## FMSP Lectures

Seminar information archive ～09/27｜Next seminar｜Future seminars 09/28～

**Seminar information archive**

### 2013/05/29

13:00-14:30 Room #122 (Graduate School of Math. Sci. Bldg.)

Low-dimensional linear representations of mapping class groups (II) (ENGLISH)

[ Reference URL ]

http://faculty.ms.u-tokyo.ac.jp/~topology/Korkmaz.pdf

**Mustafa Korkmaz**(Middle East Technical University)Low-dimensional linear representations of mapping class groups (II) (ENGLISH)

[ Reference URL ]

http://faculty.ms.u-tokyo.ac.jp/~topology/Korkmaz.pdf

### 2013/05/28

17:10-18:40 Room #117 (Graduate School of Math. Sci. Bldg.)

Low-dimensional linear representations of mapping class groups (I) (ENGLISH)

[ Reference URL ]

http://faculty.ms.u-tokyo.ac.jp/~topology/Korkmaz.pdf

**Mustafa Korkmaz**(Middle East Technical University)Low-dimensional linear representations of mapping class groups (I) (ENGLISH)

[ Reference URL ]

http://faculty.ms.u-tokyo.ac.jp/~topology/Korkmaz.pdf

### 2013/03/08

10:30-12:00 Room #056 (Graduate School of Math. Sci. Bldg.)

Knots, algorithms and linear programming: the quest to solve unknot recognition in polynomial time (ENGLISH)

**Benjamin Burton**(The University of Queensland, Australia)Knots, algorithms and linear programming: the quest to solve unknot recognition in polynomial time (ENGLISH)

[ Abstract ]

In this talk we explore new approaches to the old and difficult computational problem of unknot recognition. Although the best known algorithms for this problem run in exponential time, there is increasing evidence that a polynomial time solution might be possible. We outline several promising approaches, in which computational geometry, linear programming and greedy algorithms all play starring roles. We finish with a new algorithm that combines techniques from topology and combinatorial optimisation, which is the first to exhibit "real world" polynomial time behaviour: although it is still exponential time in theory, exhaustive experimentation shows that this algorithm can solve unknot recognition for "practical" inputs by running just a linear number of linear programs.

This is joint work with Melih Ozlen.

In this talk we explore new approaches to the old and difficult computational problem of unknot recognition. Although the best known algorithms for this problem run in exponential time, there is increasing evidence that a polynomial time solution might be possible. We outline several promising approaches, in which computational geometry, linear programming and greedy algorithms all play starring roles. We finish with a new algorithm that combines techniques from topology and combinatorial optimisation, which is the first to exhibit "real world" polynomial time behaviour: although it is still exponential time in theory, exhaustive experimentation shows that this algorithm can solve unknot recognition for "practical" inputs by running just a linear number of linear programs.

This is joint work with Melih Ozlen.

### 2013/02/22

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

II_1 factors with a unique Cartan subalgebra (ENGLISH)

**Stefaan Vaes**(KU Leuven)II_1 factors with a unique Cartan subalgebra (ENGLISH)

### 2013/02/21

15:00-16:00 Room #056 (Graduate School of Math. Sci. Bldg.)

)

Monte Carlo Methods for Partial Differential Equations: Computing Permeability

(ENGLISH)

**Michael Mascagni**(Florida State University)

Monte Carlo Methods for Partial Differential Equations: Computing Permeability

(ENGLISH)

[ Abstract ]

We present a brief overview of Monte Carlo methods for the solution of elliptic and parabolic partial differential equations (PDEs). We begin with a review of the Feynman-Kac formula, and its use in the probabilistic representation of the solutions of elliptic and parabolic PDEs. We then consider some specific Monte Carlo methods used for obtaining the solution of simple elliptic partial differential equations (PDEs) as part of exterior boundary value problems that arise in electrostatics and flow through porous media. These Monte Carlo methods use Feynman-Kac to represent the solution of the elliptic PDE at a point as the expected value of functionals of Brownian motion trajectories started at the point of interest. We discuss the rapid solution of these equations, in complex exterior geometries, using both the "walk on spheres" and "Greens function first-passage" algorithms. We then concentrate on methods for quickly computing the isotropic permeability using the "unit

capacitance" and "penetration depth'' methods. The first of these methods, requires computing a linear functional of the solution to an exterior elliptic PDE. Both these methods for computing permeability are simple, and provide accurate solutions in a few seconds on laptop-scale computers. We then conclude with a brief look at other Monte Carlo methods and problems that arise on related application areas.

We present a brief overview of Monte Carlo methods for the solution of elliptic and parabolic partial differential equations (PDEs). We begin with a review of the Feynman-Kac formula, and its use in the probabilistic representation of the solutions of elliptic and parabolic PDEs. We then consider some specific Monte Carlo methods used for obtaining the solution of simple elliptic partial differential equations (PDEs) as part of exterior boundary value problems that arise in electrostatics and flow through porous media. These Monte Carlo methods use Feynman-Kac to represent the solution of the elliptic PDE at a point as the expected value of functionals of Brownian motion trajectories started at the point of interest. We discuss the rapid solution of these equations, in complex exterior geometries, using both the "walk on spheres" and "Greens function first-passage" algorithms. We then concentrate on methods for quickly computing the isotropic permeability using the "unit

capacitance" and "penetration depth'' methods. The first of these methods, requires computing a linear functional of the solution to an exterior elliptic PDE. Both these methods for computing permeability are simple, and provide accurate solutions in a few seconds on laptop-scale computers. We then conclude with a brief look at other Monte Carlo methods and problems that arise on related application areas.

### 2012/12/18

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

On convergence to equilibrium with applications of Lojasiewicz-Simon

inequality (II) (ENGLISH)

**Jie Jiang**(Wuhan Institute of Physics and Mathematics, Chinese Academy of Sciences)On convergence to equilibrium with applications of Lojasiewicz-Simon

inequality (II) (ENGLISH)

### 2012/12/12

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

Large subalgebras of crossed product C*-algebras (ENGLISH)

**N. Christopher Phillips**(Univ. Oregon)Large subalgebras of crossed product C*-algebras (ENGLISH)

[ Abstract ]

This is work in progress; not everything has been checked.

We define a "large subalgebra" and a "centrally large subalgebra" of a C*-algebra. The motivating example is what we now call the "orbit breaking subalgebra" of the crossed product by a minimal homeomorphism h of a compact metric space X. Let v be the standard unitary in the crossed product C* (Z, X, h). For a closed subset Y of X, we form the subalgebra of C* (Z, X, h) generated by C (X) and all elements f v for f in C (X) such that f vanishes on Y. When each orbit meets Y at most once, this subalgebra is centrally large in the crossed product. Crossed products by smooth free minimal actions of Zd also contain centrally large subalgebras which are simple direct limits, with no dimension growth, of recursive subhomogeneous algebras.

If B is a large subalgebra of A, then the Cuntz semigroups of A and B are the almost the same: if one deletes the classes of nonzero projections, then the inclusion is a bijection on what is left. Also (joint work with Dawn Archey), if B is a centrally large subalgebra of A, and B has stable rank one, then so does A. Moreover, if B is a centrally large subalgebra of A, if B is Z-stable, and if A is nuclear, then A is Z-stable.

This is work in progress; not everything has been checked.

We define a "large subalgebra" and a "centrally large subalgebra" of a C*-algebra. The motivating example is what we now call the "orbit breaking subalgebra" of the crossed product by a minimal homeomorphism h of a compact metric space X. Let v be the standard unitary in the crossed product C* (Z, X, h). For a closed subset Y of X, we form the subalgebra of C* (Z, X, h) generated by C (X) and all elements f v for f in C (X) such that f vanishes on Y. When each orbit meets Y at most once, this subalgebra is centrally large in the crossed product. Crossed products by smooth free minimal actions of Zd also contain centrally large subalgebras which are simple direct limits, with no dimension growth, of recursive subhomogeneous algebras.

If B is a large subalgebra of A, then the Cuntz semigroups of A and B are the almost the same: if one deletes the classes of nonzero projections, then the inclusion is a bijection on what is left. Also (joint work with Dawn Archey), if B is a centrally large subalgebra of A, and B has stable rank one, then so does A. Moreover, if B is a centrally large subalgebra of A, if B is Z-stable, and if A is nuclear, then A is Z-stable.

### 2012/12/11

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

On convergence to equilibrium with applications of Lojasiewicz-Simon

inequality (I) (ENGLISH)

**Jie Jiang**(Wuhan Institute of Physics and Mathematics, Chinese Academy of Sciences)On convergence to equilibrium with applications of Lojasiewicz-Simon

inequality (I) (ENGLISH)