## Seminar information archive

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

### 2012/11/19

#### Lectures

16:45-17:45 Room #126 (Graduate School of Math. Sci. Bldg.)

Invariant measure of the stochastic Allen-Cahn equation: the regime of small noise and large system size (ENGLISH)

**Hendrik Weber**(University of Warwick)Invariant measure of the stochastic Allen-Cahn equation: the regime of small noise and large system size (ENGLISH)

[ Abstract ]

We study the invariant measure of the one-dimensional stochastic Allen-Cahn equation for a small noise strength and a large but finite system. We endow the system with inhomogeneous Dirichlet boundary conditions that enforce at least one transition from -1 to 1. We are interested in the competition between the ``energy'' that should be minimized due to the small noise strength and the ``entropy'' that is induced by the large system size.

Our methods handle system sizes that are exponential with respect to the inverse noise strength, up to the ``critical'' exponential size predicted by the heuristics. We capture the competition between energy and entropy through upper and lower bounds on the probability of extra transitions between $\\pm 1$. These bounds are sharp on the exponential scale and imply in particular that the probability of having one and only one transition from -1 to +1 is exponentially close to one. In addition, we show that the position of the transition layer is uniformly distributed over the system on scales larger than the logarithm of the inverse noise strength.

Our arguments rely on local large deviation bounds, the strong Markov property, the symmetry of the potential, and measure-preserving reflections.

This is a joint work with Felix Otto and Maria Westdickenberg.

We study the invariant measure of the one-dimensional stochastic Allen-Cahn equation for a small noise strength and a large but finite system. We endow the system with inhomogeneous Dirichlet boundary conditions that enforce at least one transition from -1 to 1. We are interested in the competition between the ``energy'' that should be minimized due to the small noise strength and the ``entropy'' that is induced by the large system size.

Our methods handle system sizes that are exponential with respect to the inverse noise strength, up to the ``critical'' exponential size predicted by the heuristics. We capture the competition between energy and entropy through upper and lower bounds on the probability of extra transitions between $\\pm 1$. These bounds are sharp on the exponential scale and imply in particular that the probability of having one and only one transition from -1 to +1 is exponentially close to one. In addition, we show that the position of the transition layer is uniformly distributed over the system on scales larger than the logarithm of the inverse noise strength.

Our arguments rely on local large deviation bounds, the strong Markov property, the symmetry of the potential, and measure-preserving reflections.

This is a joint work with Felix Otto and Maria Westdickenberg.

#### Algebraic Geometry Seminar

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

Stability conditions and birational geometry (JAPANESE)

**Yukinobu Toda**(IPMU)Stability conditions and birational geometry (JAPANESE)

[ Abstract ]

I propose a conjecture which claims that MMP for a smooth projective variety is realized as a variation of Bridgeland moduli spaces of semistable objects in the derived category of coherent sheaves. I will discuss the surface case and extremal contractions for 3-folds. In the former case, the conjecture is completely solved. In the latter case, I will construct the perverse t-structure associated to the extremal contraction, and construct a candidate of the desired stability condition as a double tilting of the perverse heart.

I propose a conjecture which claims that MMP for a smooth projective variety is realized as a variation of Bridgeland moduli spaces of semistable objects in the derived category of coherent sheaves. I will discuss the surface case and extremal contractions for 3-folds. In the former case, the conjecture is completely solved. In the latter case, I will construct the perverse t-structure associated to the extremal contraction, and construct a candidate of the desired stability condition as a double tilting of the perverse heart.

#### GCOE Seminars

15:30-17:00 Room #056 (Graduate School of Math. Sci. Bldg.)

Linearisable mappings (ENGLISH)

**Alfred Ramani**(Ecole Polytechnique)Linearisable mappings (ENGLISH)

[ Abstract ]

We present a series of results on linearisable second-order mappings.

Three distinct families of such mappings do exist: projective, mappings of Gambier type and mappings which we have dubbed "of third kind".

Our starting point are the linearisable mappings belonging to the QRT family. We show how they can be linearised and how in some cases their explicit solution can be constructed. We discuss also the growth property of these mappings, a property intimately related to linearisability.

In the second part of the talk we address the question of the extension of these mappings to a non-autonomous form.

We show that the QRT invariant can also be extended (to a quantity which depends explicitly on the independent variable). Using this non-autonomous form we show that it is possible to construct the explicit solution of all third-kind mappings. We discuss also the relation of mappings of the third kind to Gambier-type mappings. We show that a large subclass of third-kind mappings can be considered as the discrete derivative of Gambier-type ones.

We present a series of results on linearisable second-order mappings.

Three distinct families of such mappings do exist: projective, mappings of Gambier type and mappings which we have dubbed "of third kind".

Our starting point are the linearisable mappings belonging to the QRT family. We show how they can be linearised and how in some cases their explicit solution can be constructed. We discuss also the growth property of these mappings, a property intimately related to linearisability.

In the second part of the talk we address the question of the extension of these mappings to a non-autonomous form.

We show that the QRT invariant can also be extended (to a quantity which depends explicitly on the independent variable). Using this non-autonomous form we show that it is possible to construct the explicit solution of all third-kind mappings. We discuss also the relation of mappings of the third kind to Gambier-type mappings. We show that a large subclass of third-kind mappings can be considered as the discrete derivative of Gambier-type ones.

#### Seminar on Geometric Complex Analysis

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

On a degenerate family of Riemann surfaces of genus two over an elliptic curve (JAPANESE)

**Yohei Komori**(Waseda University)On a degenerate family of Riemann surfaces of genus two over an elliptic curve (JAPANESE)

[ Abstract ]

We construct a degenerate family of Riemann surfaces of genus two constructed as double branched covering surfaces of a fixed torus. We determine its singular fibers and holomorphic sections.

We construct a degenerate family of Riemann surfaces of genus two constructed as double branched covering surfaces of a fixed torus. We determine its singular fibers and holomorphic sections.

### 2012/11/16

#### Colloquium

16:30-17:30 Room #002 (Graduate School of Math. Sci. Bldg.)

Integral invariants in complex differential geometry (JAPANESE)

**Akito FUTAKI**(University of Tokyo)Integral invariants in complex differential geometry (JAPANESE)

#### GCOE Seminars

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

Subfactors with small Jones index (ENGLISH)

**Dietmar Bisch**(Vanderbilt University)Subfactors with small Jones index (ENGLISH)

### 2012/11/14

#### Number Theory Seminar

18:00-19:00 Room #002 (Graduate School of Math. Sci. Bldg.)

De Rham-Witt complexes with coefficients and rigid cohomology

(ENGLISH)

**Pierre Berthelot**(Université de Rennes 1)De Rham-Witt complexes with coefficients and rigid cohomology

(ENGLISH)

[ Abstract ]

For a smooth scheme over a perfect field of characteristic p, we will explain a generalization of the classical comparison theorem between crystalline cohomology and de Rham-Witt cohomology to the case of cohomologies with coefficients in a p-torsion free crystal. This provides in particular a description of the rigid cohomology of a proper singular scheme in terms of a de Rham-Witt complex built from a closed immersion into a smooth scheme.

For a smooth scheme over a perfect field of characteristic p, we will explain a generalization of the classical comparison theorem between crystalline cohomology and de Rham-Witt cohomology to the case of cohomologies with coefficients in a p-torsion free crystal. This provides in particular a description of the rigid cohomology of a proper singular scheme in terms of a de Rham-Witt complex built from a closed immersion into a smooth scheme.

#### Geometry Colloquium

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

Construction of coassociative submanifolds (JAPANESE)

**Kotaro Kawai**(Tohoku University)Construction of coassociative submanifolds (JAPANESE)

[ Abstract ]

The notion of coassociative submanifolds is defined as the special class of the minimal submanifolds in G_2 manifolds. In this talk, we introduce the method to construct coassociative submanifolds by using the symmetry of the Lie group action. As an application, we give explicit examples in the 7-dimensional Euclidean space and in the anti-self-dual bundle over the 4-sphere.

The notion of coassociative submanifolds is defined as the special class of the minimal submanifolds in G_2 manifolds. In this talk, we introduce the method to construct coassociative submanifolds by using the symmetry of the Lie group action. As an application, we give explicit examples in the 7-dimensional Euclidean space and in the anti-self-dual bundle over the 4-sphere.

### 2012/11/13

#### Tuesday Seminar on Topology

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

The virtual fibering theorem and sutured manifold hierarchies (JAPANESE)

**Takahiro Kitayama**(RIMS, Kyoto University,JSPS PD)The virtual fibering theorem and sutured manifold hierarchies (JAPANESE)

[ Abstract ]

In 2007 Agol showed that every irreducible 3-manifold whose fundamental

group is nontrivial and virtually residually finite rationally solvable

(RFRS) is virtually fibered. In the proof he used the theory of

least-weight taut normal surfaces introduced and developed by Oertel and

Tollefson-Wang. We give another proof using complexities of sutured

manifolds. This is a joint work with Stefan Friedl (University of

Cologne).

In 2007 Agol showed that every irreducible 3-manifold whose fundamental

group is nontrivial and virtually residually finite rationally solvable

(RFRS) is virtually fibered. In the proof he used the theory of

least-weight taut normal surfaces introduced and developed by Oertel and

Tollefson-Wang. We give another proof using complexities of sutured

manifolds. This is a joint work with Stefan Friedl (University of

Cologne).

#### Lie Groups and Representation Theory

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

Tight maps, a classification (ENGLISH)

**Oskar Hamlet**(Chalmers University)Tight maps, a classification (ENGLISH)

[ Abstract ]

Tight maps/homomorphisms were introduced during the study of rigidity properties of surface groups in Hermitian Lie groups. In this talk I'll discuss the properties of tight maps, their connection to rigidity theory and my work classifying them.

Tight maps/homomorphisms were introduced during the study of rigidity properties of surface groups in Hermitian Lie groups. In this talk I'll discuss the properties of tight maps, their connection to rigidity theory and my work classifying them.

### 2012/11/12

#### Algebraic Geometry Seminar

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

On Fano fourfolds with nef vector bundles $Λ^2T_X$ (JAPANESE)

**Kazunori Yasutake**(Kyushu University)On Fano fourfolds with nef vector bundles $Λ^2T_X$ (JAPANESE)

[ Abstract ]

By using results about extremal contractions on smooth fourfolds, we give a classification of fano fourfolds whose the second exterior power of tangent bundles are numerically effective.

By using results about extremal contractions on smooth fourfolds, we give a classification of fano fourfolds whose the second exterior power of tangent bundles are numerically effective.

#### Seminar on Geometric Complex Analysis

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

Residues of meromorphic differential forms (ENGLISH)

**A.G. Aleksandrov**(Institute of Control Sciences, Russian Acad. of Sci.)Residues of meromorphic differential forms (ENGLISH)

[ Abstract ]

The purpose of the talk is to discuss several interesting aspects

of the classical residue theory originated by H. Poincar\\'e, J. de Rham and J. Leray and their followers. Focus topics of our studies are some of the less known applications, developed by the author in the past few years in complex analysis, topology and geometry of singular varieties and in the theory of differential equations. Almost all considerations are based essentially on properties of a special class of meromorphic differential forms called logarithmic or multi-logarithmic forms.

The purpose of the talk is to discuss several interesting aspects

of the classical residue theory originated by H. Poincar\\'e, J. de Rham and J. Leray and their followers. Focus topics of our studies are some of the less known applications, developed by the author in the past few years in complex analysis, topology and geometry of singular varieties and in the theory of differential equations. Almost all considerations are based essentially on properties of a special class of meromorphic differential forms called logarithmic or multi-logarithmic forms.

### 2012/11/10

#### Harmonic Analysis Komaba Seminar

13:00-18:00 Room #128 (Graduate School of Math. Sci. Bldg.)

Perturbed Besov spaces by short-range type potential

in exterior domains (JAPANESE)

Optimal constants and extremisers for some smoothing estimates (JAPANESE)

Spectral stability of the p-Laplacian (JAPANESE)

**Tokio Matsuyama**(Chuo University) 13:00-14:20Perturbed Besov spaces by short-range type potential

in exterior domains (JAPANESE)

[ Abstract ]

In this talk we will define perturbed Besov spaces by a short-range potential over exterior domains. These spaces will be available for obtaining the Strichartz estimates of wave equation with a potential in exterior domains.

We will pay attention to observe the equivalence relation between the perturbed Besov spaces and the free ones.

In this talk we will define perturbed Besov spaces by a short-range potential over exterior domains. These spaces will be available for obtaining the Strichartz estimates of wave equation with a potential in exterior domains.

We will pay attention to observe the equivalence relation between the perturbed Besov spaces and the free ones.

**Sugimoto Mitsuru**(Nagoya University) 14:40-16:00Optimal constants and extremisers for some smoothing estimates (JAPANESE)

[ Abstract ]

Our purpose is to study the optimal constant and extremising initial data for a broad class of smoothing estimates for solutions of linear dispersive equations.

Firstly, we discuss the existence/nonexistence of extremisers and then we provide an explicit formula and new observations for the optimal constant.

The talk is based on joint work with Neal Bez (University of Birmingham).

Our purpose is to study the optimal constant and extremising initial data for a broad class of smoothing estimates for solutions of linear dispersive equations.

Firstly, we discuss the existence/nonexistence of extremisers and then we provide an explicit formula and new observations for the optimal constant.

The talk is based on joint work with Neal Bez (University of Birmingham).

**Victor I. Burenkov**(Russia/United Kingdom) 16:30-17:50Spectral stability of the p-Laplacian (JAPANESE)

[ Abstract ]

Dependence of the eigenvalues of the p-Laplacian upon domain perturbation will be under discussion. Namely Lipschitz-type estimates for deviation of the eigenvalues following a domain perturbation will be presented. Such estimates are obtained for the class of open sets admitting open sets with arbitrarily strong degeneration and are expressed in terms of suitable measures of vicinity of two open sets, such as the \\lq\\lq atlas distance" between these sets or the \\lq\\lq lower Hausdor-Pompeiu

deviation" of their boundaries. In the case of open sets with Holder continuous boundaries, our results essentially improve a result known for the rst eigenvalue [2].

Joint work with P. D. Lamberti. The results were recently published in [1].

Supported by the grant of RFBR (project 08-01-00443).

References:

[1] V.I. Burenkov, P.D. Lamberti, Spectral stability of the p-Laplacian, Nonlinear Analysis, 71, 2009, 2227-2235.

[2] J. Fleckinger, E.M. Harrell and F. de Thelin, Boundary behaviour and estimates for solutions for equations containing the p-Laplacian, Electronic Journal of Dierential Equations, 38, 1999, 1-19.

Dependence of the eigenvalues of the p-Laplacian upon domain perturbation will be under discussion. Namely Lipschitz-type estimates for deviation of the eigenvalues following a domain perturbation will be presented. Such estimates are obtained for the class of open sets admitting open sets with arbitrarily strong degeneration and are expressed in terms of suitable measures of vicinity of two open sets, such as the \\lq\\lq atlas distance" between these sets or the \\lq\\lq lower Hausdor-Pompeiu

deviation" of their boundaries. In the case of open sets with Holder continuous boundaries, our results essentially improve a result known for the rst eigenvalue [2].

Joint work with P. D. Lamberti. The results were recently published in [1].

Supported by the grant of RFBR (project 08-01-00443).

References:

[1] V.I. Burenkov, P.D. Lamberti, Spectral stability of the p-Laplacian, Nonlinear Analysis, 71, 2009, 2227-2235.

[2] J. Fleckinger, E.M. Harrell and F. de Thelin, Boundary behaviour and estimates for solutions for equations containing the p-Laplacian, Electronic Journal of Dierential Equations, 38, 1999, 1-19.

### 2012/11/09

#### Seminar on Probability and Statistics

14:50-16:00 Room #006 (Graduate School of Math. Sci. Bldg.)

Tuning parameter selection in sparse regression modeling (JAPANESE)

http://www.sigmath.es.osaka-u.ac.jp/~kamatani/statseminar/2012/10.html

**HIROSE, Kei**(Graduate School of Engineering Science, Osaka University)Tuning parameter selection in sparse regression modeling (JAPANESE)

[ Abstract ]

In sparse regression modeling via regularization such as the lasso, it is important to select appropriate values of tuning parameters including regularization parameters. The choice of tuning parameters can be viewed as a model selection and evaluation problem. Mallows' Cp type criteria may be used as a tuning parameter selection tool in lasso type regularization methods, for which the concept of degrees of freedom plays a key role. In this talk, we propose an efficient algorithm that computes the degrees of freedom by extending the generalized path seeking algorithm. Our procedure allows us to construct model selection criteria for evaluating models estimated by regularization with a wide variety of convex and nonconvex penalties. The proposed methodology is investigated through the analysis of real data and Monte Carlo simulations. Numerical results show that Cp criterion based on our algorithm performs well in various situations.

[ Reference URL ]In sparse regression modeling via regularization such as the lasso, it is important to select appropriate values of tuning parameters including regularization parameters. The choice of tuning parameters can be viewed as a model selection and evaluation problem. Mallows' Cp type criteria may be used as a tuning parameter selection tool in lasso type regularization methods, for which the concept of degrees of freedom plays a key role. In this talk, we propose an efficient algorithm that computes the degrees of freedom by extending the generalized path seeking algorithm. Our procedure allows us to construct model selection criteria for evaluating models estimated by regularization with a wide variety of convex and nonconvex penalties. The proposed methodology is investigated through the analysis of real data and Monte Carlo simulations. Numerical results show that Cp criterion based on our algorithm performs well in various situations.

http://www.sigmath.es.osaka-u.ac.jp/~kamatani/statseminar/2012/10.html

### 2012/11/07

#### Classical Analysis

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

WKB analysis of the Painlev\\'e functions and parameteric Stokes phenomena (JAPANESE)

**Kohei IWAKI**(Kyoto University)WKB analysis of the Painlev\\'e functions and parameteric Stokes phenomena (JAPANESE)

### 2012/11/06

#### Tuesday Seminar on Topology

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

Galois action on knots (JAPANESE)

**Furusho Hidekazu**(Nagoya University)Galois action on knots (JAPANESE)

[ Abstract ]

I will explain a motivic structure on knots.

Then I will explain that the absolute Galois group of

the rational number field acts non-trivially

on 'the space of knots' in a non-trivial way.

I will explain a motivic structure on knots.

Then I will explain that the absolute Galois group of

the rational number field acts non-trivially

on 'the space of knots' in a non-trivial way.

#### Tuesday Seminar of Analysis

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

Resonance free domains for homoclinic orbits (ENGLISH)

**Thierry Ramond**(Univ. Paris, Orsay)Resonance free domains for homoclinic orbits (ENGLISH)

#### Lie Groups and Representation Theory

16:30-17:30 Room #126 (Graduate School of Math. Sci. Bldg.)

An explicit construction of spherical designs on S^3 (JAPANESE)

**Takayuki Okuda**(the University of Tokyo)An explicit construction of spherical designs on S^3 (JAPANESE)

[ Abstract ]

The existence of spherical t-designs on S^d for any t and d are proved by Seymour--Zaslavsky in 1984.

However, explicit constructions of spherical designs were not known for d > 2 and large t.

In this talk, for a given spherical t-design Y on S^2, we give an

algorithm to make a spherical 2t-design X on S^3 which maps Y by a Hopf map. In particular, by combining with the results of Kuperberg in 2005, we have an explicit construction of spherical t-designs on S^3 for any t.

The existence of spherical t-designs on S^d for any t and d are proved by Seymour--Zaslavsky in 1984.

However, explicit constructions of spherical designs were not known for d > 2 and large t.

In this talk, for a given spherical t-design Y on S^2, we give an

algorithm to make a spherical 2t-design X on S^3 which maps Y by a Hopf map. In particular, by combining with the results of Kuperberg in 2005, we have an explicit construction of spherical t-designs on S^3 for any t.

### 2012/11/05

#### Algebraic Geometry Seminar

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

The rationality of the moduli spaces of trigonal curves (JAPANESE)

**Shouhei Ma**(Nagoya University)The rationality of the moduli spaces of trigonal curves (JAPANESE)

### 2012/10/31

#### Geometry Colloquium

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

Hofer-Zehnder capacity and a Hamiltonian circle action with noncontractible orbits (JAPANESE)

**Kei Irie**(Kyoto University)Hofer-Zehnder capacity and a Hamiltonian circle action with noncontractible orbits (JAPANESE)

[ Abstract ]

Hofer-Zehnder (HZ) capacity is an invariant of symplectic manifolds, which is important in symplectic topology and Hamiltonian dynamics. The energy-capacity inequality (due to Hofer and many others) claims that HZ capacity of a domain is bounded from above by its dispalcement energy.

In this talk, we prove a variant of this inequality, which is applicable to nondisplaceable domains. We also give some applications, including case of disc cotangent bundles.

Hofer-Zehnder (HZ) capacity is an invariant of symplectic manifolds, which is important in symplectic topology and Hamiltonian dynamics. The energy-capacity inequality (due to Hofer and many others) claims that HZ capacity of a domain is bounded from above by its dispalcement energy.

In this talk, we prove a variant of this inequality, which is applicable to nondisplaceable domains. We also give some applications, including case of disc cotangent bundles.

### 2012/10/30

#### Numerical Analysis Seminar

16:30-18:00 Room #002 (Graduate School of Math. Sci. Bldg.)

Error analysis of Galerkin's method for semilinear partial differential equations (JAPANESE)

[ Reference URL ]

http://www.infsup.jp/utnas/

**Tadashi Kawanago**(Tokyo Institute of Technology)Error analysis of Galerkin's method for semilinear partial differential equations (JAPANESE)

[ Reference URL ]

http://www.infsup.jp/utnas/

#### Tuesday Seminar on Topology

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

Applications of knot theory to molecular biology (JAPANESE)

**Koya Shimokawa**(Saitama University)Applications of knot theory to molecular biology (JAPANESE)

[ Abstract ]

In this talk we discuss applications of knot theory to studies of DNA

and proteins.

Especially we will consider (1)topological characterization of

mechanisms of site-specific recombination systems,

(2)modeling knotted DNA and proteins in confined regions using lattice

knots, and

(3)mechanism of topoisomerases and signed crossing changes.

In this talk we discuss applications of knot theory to studies of DNA

and proteins.

Especially we will consider (1)topological characterization of

mechanisms of site-specific recombination systems,

(2)modeling knotted DNA and proteins in confined regions using lattice

knots, and

(3)mechanism of topoisomerases and signed crossing changes.

#### Tuesday Seminar of Analysis

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

About the method of characteristics (ENGLISH)

**Francis Nier**(Univ. Rennes 1)About the method of characteristics (ENGLISH)

[ Abstract ]

While studying the mean field dynamics of a systems of bosons, one is led to solve a transport equation for a probability measure in an infinite dimensional phase-space. Since those probability measures are characterized after testing with cylindrical or polynomial observables, which make classes which are not invariant after composing with a nonlinear flow. Thus the standard method of characteristics for transport equations cannot be extended at once to the infinite dimensional case. A solution comes from techniques developed for optimal transport and a probabilistic interpretation of trajectories.

While studying the mean field dynamics of a systems of bosons, one is led to solve a transport equation for a probability measure in an infinite dimensional phase-space. Since those probability measures are characterized after testing with cylindrical or polynomial observables, which make classes which are not invariant after composing with a nonlinear flow. Thus the standard method of characteristics for transport equations cannot be extended at once to the infinite dimensional case. A solution comes from techniques developed for optimal transport and a probabilistic interpretation of trajectories.

#### Lectures

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

Discrete Topologgy of Cellular Microstructures

and Complicatedness Measurements for Cell Complexes (JAPANESE)

**Frank Lutz**(Technische Universität Berlin)Discrete Topologgy of Cellular Microstructures

and Complicatedness Measurements for Cell Complexes (JAPANESE)

[ Abstract ]

Our first aim is to use methods from discrete and geometric topology

to recover structural information from the composition of

monocrystalline materials that have a periodic foam structure

(such as gas hydrates and transition metal alloys) and also of

polycrystalline materials (such as metals and certain ceramics).

For more general complexes, even with a billion of faces, homological

information can be obtained with computational homology packages

such as CHomP or RedHom. These packages extensively use discrete Morse

theory as a preprocessing step. Although it is NP-hard to find optimal

discrete Morse functions, most data appears to be easy and it is

in fact hard to construct ``complicated'' examples. As we will see,

random discrete Morse theory will allow us to measure the

``complicatedness'' of complexes.

Our first aim is to use methods from discrete and geometric topology

to recover structural information from the composition of

monocrystalline materials that have a periodic foam structure

(such as gas hydrates and transition metal alloys) and also of

polycrystalline materials (such as metals and certain ceramics).

For more general complexes, even with a billion of faces, homological

information can be obtained with computational homology packages

such as CHomP or RedHom. These packages extensively use discrete Morse

theory as a preprocessing step. Although it is NP-hard to find optimal

discrete Morse functions, most data appears to be easy and it is

in fact hard to construct ``complicated'' examples. As we will see,

random discrete Morse theory will allow us to measure the

``complicatedness'' of complexes.

### 2012/10/29

#### Algebraic Geometry Seminar

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

The Mukai conjecture for log Fano manifolds (JAPANESE)

**Kento Fujita**(RIMS)The Mukai conjecture for log Fano manifolds (JAPANESE)

[ Abstract ]

The concept of log Fano manifolds is one of the most natural generalization of the concept of Fano manifolds. We will give some structure theorems of log Fano manifolds. For example, we will show that the Mukai conjecture for Fano manifolds implies the `log Mukai conjecture' for log Fano manifolds.

The concept of log Fano manifolds is one of the most natural generalization of the concept of Fano manifolds. We will give some structure theorems of log Fano manifolds. For example, we will show that the Mukai conjecture for Fano manifolds implies the `log Mukai conjecture' for log Fano manifolds.

< Previous 123456789101112131415161718192021222324252627282930313233343536373839404142434445464748495051525354555657585960616263646566676869707172737475767778798081828384858687888990919293949596979899100101102103104105106107108109110111112113114115116117118119120121122123124125126127128129130131132133134135136137138139140141142143144145146147148149150151152153154155156157158159160161162163164165166167168169170171172173174175176 Next >