広島大学理学部数学科 代数数理グループ

2023年度の代数学セミナー

通常の講演時間はおよそ 1 時間半です。

今後の予定

209(金)1500 分  於 広島大学先端研 N 棟 404N 号室
Dino Festi 氏 (The University of Padua)
Black holes, rationalizability and modularity
Physicists interested in high energy physics often encounter Feynman integrals presenting square roots in their argument. Exact solutions of these integrals are normally out of reach and so they are usually solved numerically. In order to achieve higher precision in the numeric evaluationit is necessary to find a change of the variables of the integral that makes the square root disappear. Deciding the existence of such a change of variable is an algebraic problem that can be naturally translated into investigating the unirationality of a variety. The original problem can be generalized to sets of square roots and to algebraic extensions of function fields. We will finally present a case coming from the study of two black holes, treated also using modularity results.

The content of this talk is the fruit of a series of joint works with Marco Besier, Andreas Hochenegger, and Bert van Geemen.

これまでの記録

112(金)1500 分  於 B702 号室
松澤 陽介 氏(大阪公立大学)
Preimages question of self-morphisms on projective varieties over number fields
Pulling back an invariant subvariety by a self-morphism on projective variety, you will get a tower of increasing closed subsets. Working over a number field, we expect that the set of rational points (of bounded degree) contained in this increasing subsets eventually stabilizes. I will explain why this expectation seems to be reasonable and introduce several affirmative cases, such as the case of etale morphisms and morphisms on the product of two P^1. I will also present some counterexamples that occur when we drop some of the assumptions.

This work is based on a joint work with Matt Satriano and Jason Bell, and recent work with Kaoru Sano.

106(金)1500 分  於 B702 号室
Pho Duc Tai 氏(VNU University of Science, Hanoi)
On the arithmetic of Edwards curves
In this talk we recall the history of elliptic curves in Weierstrass normal form, then we will explain the construction of the arithmetic (point addition and point doubling) on elliptic curves in Edwards normal form. Later we discuss the geometry of Edwards curves and applications.

929(金)1500 分  於 B702 号室
Davide Cesare Veniani 氏(Stuttgart University)
Non-degeneracy of Enriques surfaces
Enriques' original construction of Enriques surfaces dates back to 1896. It involves a 10-dimensional family of sextic surfaces in the projective space which are non-normal along the edges of a tetrahedron. The question whether all Enriques surfaces arise through Enriques' construction has remained open for more than a century.

In two joint works with G. Martin and G. Mezzedimi, we have now settled this question in all characteristics by studying particular configurations of genus one fibrations, and two invariants called maximal and minimal non-degeneracy. The proof involves so-called `triangle graphs' and the distinction between special and non-special 3- sequences of half-fibers.

In this talk, I will present the classification of Enriques surfaces of low non-degeneracy and explain how this classification solves this long- standing problem.

84(金)1500 分  於 B702 号室
Simon Brandhorst 氏(Saarland University)
K3 surfaces of zero entropy (Joint work with Giacomo Mezzedimi)
Automorphisms of K3 surfaces come in 3 flavors: 1) The orbit of every point is finite. 2) There exists a point with an infinite orbit, but no orbit is Zariski dense. 3) There is a Zariski dense orbit. In the first and second case the automorphism has zero topological entropy while in the last case it is of positive entropy. We say that a surface has zero entropy if every of its automorphisms has zero entropy.

In this talk we classify K3 surfaces of zero entropy yet with infinite automorphism group, equivalently, which have a unique elliptic fibration whose Jacobian has infinite Mordell-Weil group.

728(金)1500 分  於 B702 号室
山田 裕史 氏(岡山大学)
Virasoro代数とユニタリ群上のベクトル場
円周上のベクトル場のなす無限次元リー環はWitt代数と呼ばれる. その1次元中心拡大が表題のVirasoro代数である. 40年ほど前に脇本實氏と私は,そのFock表現について 特異ベクトルが特別なシューア函数として表示されることに気がついた. 長方形のヤング図形に対応するシューア函数である. なぜ長方形が登場するのかという疑問には,脇本ー山田のプレプリントが 出回った直後にN. Wallachが明確に答えてくれた. 回はこのWallachの仕事を大雑把に紹介したい.

710(月)1600 分  於 B701 号室
木谷 裕紀 氏(大阪公立大学)
手札消費型組合せゲームにおける勝敗判定アルゴリズム
将棋,囲碁などの偶然の要素を含まない完全情報ゲームを組合せゲームと呼ぶ.組合せゲームに対する理論計算機科学的興味の一つがお互いが最善手を続けた場合どちらが勝つかという判定問題における計算量である.本講演では,「手札を出し切ったプレイヤが勝ち」である手札消費型組合せゲームをいくつか取り上げ,それぞれのゲームに対する勝者判定アルゴリズムを紹介する.

710(月)1435 分  於 B701 号室
安福 智明 氏(国立情報学研究所)
不偏ゲームと代数の関係
組合せゲーム理論の分野において,不偏ゲームは古くから活発に研究されており,不偏ゲームのもつ代数的構造は興味深いものとして知られている.本講演では,不偏ゲームと代数との関係について,いくつかの具体的なゲームに触れながら紹介する.また,時間が許せば最新の研究成果についても紹介する.

77(金)1500 分  於 B702 号室
末續 鴻輝 氏(国立情報学研究所)
非不偏ゲームと全象ルール
本講演では、第一に組合せゲーム理論について簡単に説明し、次いで全象ルールに関する研究を紹介する。本研究で扱う全象ルールとは、正規形非不偏ゲームの(TransfiniteやLoopyではない)ルールセットであって、任意の「ゲームの値」を取ることができるものである。このようなルールはこれまで一つしか見つかっていなかったが、講演者の研究で新たに三つの全象ルールを発見した。それらの全象ルールと全象性の証明手法について本講演では紹介する。さらに時間が許せば、聴講者の興味に応じて講演者の過去の研究について紹介する。

623(金)1500 分  於 B702 号室
山下 貴央 氏(広島大学)
Yama Nim and a comply/constrain operator of combinatorial games
We introduce Yama Nim, a variation of Nim played on a two-dimensional semi-infinite game board, with terminal positions in the upper left corner. The player can move two or more up steps and one right step, or two or more left steps and one down step. If a player cannot move, they lose. We find the solution to this game. We also consider a comply/constrain operator on impartial rulesets. Applied to the rulesets A and B, on each turn the opponent proposes one of the rulesets and the current player complies, by playing a move in that ruleset. If the outcome table of the comply/constrain variation of A and B is the same as the outcome table of A, then we say that B is dominated by A. We show necessary and sufficient conditions of "A dominates B". Yama Nim is a good example that dominates classical rulesets such as Nim and Wythoff Nim.