iTHEMS Math Seminar
122 events
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Seminar
Introduction to Singularity Theory in Algebraic Geometry
May 16 (Thu) 16:00 - 18:10, 2019
Kenta Sato (Special Postdoctoral Researcher, RIKEN Interdisciplinary Theoretical and Mathematical Sciences Program (iTHEMS))
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. 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.
Venue: Seminar Room #160
Event Official Language: English
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Seminar
Gauge Theory and Symmetries of 4-Dimensional Spaces
April 26 (Fri) 16:00 - 18:10, 2019
Hokuto Konno (Special Postdoctoral Researcher, RIKEN Interdisciplinary Theoretical and Mathematical Sciences Program (iTHEMS))
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. 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.
Venue: Seminar Room #160
Event Official Language: English
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Seminar
Introduction to Galois Theory and Class Field Theory
April 18 (Thu) 16:00 - 18:00, 2019
Hiroyasu Miyazaki (Special Postdoctoral Researcher, RIKEN Interdisciplinary Theoretical and Mathematical Sciences Program (iTHEMS))
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. 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.
Venue: Seminar Room #160
Event Official Language: English
122 events