セミナー

Relations between fractal dimensions and arithmetic progressions

2018年10月23日(火) 11:35 - 12:35

Kota Saito (名古屋大学)

In this talk we give estimates for the dimensions of sets in real numbers which uniformly avoid finite arithmetic progressions. More precisely, we say that $F$ uniformly avoids arithmetic progressions of length $k\geq 3$ if there is an $\epsilon>0$ such that one cannot find an arithmetic progression of length $k$ and gap length $\Delta>0$ inside the $\epsilon\Delta$ neighbourhood of $F$. Our main result is an explicit upper bound for the Assouad (and thus Hausdorff) dimension of such sets in terms of $k$ and $\epsilon$. In the other direction, we give examples of sets which uniformly avoid arithmetic progressions of a given length. We also consider higher dimensional analogues of these problems, where arithmetic progressions are replaced with arithmetic patches lying in a hyperplane. As a consequence, we obtain a discretised version of a `reverse Kakeya problem': we show that if the dimension of a set in $\mathbb{R}^d$ is sufficiently large, then it closely approximates arithmetic progressions in every direction. The above is a joint work with Fraser and Yu. Finally we show that the converse of `reverse Kakeya problem' does not hold. This is a single-author work.

会場: 統合支援施設（第一食堂） 2階 大会議室

イベント公式言語: 英語

Generalized Erdös and Obláth theorem for polynomial-factorial Diophantine equations

2018年10月23日(火) 10:30 - 11:30

武田 渉 (名古屋大学)

Diophantine equations are equations where only integer solutions are accepted. There are many types of Diophantine equations and many results are known. Our Diophantine equation is of the form x^n+y^n=m!. Erdös and Obláth showed that the Diophantine equation x^2+y^2=m! has only two positive integer solutions (x,y,m)=(1,1,2),(12,24,6). In this talk, the factorial function m! is replaced with a generalized factorial function Π(m) over number fields. Then whether there are infinitely many solutions or not depends on the number field. We give necessary and sufficient condition for existence of infinitely many solutions of x^2+y^2=Π(m). More generally, we introduce an observation for higher degree equation x^n+y^n=Π(m).

会場: 統合支援施設（第一食堂） 2階 大会議室

イベント公式言語: 英語

励起子ポラリトンのボーズ・アインシュタイン凝縮から半導体レーザーへの非エルミート相転移

2018年9月28日(金) 13:30 - 17:00

花井 亮 (大阪大学)

超伝導や超流動などの巨視的スケールで自発的にコヒーレンスが現れる現象は、現代物性物理学の中心的トピックのひとつに数えられる。ボーズアインシュタイン凝縮(BEC)は、熱化により巨視的な数の粒子が単一エネルギーレベルに占有する現象で、液体ヘリウム、冷却原子気体、マグノン、そして最近では光子に至るまでの様々な系で観測されている。また、光子レーザーは媒質中の反転分布がもたらす光学利得により光の巨視的コヒーレンスが現れる、非平衡凝縮現象である。 励起子ポラリトン系では、ポンプ強度の強弱によって、これら２種類の凝縮現象（ポラリトンBECとVertical cavity surface emitting laser, VCSEL)を単一デバイス中で実現することが可能で、両者の類似性・相違性が盛んに研究されている。興味深いことに、BECからレーザーへと系を変化させると、放射エネルギーに飛びが見られ、両者の間に相境界が存在することが多くの実験で示されている。しかし、どちらも凝縮相にあり 、系の対称性が同一であるため、このような相境界が現れる積極的な理由は一見見当たらず、実際、既存の理論は（我々の知る限り）すべて、クロスオーバーを予言する。 本講演では、このような凝縮相内で現れる相転移が、非エルミート性により引き起こされている、というシナリオを提案する。励起子ポラリトン系の最近の発展について簡単なレビューを行なった後、流入と流出のある量子多体系の理論的な枠組みを説明する。そこで得られる運動方程式の構造から、電子正孔光子凝縮体の非平衡定常解は必ず2種類に分類できることを示し、非エルミート性が相転移をもたらしうることを示す。特に、2種類の解が「合体する」点（例外点）が 、気液相転移の臨界点と同様、相転移線の終端点になることを示す。以上の結果から、予想される相図を提案する。 本講演で提案するシナリオは、非エルミート性のみが本質的であるため、流入と流出のある二成分凝縮体に一般に適用可能である。

会場: セミナー室 (160号室)

イベント公式言語: 日本語

Number Theory Seminar: 3 Talks

2018年7月24日(火) 10:00 - 12:35

小野 雅隆 (慶應義塾大学)
杉山 真吾 (日本大学)
平川 義之輔 (慶應義塾大学)

This seminar is aimed at scientists in general, not only to mathematicians. 10:00-10:45 Title: Multiple zeta functions associated with 2-colored rooted trees Speaker: Dr. Masataka Ono (Keio University) Abstract: In our recent work, we introduced a combinatorial object and finite sum associated with them which we call finite multiple zeta values associated with 2-colored rooted trees and gave a unified interpretation to some types of finite multiple zeta values. In this talk, we introduce multiple zeta function associated with 2-colored rooted tree and discuss its analytic properties, for example, the possible singularities and functional equations. 10:55-11:40 Title: Modular forms and trace formulas with applications to equidistributions of their Fourier coefficients Speaker: Dr. Shingo Sugiyama (Nihon University) Abstract: Modular forms are interesting objects in number theory as they are related to arithmetic problems. Trace formulas of Hecke operators acting on modular forms are very useful tools to study arithmetic invariants: Fourier coefficients, special values of L-functions, Hurwitz class numbers. We will start fundamental notions on modular forms and trace formulas of Hecke operators for non-experts, and introduce our results on a generalization of Serre’s vertical Sato-Tate law. Some results in this talk are based on a joint work with Masao Tsuzuki (Sophia University). 11:50-12:35 Title: On a generalization of Dobinski's formula Speaker: Yoshinosuke Hirakawa (Keio University) Abstract: Dobinski's formula is a very classical formula, which expresses the Bell number as an infinite series. Here, the Bell number is the number of partitions of a finite set. Such a "combinatorial-analytic" formula should lead us to more beautiful number theory. In this talk, we introduce a generalization of Dobinski's formula by means of a certain multiple generalization of the exponential function.

会場: セミナー室 (160号室)

イベント公式言語: 英語

Introduction to topological band structure

2018年7月6日(金) 10:30 - 17:30

小澤 知己 (理化学研究所 数理創造プログラム (iTHEMS) 上級研究員)

In this seminar, I give an introduction to topological band structures. Topological band structure is the fundamental idea to understand phenomena such as the quantum Hall effect, topological insulators, and topological superconductors. Although originally found in fermionic electron systems, topological band structure is essentially a single-particle property, and thus the same phenomenon can arise also in bosonic systems. In fact, topological band structure does not even need quantum mechanics; topological band structure can arise in classical waves inside a periodic medium, such as electromagnetic waves or classical mechanical waves. In this seminar, I first give a brief introduction to key concepts in topological band structures, such as Berry curvature, Chern number, bulk-edge correspondence, and ten-fold way classification of topological insulators. I then explain how topological band structure can arise in classical systems, giving a brief introduction to the field of topological photonics and topological mechanics.

会場: セミナー室 (160号室)

イベント公式言語: 日本語

Topological Photonics and the four-dimensional quantum Hall effect

2018年7月3日(火) 15:00 - 17:00

小澤 知己 (理化学研究所 数理創造プログラム (iTHEMS) 上級研究員)

In this seminar, I will give an introduction to topological photonics, that is, the study of topological band structures and resulting topological phenomena in photonic systems. I will first review basic concepts of topological band structures, and then explain what it means to realize topological band structures in photonic systems. I will particularly emphasize some important differences with respect to solid-state electron systems. I then present some of my own works in topological photonics, such as the synthetic dimensions in photonic systems, which allows one to explore models and phenomena in high dimensions including the four-dimensional quantum Hall effect.

会場: 東北大学 AIMR本館 1階会議室

イベント公式言語: 英語

デンソーITラボ - 理研 産学連携交流会

2018年6月29日(金) 16:00 - 17:40

16:00-17:00 デンソーITラボ 「ITラボの会社紹介」CTO岩崎弘利 「人とクルマの会話インタラクション」シニアリサーチャ 塚原裕史 「機械学習とコンピュータビジョンにも役立つ数学」リサーチャ 石川康太 17:00-17:40 理化学研究所 「古典物理と量子物理を結ぶ数理：古典系での量子ホール効果」iTHEMS上級研究員 小澤知己 「新粒子「ダイオメガ」－スパコン「京」と数理で予言するクォーク6個の新世界－」RNC基礎科学特別研究員 権業慎也

会場: コモンルーム (246-248号室)

イベント公式言語: 日本語

スピンボゾンモデルの基礎

2018年6月22日(金) 10:30 - 17:30

加藤 岳生 (東京大学 物性研究所)

10:30-12:30 スピン ボゾンモデルの基礎 @DR3 (14棟213) 14:00-15:30 近藤効果 @DR7 (14棟217) 16:00-17:30 NIBA formalism の詳細 @DR7 (14棟217)

会場: 慶應義塾大学 矢上キャンパス

イベント公式言語: 日本語

相対かつ同変ラグランジアンフレアーホモロジーとアティヤ-フレアー予想

2018年6月19日(火) 14:30 - 16:00

深谷 賢治 (理化学研究所 数理創造プログラム (iTHEMS) 客員主管研究員 / Permanent Member, Simons Center for Geometry and Physics, Stony Brook University, New York, USA)

アティヤ-フレアー予想は、 ゲージ理論におけるフレアーホモロジーとラグランジアンフレアーホモロジーの間の関係に関するものである。 その一つの困難は、ラグランジアンフレアーホモロジーを考えるシンプレクティック多様体が特異点を持つことである。双対かつ同変ラグランジアンフレアーホモロジーを考えることで、この困難が解消し、少なくともアティヤ-フレアー予想を数学的に厳密な予想として定式化できることを説明する。 RIKEN iTHEMS、東京大学数理科学研究科トポロジー火曜セミナー共催

会場: 東京大学数理科学研究科数理科学研究科棟 056号室

イベント公式言語: 日本語

1-form Lieb-Schultz-Mattis theorem and anomaly matching in quantum dimer model

2018年6月8日(金) 13:30 - 17:00

菊池 勇太 (理化学研究所 仁科加速器科学研究センター (RNC) 理研BNL研究センター 理論研究グループ 理研BNLセンター研究員)

The Lieb-Schultz-Mattis theorem dictates that a trivial symmetric insulator in lattice models is prohibited if lattice translation symmetry and U(1) charge conservation are both preserved. In this talk, we discuss the generalization of the Lieb-Schultz-Mattis theorem to systems with higher-form symmetries, which act on extended objects of dimension n > 0. The prototypical lattice system with higher-form symmetry is the pure abelian lattice gauge theory whose action consists only of the field strength. We first construct the higher-form generalization of the Lieb-Schultz-Mattis theorem with a proof. We then apply it to the U(1) lattice gauge theory description of the quantum dimer model on bipartite lattices. Finally, using the continuum field theory description in the vicinity of the Rokhsar-Kivelson point of the quantum dimer model, we diagnose and compute the mixed ’t Hooft anomaly corresponding to the higher-form Lieb-Schultz-Mattis theorem.

会場: SUURI-COOL (Kyoto)

イベント公式言語: 日本語

Abelian-Higgs dualities in quantum defect-mediated melting phase transitions

2018年4月27日(金) 14:30 - 17:30

Aron Beekman (慶應義塾大学)

In the study of zero-temperature quantum phase transitions, instead of looking how symmetry is broken, it is often useful to see how symmetry can be restored by the condensation of topological defects. Through a duality mapping, Nambu-Goldstone modes are represented as gauge bosons, mediating long-range interactions between topological defects. When the latter condense, those bosons get as mass via the Anderson-Higgs mechanism, which signals the loss of rigidity and the restoration of symmetry. I will first review the best-studied case: the 2+1D superfluid-insulator transitions where the defects are U(1) vortices. Consecutively several extensions are discussed: going to 3+1D where the defects are not point particles but strings, and quantum elasticity, which studies breaking of spatial translations and rotations.

会場: セミナー室 (160号室)

イベント公式言語: 英語

Thermodynamic of a one-dimensional Bose gas at low temperature & superfluidity in neutron-star matter

2018年4月24日(火) 15:30 - 16:30

Giulia De Rosi (The Institute of Photonic Sciences, Barcelona, Spain)

Ultracold atomic gases and neutron-star matter play a paramount importance in many-body physics, by exhibiting important analogies and differences. One of the most interesting phenomena shared by both systems is provided by superfluidity. In the first part of the talk, I will show that the chemical potential of a one-dimensional (1D) interacting Bose gas exhibits a non-monotonic temperature dependence which is peculiar of superfluids. The effect is a direct consequence of the phononic nature of the excitation spectrum at large wavelengths exhibited by 1D Bose gases. For low temperatures T, I demonstrate that the coefficient in the $T^2$ expansion of the chemical potential is defined by the zero-temperature density dependence of the sound velocity and it has been calculated along the crossover between the Bogoliubov weakly-interacting gas and the Tonks-Girardeau gas of impenetrable bosons. The theoretical predictions along the crossover are confirmed by comparison with the exactly solvable Yang-Yang model in which the finite-temperature equation of state is obtained numerically by solving Bethe-ansatz equations. In the second part of the talk, I will show the results of a study of the superfluid gap in pure neutron matter, associated with the formation of Cooper pairs in the $^1S_0$ channel. The interaction responsible of the onset of superfluidity is an effective interaction coming from a nuclear Hamiltonian strongly constrained by phenomenology and obtained from the correlated basis function (CBF) perturbation theory and the formalism of cluster expansions. The calculations have been carried out using an improved version of the CBF effective interaction, in which three-nucleon forces are taken into account using a microscopic model. Our results show that the superfluid transition occurs at values of densities corresponding to the neutron-star inner crust.

会場: セミナー室 (160号室)

イベント公式言語: 英語

基本解近似解法による点渦力学系の数値解析 

2018年4月17日(火) 16:00 - 17:30

榊原 航也 (京都大学 大学院理学研究科 特定助教)

iTHEMS / KUAMSジョイントセミナー

会場: 京都大学 北部総合教育研究棟 1階 益川ホール

イベント公式言語: 日本語

Including an ocean mixing model in atmospheric data assimilation: a case of Typhoon Soudelor 2015

2018年4月17日(火) 15:00 - 15:30

高玉 孝平 (理化学研究所 計算科学研究センター (R-CCS) データ同化研究チーム 特別研究員)

会場: 京都大学 北部総合教育研究棟 1階 益川ホール

イベント公式言語: 英語

Resurgence Theory for Non-Perturbative Quantum Analysis

2018年3月28日(水) 13:00 - 15:00

三角 樹弘 (理化学研究所 数理創造プログラム (iTHEMS) 客員研究員 / 秋田大学 専任講師)

会場: セミナー室 (160号室)

イベント公式言語: 英語

Introduction to exact WKB analysis

2018年3月27日(火) 10:00 - 12:00

岩木 耕平 (名古屋大学)

会場: セミナー室 (160号室)

イベント公式言語: 英語

Anomaly matching for spin chains

2018年3月19日(月) 14:00 - 16:00

谷崎 佑弥 (理化学研究所 仁科加速器科学研究センター (RNC) 理論研究グループ 基礎科学特別研究員)

会場: セミナー室 (160号室)

イベント公式言語: 英語

Several identities related to the degenerate Bernoulli polynomials and numbers

2018年2月24日(土) 16:00 - 17:00

Takao Komatsu (Professor, Wuhan University, China)

In this talk we demonstrate some relations in degenerate Bernoulli polynomials, which may be expressed as a general convolution identity. We also show some properties of hypergeometric degenerate Bernoulli polynomials and numbers.

会場: セミナー室 (160号室)

イベント公式言語: 英語

Diophantine Frobenius Problems — counting theory, generating function and zeta functions

2018年2月23日(金) 16:00 - 17:00

Takao Komatsu (Professor, Wuhan University, China)

When a_1, ...,a_m are relatively prime positive integers, the number of solutions of the linear Diophantine equation a_1 x_1 + … + a_m x_m = b in non-negative integers x_1, ...,x_m, for any integer b, is our concern. We show several formulas to give the largest integer b without solution. Then we discuss the generating function of the number of solutions. Finally, we derive an explicit expression for an inverse power series over the gaps values of numerical semigroups generated by two integers. It implies a set of new identities for the Hurwitz zeta function.

会場: セミナー室 (160号室)

イベント公式言語: 英語

Statistical mechanics of tissue homeostasis

2018年1月22日(月) 14:00 - 15:00

川口 喬吾 (東京大学)

Adult tissues undergo rapid turnover as mature cells are continuously lost and new cells arise through cell division. The balance between the gain and loss of cells must be finely orchestrated to maintain tissues, but how this balance is achieved remains largely unknown. Previous works [1] have used universal scaling laws to claim that the fate choice of stem cells (division or differentiation) are made strictly cell-autonomously. However, we recently recorded every stem cell fate choice within the mouse skin epidermal regions over one week and found that, far from being cell-autonomous, stem cell loss by differentiation is compensated by direct neighboring division [2]. In this talk, I will describe a model of tissue homeostasis using a macroscopic nonequilibrium setup, and explain how the coarse-graining of this model will lead to the effective dynamics of the voter model (DP2). I will show how we can use the property of dynamical crossover in the model -from the cell-autonomous regime (critical birth-death model) to the fate-coordinated regime (voter model)- to measure the length and time scales of stem cell coordination. I will then explain the pitfall in two-dimensions of using scaling relations for the clonal fate trace of cells, and present the workaround we used in the data analysis to definitively show the existence of cell-to-cell fate correlation.

会場: コモンルーム (246-248号室)

イベント公式言語: 英語