## Colloquium

Seminar information archive ～06/09｜Next seminar｜Future seminars 06/10～

Organizer(s) | ABE Noriyuki, IWAKI Kohei, KAWAZUMI Nariya (chair), KOIKE Yuta |
---|---|

URL | https://www.ms.u-tokyo.ac.jp/seminar/colloquium/index_e.html |

**Seminar information archive**

### 2014/09/19

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

William Thurston and foliation theory (ENGLISH)

**Etienne Ghys**(École normale supérieure de Lyon)William Thurston and foliation theory (ENGLISH)

[ Abstract ]

Between 1972 and 1976, William Thurston revolutionized foliation theory. Twenty years later, he described this period of his mathematical life in a remarkable paper « On proofs and progress in mathematics ». In this talk, I will begin by a general overview of some of Thurston's contribution to this theory. I will then describe some of the current development.

Between 1972 and 1976, William Thurston revolutionized foliation theory. Twenty years later, he described this period of his mathematical life in a remarkable paper « On proofs and progress in mathematics ». In this talk, I will begin by a general overview of some of Thurston's contribution to this theory. I will then describe some of the current development.

### 2014/07/25

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

Mathematical modelling of Tumor Immune System Interaction (JAPANESE)

**Yasuhiro Takeuchi**(Aoyama Gakuin University)Mathematical modelling of Tumor Immune System Interaction (JAPANESE)

[ Abstract ]

We study the dynamical behavior of a tumor-immune system (T-IS) interaction model with two discrete delays,

namely the immune activation delay for effector cells (ECs) and activation delay for Helper T cells (HTCs).

By analyzing the characteristic equations, we establish the stability of two equilibria (tumor-free equilibrium and immune-control equilibrium) and the existence of Hopf bifurcations when two delays are used as the bifurcation parameter.

Our results exhibit that both delays do not affect the stability of tumor-free equilibrium.

However, they are able to destabilize the immune-control equilibrium and cause periodic solutions.

We numerically illustrate how these two delays can change the stability region of the immune-control equilibrium and display the different impacts to the control of tumors.

The numerical simulation results show that the immune activation delay for HTCs can induce heteroclinic cycles to connect the tumor-free equilibrium and immune-control equilibrium.

Furthermore, we observe that the immune activation delay for HTCs can even stabilize the unstable immune-control equilibrium.

We study the dynamical behavior of a tumor-immune system (T-IS) interaction model with two discrete delays,

namely the immune activation delay for effector cells (ECs) and activation delay for Helper T cells (HTCs).

By analyzing the characteristic equations, we establish the stability of two equilibria (tumor-free equilibrium and immune-control equilibrium) and the existence of Hopf bifurcations when two delays are used as the bifurcation parameter.

Our results exhibit that both delays do not affect the stability of tumor-free equilibrium.

However, they are able to destabilize the immune-control equilibrium and cause periodic solutions.

We numerically illustrate how these two delays can change the stability region of the immune-control equilibrium and display the different impacts to the control of tumors.

The numerical simulation results show that the immune activation delay for HTCs can induce heteroclinic cycles to connect the tumor-free equilibrium and immune-control equilibrium.

Furthermore, we observe that the immune activation delay for HTCs can even stabilize the unstable immune-control equilibrium.

### 2014/07/11

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

Global Geometry and Analysis on Locally Symmetric Spaces with

Indefinite-metric (JAPANESE)

**Toshiyuki Kobayashi**(Graduate School of Mathematical Sciences, University of Tokyo)Global Geometry and Analysis on Locally Symmetric Spaces with

Indefinite-metric (JAPANESE)

[ Abstract ]

The local to global study of geometries was a major trend of 20th

century geometry,

with remarkable developments achieved particularly in Riemannian geometry.

In contrast, in areas such as pseudo-Riemannian geometry, familiar to us

as the space-time of relativity theory, and more generally in

pseudo-Riemannian geometry of general signature, surprising little is

known about global properties of the geometry even if we impose a

locally homogeneous structure.

I plan to explain two programs:

1. (global shape) Existence problem of compact locally homogeneous spaces,

and deformation theory.

2. (spectral analysis) Construction of the spectrum of the Laplacian,

and its stability under the deformation of the geometric structure.

by taking anti-de Sitter manifolds as a typical example.

The local to global study of geometries was a major trend of 20th

century geometry,

with remarkable developments achieved particularly in Riemannian geometry.

In contrast, in areas such as pseudo-Riemannian geometry, familiar to us

as the space-time of relativity theory, and more generally in

pseudo-Riemannian geometry of general signature, surprising little is

known about global properties of the geometry even if we impose a

locally homogeneous structure.

I plan to explain two programs:

1. (global shape) Existence problem of compact locally homogeneous spaces,

and deformation theory.

2. (spectral analysis) Construction of the spectrum of the Laplacian,

and its stability under the deformation of the geometric structure.

by taking anti-de Sitter manifolds as a typical example.

### 2014/06/06

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

Lie algebras from secondary polytopes (ENGLISH)

**Mikhail Kapranov**(Kavli IPMU)Lie algebras from secondary polytopes (ENGLISH)

[ Abstract ]

The secondary polytope of a point configuration

in the Euclidean space was introduced by Gelfand, Zelevinsky

and the speaker long time ago in order to understand discriminants

of multi-variable polynomials. These polytopes have

a remarkable factorization (or operadic) property: each

face of any secondary polytope is isomorphic to the

product of several other secondary polytopes.

The talk, based on joint work in progress with M. Kontsevich

and Y. Soibelman, will explain how the factorization property

can be used to construct Lie algebra-type objects:

$L_¥infty$ and $A_¥infty$-algebras. These algebras

turn out to be related to the problem of deformation

of triangulated categories with semiorthogonal decompositions.

The secondary polytope of a point configuration

in the Euclidean space was introduced by Gelfand, Zelevinsky

and the speaker long time ago in order to understand discriminants

of multi-variable polynomials. These polytopes have

a remarkable factorization (or operadic) property: each

face of any secondary polytope is isomorphic to the

product of several other secondary polytopes.

The talk, based on joint work in progress with M. Kontsevich

and Y. Soibelman, will explain how the factorization property

can be used to construct Lie algebra-type objects:

$L_¥infty$ and $A_¥infty$-algebras. These algebras

turn out to be related to the problem of deformation

of triangulated categories with semiorthogonal decompositions.

### 2014/05/02

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

From hyperplane arrangements to Deligne-Mumford moduli spaces: Kohno-Drinfeld way (ENGLISH)

**A.P. Veselov**(Loughborough, UK and Tokyo, Japan)From hyperplane arrangements to Deligne-Mumford moduli spaces: Kohno-Drinfeld way (ENGLISH)

[ Abstract ]

Gaudin subalgebras are abelian Lie subalgebras of maximal

dimension spanned by generators of the Kohno-Drinfeld Lie algebra t_n,

associated to A-type hyperplane arrangement.

It turns out that Gaudin subalgebras form a smooth algebraic variety

isomorphic to the Deligne-Mumford moduli space \\bar M_{0,n+1} of

stable genus zero curves with n+1 marked points.

A real version of this result allows to describe the

moduli space of integrable n-dimensional tops and

separation coordinates on the unit sphere

in terms of the geometry of Stasheff polytope.

The talk is based on joint works with L. Aguirre and G. Felder and with K.

Schoebel.

Gaudin subalgebras are abelian Lie subalgebras of maximal

dimension spanned by generators of the Kohno-Drinfeld Lie algebra t_n,

associated to A-type hyperplane arrangement.

It turns out that Gaudin subalgebras form a smooth algebraic variety

isomorphic to the Deligne-Mumford moduli space \\bar M_{0,n+1} of

stable genus zero curves with n+1 marked points.

A real version of this result allows to describe the

moduli space of integrable n-dimensional tops and

separation coordinates on the unit sphere

in terms of the geometry of Stasheff polytope.

The talk is based on joint works with L. Aguirre and G. Felder and with K.

Schoebel.

### 2014/01/31

16:30-17:30 Room #122 (Graduate School of Math. Sci. Bldg.)

Controllability of fluid flows (ENGLISH)

**Jean-Pierre Puel**(Université de Versailles Saint-Quentin-en-Yvelines)Controllability of fluid flows (ENGLISH)

[ Abstract ]

First of all we will describe in an abstract situation the various concepts

of controllability for evolution equations.

We will then present some problems and results concerning the

controllability of systems modeling fluid flows.

First of all we will consider the Euler equation describing the motion of an

incompressible inviscid fluid.

Then we will give some results concerning the Navier-Stokes equations,

modeling an incompressible viscous fluid, and some related systems.

Finally we will give a first result of controllability for the case of a

compressible fluid (in dimension 1) and some important open problems.

First of all we will describe in an abstract situation the various concepts

of controllability for evolution equations.

We will then present some problems and results concerning the

controllability of systems modeling fluid flows.

First of all we will consider the Euler equation describing the motion of an

incompressible inviscid fluid.

Then we will give some results concerning the Navier-Stokes equations,

modeling an incompressible viscous fluid, and some related systems.

Finally we will give a first result of controllability for the case of a

compressible fluid (in dimension 1) and some important open problems.

### 2014/01/24

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

Complex Brunn-Minkowski theory (ENGLISH)

**Bo Berndtsson**(Chalmers University of Technology)Complex Brunn-Minkowski theory (ENGLISH)

[ Abstract ]

The classical Brunn-Minkowski theory deals with the volume of convex sets.

It can be formulated as a statement about how the volume of slices of a convex set varies when the slice changes. Its complex counterpart deals with slices of pseudo convex sets, or more generally fibers of a complex fibration. It describes how $L^2$-norms of holomorphic functions, or sections of a line bundle, vary when the fibers change, and says essentially that a certain associated vector bundle has positive curvature. In the presence of enough symmetry this implies convexity properties of volumes; the real Brunn-Minkowski theorem corresponding to maximal symmetry. There are also applications and relations in other directions, like variations of Kahler metrics, variations of complex structures and the study of plurisubharmonic functions.

The classical Brunn-Minkowski theory deals with the volume of convex sets.

It can be formulated as a statement about how the volume of slices of a convex set varies when the slice changes. Its complex counterpart deals with slices of pseudo convex sets, or more generally fibers of a complex fibration. It describes how $L^2$-norms of holomorphic functions, or sections of a line bundle, vary when the fibers change, and says essentially that a certain associated vector bundle has positive curvature. In the presence of enough symmetry this implies convexity properties of volumes; the real Brunn-Minkowski theorem corresponding to maximal symmetry. There are also applications and relations in other directions, like variations of Kahler metrics, variations of complex structures and the study of plurisubharmonic functions.

### 2013/12/06

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

Local Langlands correspondence and Lubin-Tate perfectoid spaces (JAPANESE)

**Naoki Imai**(Graduate School of Mathematical Scinences, The University of Tokyo)Local Langlands correspondence and Lubin-Tate perfectoid spaces (JAPANESE)

### 2013/12/06

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

Local Langlands correspondence and Lubin-Tate perfectoid spaces (JAPANESE)

**Naoki Imai**(Graduate School of Mathematical Sciences, The University of Tokyo)Local Langlands correspondence and Lubin-Tate perfectoid spaces (JAPANESE)

### 2013/11/08

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

Ext Analogues of Branching laws (ENGLISH)

**Dipendra Prasad**(Tata Institute of Fundamental Research)Ext Analogues of Branching laws (ENGLISH)

[ Abstract ]

The decomposition of a representation of a group when restricted to a

subgroup is an important problem well-studied for finite and compact Lie

groups, and continues to be of much contemporary interest in the context

of real and $p$-adic groups. We will survey some of the questions that have

recently been considered, and look at a variation of these questions involving concepts in homological algebra which gives rise to interesting newer questions.

The decomposition of a representation of a group when restricted to a

subgroup is an important problem well-studied for finite and compact Lie

groups, and continues to be of much contemporary interest in the context

of real and $p$-adic groups. We will survey some of the questions that have

recently been considered, and look at a variation of these questions involving concepts in homological algebra which gives rise to interesting newer questions.

### 2013/07/26

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

Analysis of the Navier-Stokes and Complex Fluids Flow (ENGLISH)

**Matthias Hieber**(TU Darmstadt, Germany)Analysis of the Navier-Stokes and Complex Fluids Flow (ENGLISH)

[ Abstract ]

In this talk, we discuss the dynamics of fluid flow generated by the Navier-Stokes equations or, more generally, by models describing complex fluid flows. Besides classical questions concerning well-posedness of the underlying equations, we investigate analytically models arising in the theory of free boundary value problems, viscoelastic fluids and liquid crystals.

In this talk, we discuss the dynamics of fluid flow generated by the Navier-Stokes equations or, more generally, by models describing complex fluid flows. Besides classical questions concerning well-posedness of the underlying equations, we investigate analytically models arising in the theory of free boundary value problems, viscoelastic fluids and liquid crystals.

### 2013/06/28

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

The Geometry of protain modelling (JAPANESE)

**Hiroki Kodama**(Graduate School of Mathematical Sciences, The University of Tokyo)The Geometry of protain modelling (JAPANESE)

### 2013/05/31

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

Stable patterns and the nonlinear ``hot spots'' conjecture (JAPANESE)

**Yasuhito MIYAMOTO**(Graduate School of Mathematical Sciences, The University of Tokyo)Stable patterns and the nonlinear ``hot spots'' conjecture (JAPANESE)

### 2013/03/18

15:00-17:30 Room #050 (Graduate School of Math. Sci. Bldg.)

Value distribution theory and analytic function theory in several variables (JAPANESE)

My fifty years of differential equations (JAPANESE)

**NOGUCHI, Junjiro**(University of Tokyo) 15:00-16:00Value distribution theory and analytic function theory in several variables (JAPANESE)

**OSHIMA, Toshio**(University of Tokyo) 16:30-17:30My fifty years of differential equations (JAPANESE)

### 2013/01/25

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

State of the art in numerical verification methods of solutions for partial differential equations (JAPANESE)

**Mitsuhiro T. Nakao**(Sasebo National College of Technology)State of the art in numerical verification methods of solutions for partial differential equations (JAPANESE)

### 2012/11/30

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

What do Siegel Eisenstein series know about all modular forms? (ENGLISH)

**Siegfried BOECHERER**(University of Tokyo)What do Siegel Eisenstein series know about all modular forms? (ENGLISH)

[ Abstract ]

Eisenstein series came up in C.L.Siegel's famous work on quadratic forms. The main properties of such Eisensetin series such as analytic continuation and explict form of Fourier expansion are well understood. Nowadays, we use Eisenstein series of higher rank symplectic groups and their restrictions to study properties of all modular forms. I will try to survey the use of “pullbacks of Eisenstein series”: Basis problem, L-functions, p-adic properties, rationality and integrality questions.

Eisenstein series came up in C.L.Siegel's famous work on quadratic forms. The main properties of such Eisensetin series such as analytic continuation and explict form of Fourier expansion are well understood. Nowadays, we use Eisenstein series of higher rank symplectic groups and their restrictions to study properties of all modular forms. I will try to survey the use of “pullbacks of Eisenstein series”: Basis problem, L-functions, p-adic properties, rationality and integrality questions.

### 2012/11/16

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)

### 2012/10/12

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

Homogenization on arbitrary manifolds (ENGLISH)

**Antonio Siconolfi**(La Sapienza - University of Rome)Homogenization on arbitrary manifolds (ENGLISH)

[ Abstract ]

We show that results on periodic homogenization for Hamilton-Jacobi equations can be generalized replacing the torus by an arbitrary compact manifold. This allows to reach a deeper understanding of the matter and unveils phenomena somehow hidden in the periodic case, for instance the fact that the ambient spaces of oscillating equations and that of the limit problem are different, and possess even different dimensions. Repetition structure for the base manifold, changes of scale in it and asymptotic analysis, which are the basic ingredients of homogenization, need substantial modification to work in the new frame, and this task is partially accomplished using tools from algebraic topology. An adapted notion of convergence allowing approximating entities and limit to lie in different spaces is also provided.

We show that results on periodic homogenization for Hamilton-Jacobi equations can be generalized replacing the torus by an arbitrary compact manifold. This allows to reach a deeper understanding of the matter and unveils phenomena somehow hidden in the periodic case, for instance the fact that the ambient spaces of oscillating equations and that of the limit problem are different, and possess even different dimensions. Repetition structure for the base manifold, changes of scale in it and asymptotic analysis, which are the basic ingredients of homogenization, need substantial modification to work in the new frame, and this task is partially accomplished using tools from algebraic topology. An adapted notion of convergence allowing approximating entities and limit to lie in different spaces is also provided.

### 2012/07/06

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

Large Deviations of Random Graphs and Random Matrices (ENGLISH)

**S.R.Srinivasa Varadhan**(Courant Institute of Mathematical Sciences, New York University)Large Deviations of Random Graphs and Random Matrices (ENGLISH)

[ Abstract ]

A random graph with $n$ vertices is a random symmetric matrix of $0$'s and $1$'s and they share some common aspects in their large deviation behavior. For random matrices it is the question of having large eigenvalues. For random graphs it is having too many or too few subgraph counts, like the number of triangles etc. The question that we will try to answer is what would a random matrix or a random graph conditioned to exhibit such a large deviation look like. Since the randomness is of size $n^2$ large deviation rates of order $n^2$ are possible.

A random graph with $n$ vertices is a random symmetric matrix of $0$'s and $1$'s and they share some common aspects in their large deviation behavior. For random matrices it is the question of having large eigenvalues. For random graphs it is having too many or too few subgraph counts, like the number of triangles etc. The question that we will try to answer is what would a random matrix or a random graph conditioned to exhibit such a large deviation look like. Since the randomness is of size $n^2$ large deviation rates of order $n^2$ are possible.

### 2012/05/25

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

Quasi-Monte Carlo methods: deterministic is often better than random (ENGLISH)

https://www.ms.u-tokyo.ac.jp/~matumoto/WORKSHOP/workshop2012.html

**Harald Niederreiter**(RICAM, Austrian Academy of Sciences)Quasi-Monte Carlo methods: deterministic is often better than random (ENGLISH)

[ Abstract ]

Quasi-Monte Carlo (QMC) methods are deterministic analogs of statistical Monte Carlo methods in computational mathematics. QMC methods employ evenly distributed low-discrepancy sequences instead of the random samples used in Monte Carlo methods. For many types of computational problems, QMC methods are more efficient than Monte Carlo methods. After a general introduction to QMC methods, the talk focuses on the problem of constructing low-discrepancy sequences which has fascinating links with subjects such as finite fields, error-correcting codes, and algebraic curves.

This talk also serves as the first talk of the four lecture series. The other three are on 5/28, 5/29, 5/30, 14:50-16:20 at room 123.

[ Reference URL ]Quasi-Monte Carlo (QMC) methods are deterministic analogs of statistical Monte Carlo methods in computational mathematics. QMC methods employ evenly distributed low-discrepancy sequences instead of the random samples used in Monte Carlo methods. For many types of computational problems, QMC methods are more efficient than Monte Carlo methods. After a general introduction to QMC methods, the talk focuses on the problem of constructing low-discrepancy sequences which has fascinating links with subjects such as finite fields, error-correcting codes, and algebraic curves.

This talk also serves as the first talk of the four lecture series. The other three are on 5/28, 5/29, 5/30, 14:50-16:20 at room 123.

https://www.ms.u-tokyo.ac.jp/~matumoto/WORKSHOP/workshop2012.html

### 2012/05/11

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

Moduli spaces and symplectic derivation Lie algebras (JAPANESE)

**SAKASAI Takuya**(University of Tokyo)Moduli spaces and symplectic derivation Lie algebras (JAPANESE)

[ Abstract ]

First we overview Kontsevich's theorem describing a deep connection between homology of certain infinite dimensional Lie algebras (symplectic derivation Lie algebras) and cohomology of various moduli spaces. Then we discuss some computational results on the Lie algebras together with their applications (joint work with Shigeyuki Morita and Masaaki Suzuki).

First we overview Kontsevich's theorem describing a deep connection between homology of certain infinite dimensional Lie algebras (symplectic derivation Lie algebras) and cohomology of various moduli spaces. Then we discuss some computational results on the Lie algebras together with their applications (joint work with Shigeyuki Morita and Masaaki Suzuki).

### 2012/03/16

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.

### 2012/03/13

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.

### 2012/01/27

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

On Topological Blow-up (JAPANESE)

**Taro YOSHINO**(Graduate School of Mathematical Sciences, University of Tokyo)On Topological Blow-up (JAPANESE)

[ Abstract ]

Topological blow-up is a method to understand non-Hausdorff spaces. We define an obstruction to Hausdorffness, and call it crack. By `blowing up' the points in crack, we can obtain better space.

Topological blow-up is a method to understand non-Hausdorff spaces. We define an obstruction to Hausdorffness, and call it crack. By `blowing up' the points in crack, we can obtain better space.

### 2011/12/16

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

Application of positive characteristic methods to singularity theory (JAPANESE)

**TAKAGI, Shunsuke**(Graduate School of Mathematical Sciences, University of Tokyo)Application of positive characteristic methods to singularity theory (JAPANESE)

[ Abstract ]

As an application of positive characteristic methods to singularity theory, I will talk about a characterization of singularities in characteristic zero using Frobenius maps. Log canonical singularities form a class of singularities associated to the minimal model program. It is conjectured that they correspond to $F$-pure singularities, which form a class of singularities defined via splitting of Frobenius maps. In this talk, I will explain a recent progress on this conjecture, especially its connection to another conjecture on ordinary reductions of Calabi-Yau varieties defined over number fields.

As an application of positive characteristic methods to singularity theory, I will talk about a characterization of singularities in characteristic zero using Frobenius maps. Log canonical singularities form a class of singularities associated to the minimal model program. It is conjectured that they correspond to $F$-pure singularities, which form a class of singularities defined via splitting of Frobenius maps. In this talk, I will explain a recent progress on this conjecture, especially its connection to another conjecture on ordinary reductions of Calabi-Yau varieties defined over number fields.