateral structures [1, 2]: assumptions on the data, (loads and geometry of the middle surface of the shell) and re-scalling of the unknowns (displacement field or stress tensor); the new feature is the special handling of the components coupling.

The main result we obtain is as follows:

i) Under the assumption of regularity of the external volume and surface loads, and of the mapping that defines the middle surface of the shell, we establish that the family of elastic displacements converges strongly as the thickness tends to zero in an appropriate set which is a convex cone.

ii) The limit elastic displacement is a Kirchhoff-Love field given by a variational problem which will be analysed into details. The contact conditions are fully explicited for any finite thickness and at the limit.

This is a joint work with Alain L´eger, CNRS, Laboratoire de M´ecanique et d’Acoustique, 13402, Marseille, France.

We consider a three-dimensional elastic shell in unilateral contact with a plane. This lecture aims at justifying the asymptotic limit of the set of equilibrium equations of the structure when the thickness of the shell goes to zero. More precisely, we start with the 3D Signorini problem (with finite thickness) and obtain at the limit an obstacle 2D problem. This problem has already been studied [4] in the Cartesian framework on the basis of the bi-lateral problem [3]. The interest and the difficulty of the approach in the curvilinear framework (more appropriate to handle general shells) is due to the coupling between the tangential and transverse covariant components of the elastic field in the expression of the nonpenetrability conditions.

The procedure is the same as the one used in the asymptotic analysis of 3D bilateral structures [1, 2]: assumptions on the data, (loads and geometry of the middle surface of the shell) and re-scalling of the unknowns (displacement field or stress tensor); the new feature is the special handling of the components coupling.

The main result we obtain is as follows:

i) Under the assumption of regularity of the external volume and surface loads, and of the mapping that defines the middle surface of the shell, we establish that the family of elastic displacements converges strongly as the thickness tends to zero in an appropriate set which is a convex cone.

ii) The limit elastic displacement is a Kirchhoff-Love field given by a variational problem which will be analysed into details. The contact conditions are fully explicited for any finite thickness and at the limit.

This is a joint work with Alain L´eger, CNRS, Laboratoire de M´ecanique et d’Acoustique, 13402, Marseille, France.

#### GCOE Seminars

16:15-17:15 Room #270 (Graduate School of Math. Sci. Bldg.)

Determination of first order coefficient in semilinear elliptic equation by partial Cauchy data. (ENGLISH)

**Oleg Emanouilov**(Colorado State University)Determination of first order coefficient in semilinear elliptic equation by partial Cauchy data. (ENGLISH)

[ Abstract ]

In a bounded domain in $R^2$, we consider a semilinear elliptic equation $¥Delta u +qu +f(u)=0$.

Under some conditions on $f$, we show that the coefficient $q$ can be uniquely determined by the following partial data

$$

{¥mathcal C}_q=¥{(u,¥frac{¥partial u}{¥partial¥nu})¥vert_{\\\\tilde Gamma}¥vert

- ¥Delta u +qu +f(u)=0, ¥,¥,¥, u¥vert_{¥Gamma_0}=0,¥,¥, u¥in H^1(¥Omega)¥}

$$

where $¥tilde ¥Gamma$ is an arbitrary fixed open set of

$¥partial¥Omega$ and $¥Gamma_0=¥partial¥Omega¥setminus¥tilde¥Gamma$.

In a bounded domain in $R^2$, we consider a semilinear elliptic equation $¥Delta u +qu +f(u)=0$.

Under some conditions on $f$, we show that the coefficient $q$ can be uniquely determined by the following partial data

$$

{¥mathcal C}_q=¥{(u,¥frac{¥partial u}{¥partial¥nu})¥vert_{\\\\tilde Gamma}¥vert

- ¥Delta u +qu +f(u)=0, ¥,¥,¥, u¥vert_{¥Gamma_0}=0,¥,¥, u¥in H^1(¥Omega)¥}

$$

where $¥tilde ¥Gamma$ is an arbitrary fixed open set of

$¥partial¥Omega$ and $¥Gamma_0=¥partial¥Omega¥setminus¥tilde¥Gamma$.

### 2012/02/21

#### Tuesday Seminar on Topology

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

Property (TT)/T and homomorphism superrigidity into mapping class groups (JAPANESE)

**Masato Mimura**(The University of Tokyo)Property (TT)/T and homomorphism superrigidity into mapping class groups (JAPANESE)

[ Abstract ]

Mapping class groups (MCG's), of compact oriented surfaces (possibly

with punctures), have many mysterious features: they behave not only

like higher rank lattices (namely, irreducible lattices in higher rank

algebraic groups); but also like rank one lattices. The following

theorem, the Farb--Kaimanovich--Masur superrigidity, states a rank one

phenomenon for MCG's: "every group homomorphism from higher rank

lattices (such as SL(3,Z) and cocompact lattices in SL(3,R)) into

MCG's has finite image."

In this talk, we show a generalization of the superrigidity above, to

the case where higher rank lattices are replaced with some

(non-arithmetic) matrix groups over general rings. Our main example of

such groups is called the "universal lattice", that is, the special

linear group over commutative finitely generated polynomial rings over

integers, (such as SL(3,Z[x])). To prove this, we introduce the notion

of "property (TT)/T" for groups, which is a strengthening of Kazhdan's

property (T).

We will explain these properties and relations to ordinary and bounded

cohomology of groups (with twisted unitary coefficients); and outline

the proof of our result.

Mapping class groups (MCG's), of compact oriented surfaces (possibly

with punctures), have many mysterious features: they behave not only

like higher rank lattices (namely, irreducible lattices in higher rank

algebraic groups); but also like rank one lattices. The following

theorem, the Farb--Kaimanovich--Masur superrigidity, states a rank one

phenomenon for MCG's: "every group homomorphism from higher rank

lattices (such as SL(3,Z) and cocompact lattices in SL(3,R)) into

MCG's has finite image."

In this talk, we show a generalization of the superrigidity above, to

the case where higher rank lattices are replaced with some

(non-arithmetic) matrix groups over general rings. Our main example of

such groups is called the "universal lattice", that is, the special

linear group over commutative finitely generated polynomial rings over

integers, (such as SL(3,Z[x])). To prove this, we introduce the notion

of "property (TT)/T" for groups, which is a strengthening of Kazhdan's

property (T).

We will explain these properties and relations to ordinary and bounded

cohomology of groups (with twisted unitary coefficients); and outline

the proof of our result.

### 2012/02/20

#### Functional Analysis Seminar

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

Microscopic derivation of the Ginzburg-Landau model (ENGLISH)

**Jan Philip SOLOVEJ**(University of Copenhagen)Microscopic derivation of the Ginzburg-Landau model (ENGLISH)

[ Abstract ]

I will discuss how the \\emph{Ginzburg-Landau} (GL) model of superconductivity arises as an asymptotic limit of the microscopic Bardeen-Cooper-Schrieffer (BCS) model. The asymptotic limit may be seen as a semiclassical limit and one of the main difficulties is to derive a semiclassical expansion with minimal regularity assumptions. It is not rigorously understood how the BCS model approximates the underlying many-body quantum system. I will formulate the BCS model as a variational problem, but only heuristically discuss its relation to quantum mechanics.

I will discuss how the \\emph{Ginzburg-Landau} (GL) model of superconductivity arises as an asymptotic limit of the microscopic Bardeen-Cooper-Schrieffer (BCS) model. The asymptotic limit may be seen as a semiclassical limit and one of the main difficulties is to derive a semiclassical expansion with minimal regularity assumptions. It is not rigorously understood how the BCS model approximates the underlying many-body quantum system. I will formulate the BCS model as a variational problem, but only heuristically discuss its relation to quantum mechanics.

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