Shuji Yamamoto (Keio University)
Plan of the seminar: we separate each talk into two. In the first 60 minutes the speaker gives an introductory talk for non-mathematicians. After a short break, the second 60 minutes is spent for a bit more detailed talk for mathematicians (working in other areas). We welcome you joining both parts of the seminar or only the first/second half.
Abstract: This is an introduction to multiple zeta values (MZVs). Although the study of MZVs is related to various areas of mathematics, we will concentrate on the algebraic structures of MZVs themselves. The key point is that MZVs have two kinds of representations: nested series and iterated integrals. We present how these two representations yield rich algebraic relations among MZVs.
Dec. 17, 16:00-18:10, Main Research Building Room 160
Wahei Hara (Waseda University)
Plan of the seminar: we separate each talk into two. In the first 60 minutes the speaker gives an introductory talk for non-mathematicians. After a short break, the second 60 minutes is spent for a bit more detailed talk for mathematicians (working in other areas). We welcome you joining both parts of the seminar or only the first/second half.
Abstract: We will talk about the theory noncommutative resolution of singularities. Noncommutative resolution is a noncommutative analog of usual (geometric) resolution of singularities, and allows us to generalise the idea of McKay correspondence to a large class of singularities.
In the first part of the talk, we discuss the classical McKay correspondence, the definition of noncommutative crepant resolution, and some known results in lower dimensions. In the second half, we will discuss some concrete examples of noncommutative crepant resolutions in higher dimensions.
Dec. 4, 16:00-18:10, Main Research Building Room 160
Atsushi Ito (Nagoya University)
We talk about Gauss maps and projective dual varieties, which are classical objects in projective geometry of algebraic varieties. In particular, we explain Gauss maps in positive characteristic and projective dual varieties of toric varieties in characteristic 0.
Nov. 8, 16:00-18:10, Main Research Building Room 160
Ken Shiozaki (Yukawa Institute for Theoretical Physics, Kyoto University)
The topological nature of the band theory in crystalline systems can be well described by the topological K-theory over the Brillouin zone torus. In the first part of my talk, I will present the band-theory understanding of the grading of the K-group, and how the exactness axiom and the Mayer-Vietoris sequence are naturally understood. In the second part, I discuss how to compute the differentials of the Atiyah-Hirzebruch spectral sequence associated with a cell decomposition.
Oct. 24, 16:00-18:10, Main Research Building Room 160
Masaki Tsukamoto (Kyusyu University)
Mean dimension is a topological invariant of dynamical systems, which counts the number of parameters per second for describing the orbits of dynamical systems.
I will explain the connections between this quantity and information theory:
Rate distortion theory is responsible for such lossy data compression. We found that if we apply a kind of mini-max principle to rate distortion theory then we get mean dimension.
On 21st October, I will explain the outline of the above (1) and (2). I will explain more details about (1) on 22nd and about (2) on 23th. This plan of talks may be changed according to requests of participants of the seminar.
Oct. 21, 15:30-17:40, Okochi Hall.
Oct. 22, 13:30-14:30, Main Research Building Room 437
Oct. 23, 13:30-14:30, Main Research Building Room 437
Takayuki Koike (Osaka City University)
We explain our recent study on the complex analytic structure of a small tubular neighborhood of a complex submanifold, which is based on T. Ueda's classification theory. We also explain how to apply them to: (i) a study on (non-) existence of a smooth Hermitian metric on a nef line bundle over a projective manifold with semi-positive curvature, and (ii) a study on non-projective and non-Kummer K3 surfaces.
July 30 16:00-18:10, Main Research Building Room 160
Keita Mikami (iTHEMS)
In this seminar, I will talk about mathematical study of Schroedinger operators (or Schroedinger equation).
In the first part, I will talk about what mathematicians do to find a solution to Schroedinger equation. The goal of the first part is to be able to check the existence of solutions of Schroedinger equations in terms of decay/growth rate of potentials.
In the second part, I will talk about what can we say about solutions to Schroedinger equation constructed in the first part. Especially, the relationship to the corresponding classical mechanic is introduced.
July 12, 16:00-18:10, Main Research Building Room 160
Hiroki Kodama (iTHEMS/Tohoku University)
For most of mathematicians, metric is symmetric. However, we can define asymmetric metric without any difficulty. "Coarse" is a notion to describe some large scale viewpoint. For example, the set of real numbers is coarse equivalent to the set of integers (with respect to standard metric). I will discuss asymmetric metric space in "coarse" sense.
[Part 1.] I will define metric space and asymmetric metric space. I will also explain a notion of coarse equivalence.
[Part 2.] I will discuss what kind of asymmetric metric space is not coarse equivalent to (symmetric) metric space. I also would like to give other generalizations of metric.
June 20, 16:00-18:10, Main Research Building Room 160
Genki Oouchi (iTHEMS)
In this talk, I would like to talk about certain invariants that look like dimension. This talk has independent two parts.
In part 1, I will talk about finite metric spaces. In 2013, Leinster introduced the notion of magnitude of finite metric spaces. It measures effective number of points in finite metric spaces. Considering magnitude and scale transformation, Leinster and Willerton defined dimension of finite metric space with scale. I will explain the definition of magnitude of finite metric spaces and see examples.
In part 2, I will talk about derived categories of smooth projective varieties or finite dimensional algebras. In 2014, Dimitrov, Heiden, Katzarkov and Kontsevich introduced the notion of entropy of endofunctors of derived categories. It measures complexity of endofunctors under iteration. Serre functor is an autoequivalence of derived category, that describes Serre duality. Entropy of Serre functor looks like dimension of derived categories. I will talk about known results for entropy of Serre functors and some related topics.
May 24, 16:00-18:10, Main Research Building Room 160
Kenta Sato (iTHEMS)
In this talk, I will explain for all scientists how singularities are studied in algebraic geometry.
In algebraic geometry, we study algebraic varieties, which are figures defined as the zero sets of polynomial equations. To study an algebraic variety, we often expect that the variety is smooth, that is, the variety locally resembles Euclidian spaces. However, even if we start from smooth varieties, we sometimes encounter non-smooth varieties.
This is one of the reasons why we need to study singularities.
(Part I)
In the first one hour, I will explain how singularities are studied. I will introduce two invariants of singularities by which we can compare singularities numerically. One invariant is defined in terms of resolution of singularities and the other is defined in terms of positive characteristic methods. I also explain a surprising relation of these invariants.
(Part II)
In the second one hour, I will explain how singularity theory is used to study smooth projective varieties. I will introduce Minimal Model Program and explain the relation with singularity theory.
May 16, 16:00-18:10, Main Research Building Room 160
Hokuto Konno (iTHEMS)
Although the term "gauge theory" is usually used in physical contexts, in the early 1980's, mathematicians found that gauge theory has many striking applications to purely mathematical problems. Most of typical applications are related to topology of 4-dimensional spaces. As a recent development in this direction, I used gauge theory to study "the shape of the space of all symmetris of a 4-dimensional space".
In the first one hour, I will explain a notion of mathematical spaces, called manifolds, and try to describe the idea: how mathematicians make use of gauge theory to study the topology of a 4-dimensional manifold.
In the second one hour, I will explain what the space of symmetries of a manifold means, and which type of theorems about the space of symmetries can be obtained using gauge theory.
Apr. 26, 16:00-18:10, Main Research Building Room 160
Hiroyasu Miyazaki (iTHEMS)
(Part I)
Galois theory is one of the most important theories in mathematics. Speaking in one phrase, it explains the correspondence between “extensions of numbers” and “subgroups of Galois group”. Basically, finding subgroups of a finite group is much easier than finding extensions of numbers. As a result, Galois theory has incredibly strong applications. For example, we can prove polynomial equations of degree greater than 4 are not always solvable by radicals, which is a celebrated result by Abel and Galois. In the first part of the talk, I will introduce Galois theory in an accessible way for all scientists.
(Part II)
Class Field Theory (CFT) is a monumental work in number theory. Given Galois theory, which is explained in Part I, classifying “extension of numbers” is reduced to classifying “subgroups of Galois group”. So, the next thing to do would be to analyze the structure of Galois groups. CFT enables us to describe the Galois group of a number field K by using only the language of K, i.e., not by using its extensions. In the second part of the talk, I will explain CFT in an as accessible way as possible for all scientists (in particular, also for mathematicians).
If time permits, I would like to explain a geometric interpretation of Galois theory, and higher dimensional CFT.
Apr. 18, 16:00-18:10, Main Research Building Room 160