## Seminar information archive

### 2007/01/16

#### Tuesday Seminar on Topology

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

An $SO(3)$-version of $2$-torsion instanton invariants
[ Abstract ]
We construct invariants for simply connected, non-spin $4$-manifolds using torsion cohomology classes of moduli spaces of ASD connections on $SO(3)$-bundles. The invariants are $SO(3)$-version of Fintushel-Stern's $2$-torsion instanton invariants. We show that this $SO(3)$-torsion invariant of $2CP^2 \\# -CP^2$ is non-trivial, while it is known that any invariants of $2CP^2 \\# - CP^2$ coming from the Seiberg-Witten theory are trivial
since $2CP^2 \\# -CP^2$ has a positive scalar curvature metric.

On the non-acyclic Reidemeister torsion for knots
[ Abstract ]
The Reidemeister torsion is an invariant of a CW-complex and a representation of its fundamental group. We consider the Reidemeister torsion for a knot exterior in a homology three sphere and a representation given by the composition of an SL(2, C)- (or SU(2)-) representation of the knot group and the adjoint action to the Lie algebra.
We will see that this invariant is expressed by the differential coefficient of the twisted Alexander invariant of the knot and investigate some properties of the invariant by using this relation.

#### Lectures

16:30-18:00   Room #122 (Graduate School of Math. Sci. Bldg.)
Mourad Bellassoued (Faculte des Sciences de Bizerte)
Recovering a potential from partial Cauchy data for the Schrödinger equation.

### 2007/01/15

#### Seminar on Geometric Complex Analysis

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

Lagragian constructions for various topological invariants of algebraic varieties (東大数理、松井優氏との共同研究)

#### Lectures

16:30-18:00   Room #122 (Graduate School of Math. Sci. Bldg.)
Mourad Bellassoued (Faculte des Sciences de Bizerte)
Recovering a potential from full Cauchy data for the Schrödinger equation.
[ Abstract ]
In this lectures we survey recent progress on the problem of determining a potential by measuring the Dirichlet to Neumann map
for the associated Schr\\"odinger equation or wave equation. We make emphasis on the new results obtained with M.Yamamoto which is concerned with the case that the measurements are made on a strict
subset of the boundary for the wave equation.

#### Lectures

16:00-17:30   Room #056 (Graduate School of Math. Sci. Bldg.)
Antonio DeSimone (SISSA (International School for Advanced Studies))
Analysis of physical systems involving multiple spatial scales: some case studies
[ Abstract ]
Variational methods have recently proved to be a powerful tool in deriving macroscopic models for phenomena whose physics is decided at the sub-miccron scale.
We will use two case studies to illustrate this point, namely, that of liquid crystal elastomers and that of superhydrophobic surfaces.

Liquid crystal elastomers are solids which combine the optical properties of liquid crystals with the mechanical properties of rubbery solids. They display phase transformations, material instabilities, and microstructures in a way simalr to shape-memory alloys.

The richness of the underlying material symmetries makes the mathematical analysis of this system particularly rewarding. Recent progress, ranging from analytical relaxation results to numerical simulations of the macroscopic mechanical response will be reviewed.

### 2007/01/12

#### Colloquium

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

1、プレート境界で砂と泥に起こる雪だるま現象
2、プレート境界地震は確率共鳴か
[ Abstract ]

### 2007/01/11

#### Lectures

16:00-17:30   Room #123 (Graduate School of Math. Sci. Bldg.)
Oleg Yu. Emanouilov (Colorado State University)
Some Problems of Global Controllability of Burgers Equation and Navier-Stokes system.

### 2007/01/10

#### Lectures

16:00-17:30   Room #118 (Graduate School of Math. Sci. Bldg.)
Oleg Yu. Emanouilov (Colorado State University)
Some Problems of Global Controllability of Burgers Equation and Navier-Stokes system.

### 2007/01/09

#### Lectures

16:00-17:30   Room #118 (Graduate School of Math. Sci. Bldg.)
Oleg Yu. Emanouilov (Colorado State University)
Some Problems of Global Controllability of Burgers Equation and Navier-Stokes system.
[ Abstract ]
We show that 1-D Burgers equation is globally uncontrollable with control acting at two endpoints. Then we establish the global controllability of the 2-D Burgers equation. Finally we show that for 2-D Navier-Stokes system the problem of global exact controllability is solvable for the dense set of the initial data with a control acting on part of the boundary.

### 2006/12/28

#### Operator Algebra Seminars

16:30-18:00   Room #126 (Graduate School of Math. Sci. Bldg.)
Roberto Longo (University of Rome)
Operator Algebras and Conformal Field Theory II

### 2006/12/25

#### Monthly Seminar on Arithmetic of Automorphic Forms

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

なし
[ Abstract ]

12月25日午後から27日午後3時くらいまでです。詳細はURL:
http://www.ms.u-tokyo.ac.jp/activity/meeting061225.htm
をご覧下さい。織田孝幸

### 2006/12/21

#### Seminar for Mathematical Past of Asia

17:00-18:30   Room #123 (Graduate School of Math. Sci. Bldg.)

インド数学における証明
[ Reference URL ]
http://www.ms.u-tokyo.ac.jp/~kawazumi/asia.html

#### Applied Analysis

16:00-17:30   Room #056 (Graduate School of Math. Sci. Bldg.)
Susan Friedlander (University of Illinois-Chicago)
An Inviscid Dyadic Model For Turbulence
[ Abstract ]
We discuss properties of a GOY type model for the inviscid fluid equations. We prove that the forced system has a unique equilibrium which a an exponential global attractor. Every solution blows up in H^5/6 in finite time . After this time, all solutions stay in H^s, s<5/6, and "turbulent" dissipation occurs. Onsager's conjecture is confirmed for the model system.

This is joint work with Alexey Cheskidov and Natasa Pavlovic.

#### Operator Algebra Seminars

14:45-18:00   Room #126 (Graduate School of Math. Sci. Bldg.)
Benoit Collins (Univ. Claude Bernard Lyon 1) 14:45-16:15
Convergence of unitary matrix integrals and free probability
Roberto Longo (University of Rome) 16:30-18:00
Operator Algebras and Conformal Field Theory

### 2006/12/20

#### Number Theory Seminar

16:30-18:45   Room #117 (Graduate School of Math. Sci. Bldg.)
On the profinite regular inverse Galois problem
[ Abstract ]
Given a field $k$ and a (pro)finite group $G$, consider the
following weak version of the regular inverse Galois problem:
(WRIGP/$G$/$k$) \\textit{there exists a smooth geometrically
irreducible curve $X_{G}/k$ and a Galois extension $E/k(X_{G})$
regular over $k$ with group $G$.} (the regular inverse Galois
problem (RIGP/$G$/$k$) corresponding to the case
$X_{G}=\\mathbb{P}^{1}_{k}$). A standard descent argument shows that
for a finite group $G$ the (WRIGP/$G$/$k$) can be deduced from the
(RIGP/$G$/$k((T))$). For
profinite groups $G$, the (WRIGP/$G$/$k((T))$) has been proved for
lots of fields (including the cyclotomic closure of characteristic $0$
fields) but the descent argument no longer works.\\\\
\\indent Let $p\\geq 2$ be a prime, then a profinite group
$G$ is said to be \\textit{$p$-obstructed} if it fits in a profinite group extension
$$1\\rightarrow K\\rightarrow G\\rightarrow G_{0}\\rightarrow 1$$
with $G_{0}$ a finite group and $K\\twoheadrightarrow \\mathbb{Z}_{p}$. Typical examples of such profinite groups $G$ are
universal $p$-Frattini covers of finite $p$-perfect groups or
pronilpotent projective groups.\\\\
\\indent I will show that the (WRIGP/$G$/$k$) - even under
its weaker formulation: (WWRIGP/$G$/$k$) \\textit{there exists a
smooth geometrically irreducible curve $X_{G}/k$ and a Galois
extension $E/k(X_{G}).\\overline{k}$ with group $G$ and field of
moduli $k$.} - fails for the whole class of $p$-obstructed profinite
groups $G$ and any field $k$ which is either a finitely generated
field of characteristic $0$ or a finite field of characteristic
$\\not= p$.\\\\
\\indent The proof uses a profinite generalization of the cohomological obstruction
for a G-cover to be defined over its field of moduli and an analysis of the constrainsts
imposed on a smooth geometrically irreducible curve $X$ by a degree $p^{n}$
cyclic G-cover $X_{n}\\rightarrow X$, constrainsts which are too rigid to allow the
existence of projective systems $(X_{n}\\rightarrow X_{G})_{n\\geq 0}$ of degree $p^{n}$ cyclic G-covers
defined over $k$. I will also discuss other implicsations of these constrainsts
for the (RIGP).
Eric Friedlander (Northwestern) 17:45-18:45
An elementary perspective on modular representation theory

### 2006/12/19

#### Tuesday Seminar on Topology

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

Poisson structures on the homology of the spaces of knots
[ Abstract ]
We study the homological properties of the space $K$ of (framed) long knots in $\\R^n$, $n>3$, in particular its Poisson algebra structures.
We had known two kinds of Poisson structures, both of which are based on the action of little disks operad. One definition is via the action on the space $K$. Another comes from the action of chains of little disks on the Hochschild complex of an operad, which appears as $E^1$-term of certain spectral sequence converging to $H_* (K)$. The main result is that these two Poisson structures are the same.
We compute the first non-trivial example of the Poisson bracket. We show that this gives a first example of the homology class of $K$ which does not directly correspond to any chord diagrams.

[ Abstract ]
Suppose $F$ is an embedded closed surface in $R^4$.
We call $F$ a pseudo-ribbon surface link
if its projection is an immersion of $F$ into $R^3$
whose self-intersection set $\\Gamma(F)$ consists of disjointly embedded circles.
H. Aiso classified pseudo-ribbon sphere-knots ($F$ is a sphere.)
when $\\Gamma(F)$ consists of less than 6 circles.
when $F$ is two spheres and $\\Gamma(F)$ consists of less than 7 circles.

### 2006/12/18

#### Seminar on Geometric Complex Analysis

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

Height functions and affine space regular automorphisms

### 2006/12/14

#### Applied Analysis

16:00-17:30   Room #056 (Graduate School of Math. Sci. Bldg.)

[ Abstract ]

を許容する存在定理の枠組みを提供する為に発展してきた。こうして

の方法論を発展させることによって試みる。また特異点周辺の面積密度の

#### Operator Algebra Seminars

16:30-18:00   Room #126 (Graduate School of Math. Sci. Bldg.)
Chongying Dong (UC Santa Cruz)
On uniqueness of the moonshine vertex operator algebra

#### Applied Analysis

16:00-17:30   Room #056 (Graduate School of Math. Sci. Bldg.)

(東北大学・大学院理学研究科)

[ Abstract ]

### 2006/12/13

#### Seminar on Mathematics for various disciplines

10:30-11:30   Room #056 (Graduate School of Math. Sci. Bldg.)
C. M. Elliott (University of Sussex)
Computational Methods for Geometric PDEs
[ Abstract ]
Computational approaches to evolutionary geometric partial differential equations such as anisotropic motion by mean curvature and surface diffusion are reviewed. We consider methods based on graph, parametric , level set and phase field descriptions of the surface. We also discuss the approximation of partial differential equations which hold on the evolving surfaces. Numerical results will be presented along with some approximation results.
[ Reference URL ]
http://coe.math.sci.hokudai.ac.jp/sympo/various/index.html

#### Mathematical Finance

17:30-19:00   Room #118 (Graduate School of Math. Sci. Bldg.)

### 2006/12/12

#### Tuesday Seminar on Topology

16:30-18:00   Room #056 (Graduate School of Math. Sci. Bldg.)
Maxim Kazarian (Steklov Math. Institute)
Thom polynomials for maps of curves with isolated singularities
(joint with S. Lando)
[ Abstract ]
Thom (residual) polynomials in characteristic classes are used in
the analysis of geometry of functional spaces. They serve as a
tool in description of classes Poincar\\'e dual to subvarieties of
functions of prescribed types. We give explicit universal
expressions for residual polynomials in spaces of functions on
complex curves having isolated singularities and
multisingularities, in terms of few characteristic classes. These
expressions lead to a partial explicit description of a
stratification of Hurwitz spaces.

### 2006/12/11

#### Seminar on Geometric Complex Analysis

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

Modified deficiencies of holomorphic curves and defect relation

### 2006/12/08

#### Lectures

10:30-12:00   Room #056 (Graduate School of Math. Sci. Bldg.)
Charles M. Elliott (University of Sussex)
Computational Methods for Surface Partial Differential Equations
[ Abstract ]
In these lectures we discuss the formulation, approximation and applications of partial differential equations on stationary and evolving surfaces. Partial differential equations on surfaces occur in many applications. For example, traditionally they arise naturally in fluid dynamics, materials science, pattern formation on biological organisms and more recently in the mathematics of images. We will derive the conservation law on evolving surfaces and formulate a number of equations.

We propose a surface finite element method (SFEM) for the numerical solution of parabolic partial differential equations on hypersurfaces $\\Gamma$ in $\\mathbb R^{n+1}$. The key idea is based on the approximation of $\\Gamma$ by a polyhedral surface $\\Gamma_h$ consisting of a union of simplices (triangles for $n=2$, intervals for $n=1$) with vertices on $\\Gamma$. A finite element space of functions is then defined by taking the continuous functions on $\\Gamma_h$ which are linear affine on each simplex of the polygonal surface. We use surface gradients to define weak forms of elliptic operators and naturally generate weak formulations of elliptic and parabolic equations on $\\Gamma$. Our finite element method is applied to weak forms of the equations. The computation of the mass and element stiffness matrices are simple and straightforward. We give an example of error bounds in the case of semi-discretization in space for a fourth order linear problem. We extend this approach to pdes on evolving surfaces. We define an Eulerian level set method for partial differential equations on surfaces. The key idea is based on formulating the partial differential equation on all level set surfaces of a prescribed function $\\Phi$ whose zero level set is $\\Gamma$. We use Eulerian surface gradients to define weak forms
of elliptic operators which naturally generate weak formulations
of Eulerian elliptic and parabolic equations. This results in a degenerate equation formulated in anisotropic Sobolev spaces based on the level set function $\\Phi$. The resulting equation is then solved in one space dimension higher but can be solved on a fixed finite element grid.

Numerical experiments are described for several linear and Nonlinear partial differential equations. In particular the power of the method is demonstrated by employing it to solve highly nonlinear second and fourth order problems such as surface Allen-Cahn and Cahn-Hilliard equations and surface level set equations for geodesic mean curvature flow. In particular we show how surface level set and phase field models can be used to compute the motion of curves on surfaces. This is joint work with G. Dziuk(Freiburg).