九大代数学セミナー：2023年度

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最終更新： 2024年4月7日

2024年2月19日 (月) 16:00-17:00 印刷用プログラム

• Asif Zaman 氏 (University of Toronto)
  "An approximate form of Artin's holomorphy conjecture and nonvanishing of Artin L-functions"


• Artin L-functions are associated to irreducible non-trivial characters of the Galois group of a normal extension of number fields. Conjecturally, Artin L-functions are holomorphic and non-vanishing except on the critical line. Together with Robert Lemke Oliver and Jesse Thorner, we unconditionally prove for many families of Artin L-functions, all except few of them are holomorphic and non-vanishing in a wide region. The number of exceptions is quantified in terms of a field counting problem which, in many cases of interest, is provably small. This has applications to extremal class numbers, counting prime ideals in degree n S_n-extensions, and the subconvexity problem for Dedekind zeta functions. I will outline the main result, some applications, and key obstacles in the proof.

2024年1月26日 (金) 16:00-17:00 印刷用プログラム

• 片田 舞 氏 (九州大学)
  "Stable rational cohomology of the IA-automorphism groups of free groups"


• The IA-automorphism group IA_n of the free group F_n is a normal subgroup of the automorphism group of F_n. Little is known about the GL(n,Z)-representation structure of the rational cohomology of IA_n except for the first and second cohomology. We study the image of the homomorphism between cohomology induced by the abelianization map of IA_n, which we call the Albanese cohomology of IA_n. In this talk, we obtain a subquotient representation of the Albanese cohomology of IA_n in a stable range, which is conjecturally equal to the entire Albanese cohomology of IA_n. Moreover, we study the whole structure of the stable rational cohomology of IA_n. This talk is partly based on joint work with Kazuo Habiro.

2023年12月8日 (金) 16:00-17:00 印刷用プログラム

• 石塚 裕大 氏 (九州大学)
  "二元四次形式の指数和とその応用" (Title: Exponential sum on the space of binary quartic forms and its application)


• 2020年、谷口隆氏（神戸大）とFrank Thorne氏（サウスカロライナ大）は判別式が平方自由、かつ三つの異なる素数の積になっている三次体を数え上げた。 この結果の鍵となるのは、二元三次形式の空間におけるある指数和の値の評価である。 今回の講演では、二元四次形式の空間における類似の結果を紹介し、楕円曲線における応用を紹介する。 この結果は谷口隆氏、Frank Thorne氏および Stanley Yao Xiao氏（UNBC）との共同研究である。
  
  On 2020, Taniguchi and Thorne gave a lower bound for the number of cubic fields whose discriminants are squarefree and has at most three prime factors. A key point is an estimate of exponential sum on the space of binary cubic forms. In this talk, I introduce an analogue result on the space of binary quartic forms, and introduce its application on elliptic curves. This is based on a joint work with Takashi Taniguchi (Kobe), Frank Thorne (South Carolina) and Stanley Yao Xiao (UNBC).

2023年11月15日 (水) 16:00-17:00 印刷用プログラム

• 岡田 拓三 氏 (九州大学)
  "Birational Mori fiber space structures of Fano varieties and rationality problem"


• Rationality problem of algebraic varieties is a classical problem in algebraic geometry and it asks whether or not a given algebraic variety is rational (i.e. birationally equivalent to a projective space) or not. There are several approaches toward this problem. Among them I will explain an approach by analyzing birational Mori fiber space structures of a given variety and then I will talk about my recent results on Fano 3-folds.

2023年10月17日 (火) 16:00-17:00 印刷用プログラム

• Jerome T. Dimabayao 氏 (University of the Philippines Diliman)
  "An irrational variant of the congruent number problem"


• A positive integer $n$ is called a $\theta$-congruent number if there is triangle with rational sides $a,b$ and $c$ for which the angle between $a$ and $b$ is equal to $\theta$ and its area is $n\sqrt{r^2 - s^2}$, where $0 < \theta < \pi$, $\cos \theta = s/r$ and $0 \leq |s| < r$ are relatively prime integers. The notion of $\theta$-congruent numbers is a natural generalization of the classical congruent numbers, which correspond to the case where $\theta = \pi/2$. It is known that the problem of classifying $\theta$-congruent numbers is related to the problem of finding non-trivial rational points on certain families of elliptic curves. In this talk, we present a certain variant of the congruent number problem. More explicitly, we discuss integers which occur as areas of triangles with two rational sides and arbitrary fixed angle $\psi$ with one adjacent side a rational multiple of a quadratic surd. We call such numbers $\psi$-congruent. We present a criterion that involves elliptic curves for deciding whether a given positive integer is $\psi$-congruent. We also discuss some results about $\pi/4$-congruent numbers from a joint work with Soma Purkait.

2023年7月25日 (火) 15:00-16:30 印刷用プログラム

• Ken Ono 氏 (University of Virginia)
  "Can't you feel the moonshine?"
講演の録画映像(映像を見るにはパスコードが必要です. パスコードが必要な方は世話人(alg-seminar "at" math.kyushu-u.ac.jp)にご連絡ください。)


• Richard Borcherds won the Fields medal in 1998 for his proof of the Monstrous Moonshine Conjecture. Loosely speaking, the conjecture asserts that the representation theory of the Monster, the largest sporadic finite simple group, is dictated by the coefficients of certain distinguished modular functions. This conjecture arose from astonishing coincidences noticed by finite group theorists and arithmetic geometers in the 1970s. Recently, mathematical physicists have revisited moonshine, and they discovered evidence of undiscovered moonshine which some believe have applications to string theory and 3d quantum gravity. The speaker and his collaborators have been developing the mathematical facets of this theory, and have proved the Umbral moonshine Conjecture.

2023年7月21日 (金) 16:00-17:15 印刷用プログラム

• 中村 勇哉 氏 (東京大学)
  "Ehrhart theory on periodic graphs"


• 周期グラフとは, 格子Z^Nが自由に作用しているグラフであってその商グラフが有限グラフになるものをいう. 周期グラフは, 数理結晶学における研究対象になっている他, 幾何学的群論においてもvirtually abelian groupのケーリーグラフとして自然に現れる対象である. グラフのgrowth sequence b(n)は, グラフのある頂点からスタートしてグラフ距離n以下の頂点の個数として定義される. 本講演ではまず, Grosse-Kunstleve, Brunner, Sloane (1996) の予想「周期グラフのgrowth sequenceがquasi-polynomial type (十分nが大きい所でquasi-polynomial) になる」の肯定的解決 (中村-坂本-間瀬-中川, 2021) について紹介する. 証明は代数的であり, 次数付き環のヒルベルト級数の理論に帰着される. 残りの時間で, エルハート理論との関係について紹介する (井上卓哉氏との共同研究). エルハート理論の場合と異なり, 周期グラフのgrowth sequenceはquasi-polynomialになるとは限らず (有限個の例外項がありうる), また相互法則を満たすとも限らない. 一方で, 対称性の高い様々な具体例において, growth sequenceはquasi-polynomialになり, 相互法則を満たすことがConway-Sloane (1997) 等により報告されている (九大のSGWでも独立に報告されています). 本講演では, growth sequenceがquasi-polynomialになり, 相互法則を満たすようなグラフのクラスを紹介し, これらのクラスをエルハート理論の拡張として捉えることが可能であることを説明する.
  
  A periodic graph is a graph with a free Z^N-action such that its quotient graph is a finite graph. Periodic graphs are the object of study in mathematical crystallography and also appear naturally in geometric group theory as Cayley graphs of virtually abelian groups. The growth sequence b(n) of a graph is defined as the number of vertices with (graph) distance n or less from a fixed vertex. In this talk, I will first discuss an affirmative solution to the conjecture of Grosse-Kunstleve, Brunner, and Sloane in 1996 that the growth sequence of a periodic graph is of quasi-polynomial type (``type" here means ``for sufficiently large n"). The proof (by Nakamura-Sakamoto-Mase-Nakagawa, 2021) is algebraic, and it is based on the theory of Hilbert series of graded rings. In the rest of the talk, we will discuss the relation with the Ehrhart theory (joint work with Takuya Inoue). Unlike the case of the Ehrhart theory, the growth sequence of a periodic graph is not necessarily quasi-polynomial in general, and it does not necessarily satisfies the reciprocity law. On the other hand, it has been observed by Conway-Sloane (1997) and others that the growth sequence becomes a quasi-polynomial and satisfies the reciprocity law in various specific cases (also reported independently by SGW at Kyushu University). In this talk, I will introduce some necessary conditions for graphs such that the growth sequence becomes a quasi-polynomial and satisfies the reciprocal law, and explain that it can be regarded as an extension of the classical Ehrhart theory.

2023年7月7日 (金) 16:00-17:00 s 印刷用プログラム

• Chieh-Yu Chang 氏 (National Tsing Hua University)
  "On Shimura’s conjecture over function fields"


• In this talk, I will present the joint work with Dale Brownawell, Matthew Papanikolas and Fu-Tsun Wei. I will first describe Shimura's conjecture on algebraic independence of period symbols as motivation, and then formulate its function field analogue in terms of t-motives. Finally, I will give the overall strategy and ideas about how to prove it.

2023年6月2日 (金) 16:00-17:00 印刷用プログラム

• Frank Thorne 氏 (University of South Carolina)
  "An Overview of Number Field Counting"


• How many number fields are there of fixed degree and Galois group, and bounded discriminant? What can we say in terms of asymptotic formulas, upper and lower bounds, conjectures, and so on? I will give an overview of what is known about this question, and what methods go into the proofs. The subject has seen an explosion of recent activity, and I will highlight some recent and ongoing work.

2023年5月26日 (金) 16:00-17:00 印刷用プログラム

• 行田 康晃 氏 (東京大学)
  "一般化マルコフ数と一般化団代数" (Generalized Markov numbers and generalized cluster algebras)
  講演スライド


• マルコフの方程式x^2+y^2+z^2=3xyzの正整数解に現れる数は「マルコフ数」と呼ばれ、有理数による実数の近似問題やモジュラー群と深く関係する興味深い研究対象として知られている。またマルコフの方程式はその自明な整数解(x,y,z)=(1,1,1)から出発して、1つの成分を一定の操作で入れ替える操作を繰り返し行うことで全ての正整数解を得られる特徴を持つが、これが団代数理論で扱われる「団構造」と呼ばれる構造を持つことが近年判明し、団代数理論の枠組みでの研究も活発化している。本講演では、マルコフの方程式と同じように正整数解が「団構造」をもつような方程式と、それに付随する性質や予想について述べる。
  
  The numbers that appear as positive integer solutions of Markov's equation x^2+y^2+z^2=3xyz are called "Markov numbers," which are known to be interesting research objects related to the approximation of real numbers by rational numbers and modular groups. Markov's equation also has the characteristic of obtaining all positive integer solutions by repeatedly performing an operation of swapping one component, starting from the trivial integer solution (x,y,z)=(1,1,1). Recently, it has been discovered that this has a structure called "cluster structure," which is treated in cluster algebra theory, and research in the framework of cluster algebra theory has become active. In this talk, we discuss equations that have positive integer solutions with a "cluster structure" similar to Markov's equation, as well as associated properties and conjectures.

2023年4月28日 (金) 16:00-17:00 印刷用プログラム

• 村上 友哉 氏 (九州大学)
  "3次元多様体の量子不変量に関するGukov-Pei-Putrov-Vafa予想の証明'' (A proof of a conjecture by Gukov-Pei-Putrov-Vafa for quantum invariants of 3-manifolds)


• 3次元多様体の量子不変量は数理物理的観点から定義されたトポロジーの対象だが、表現論や数論とも関係が深く大変興味深い。例えば量子不変量の漸近展開とモジュラー形式の関連がZagierらによって指摘されている。この観点の下、数理物理学者のGukov-Pei-Putrov-Vafaは量子不変量の間のある関係式を予想した。本講演では講演者によって得られたこの予想の証明について述べる。この証明の手法の帰結としてある種のL関数の特殊値の関係式や消滅性が得られることも紹介する。
  
  Quantum invariants of 3-manifolds are topological objects defined from a mathematical physics perspective, and they are also deeply related to representation theory and number theory, which makes them very interesting. For instance, Zagier and others have pointed out the connection between the asymptotic expansion of quantum invariants and modular forms. With this perspective, Gukov-Pei-Putrov-Vafa, mathematical physicists, proposed a certain relation between quantum invariants. In this talk, I will discuss the proof of this proposal obtained by the speaker. I will also introduce that we can obtain some relation formulas and vanishing properties of certain L-functions as a consequence of the method used in the proof.

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