Titles and Abstracts of talks

We study the modified wave equation on the sphere. In previous studies on the Cauchy problem, observed data and velocity of the wave at $t=0$ are given, and the existence and uniqueness of the solution are known. On the other hand, in this study, two observed data of the wave at $t=0$ and $t=T$ are given. In this case, as in the Cauchy problem, we consider whether the solution is uniquely determined. Then, we call this problem “Snapshot Problem”.

We distinguish between the rational and irrational cases of $T/\pi $. In the rational case, we let $T/\pi = q/p$ and present the existence and uniqueness of the solution when p and q satisfy even-odd conditions. In the irrational case, we show the existence and uniqueness of the solution under the assumption we established. Furthermore, we show that depending on the property of Liouville numbers, there are cases where the assumption holds and cases where it doesn’t.

The Cauchy problem and the Snapshot Problem look similar, but the Snapshot Problem requires number theoretical considerations such as Liouville numbers. By changing the framework, we can obtain new mathematical knowledge.

We consider the following shuffle. Let $N$ be an even number. We prepare a deck of $N$ cards, a fair coin, and two packets A and B. We repeatedly take two cards from the top of the deck and toss the coin. If the result is head, we put the first (top) card into packet A and the second (bottom) card into packet B. If the result is tail, we put the first card into packet B and the second card into packet A. After all cards are placed into either packet A or packet B, we place the cards in packet A on top of packet B, without changing their order.

This shuffle is the reverse operation of the shuffle known as the Thorp shuffle. Therefore, we call this shuffle model the “reverse Thorp shuffle”. Instead of studying the whole deck, we focus on the mixing behavior of the positions of a small number of cards under the reverse Thorp shuffle.

Letting $X_{t}^{(N)}$ be the Markov chain representing the position of a single card in a deck of $N$ cards, we show that the sequence of Markov chains $((X_{t}^{(N)})_{t})_{N}$ has a cutoff at $\log_{2} N$ with a window size $O(1)$. Moreover, we consider the subsequence $((X_{t}^{(N)})_{t})_{N}$ restricted to the form $N_{k} = 2^{k} - 2l_{k}$ $(k \ge 2,\ 0 \le l_{k} \le 2^{k-2}-1)$, and give the scaling limit of the total variation distance in a neighborhood of the number of shuffles at which the cutoff occurs. Furthermore, in the case $N = 2^{k}$, we show that the sequence of Markov chains representing the positions of two cards with adjacent initial positions exhibits a phenomenon similar to a cutoff at $k$ with a window size $O(1)$.

This talk is about the geometric properties of special spinors on Kähler manifolds, focusing on the results established by K.-D. Kirchberg. In spin geometry, the existence of twistor and Killing spinors is closely related to the geometric structure of a manifold and plays a crucial role in classification problems. However, it is known that on Kähler manifolds of dimension $n > 2$, Killing spinors are trivial. This necessitates the consideration of special spinors specific to Kähler geometry, such as "Kähler Killing spinors."

In this talk, we present the classification theorem which proves that Kähler manifolds admitting non-trivial Kähler Killing spinors are complex odd-dimensional Einstein manifolds. Furthermore, we provide the necessary and sufficient conditions for the existence of Kähler twistor spinors in terms of specific local differential equations. As a concrete application of these theorems, we demonstrate that these special spinors can be explicitly constructed on the complex projective space $\mathbb{C}P^m$ endowed with the Fubini-Study metric.

In 2018, Kashaev and Virelizier generalized Kuperberg invariant into the case of involutory Hopf algebras in symmetric monoidal categories admitting pairs of two morphisms called good pairs. A good pair can be regarded as a generalization of (co)integrals, and it has different properties. For example, a (co)integral of an involutory Hopf algebra is two-sided, while a good pair may not be two-sided.  In 2024, Shibata, Shimizu and Wakao classified non-semisimple pointed Hopf superalgebras of dimension up to 10.  Based on these two works, we classify good pairs for each involutory pointed Hopf superalgebra of dimension up to 8 over an algebraically closed field of characteristic zero. More precisely, we extract non-trivial involutory Hopf superalgebras from the paper of Shibata, Shimizu and Wakao and construct good pairs for each non-trivial involutory Hopf superalgebra and invertible object.

Various characterizations have been established for transformations defined on measure spaces, including ergodicity, weak mixing, and mixing. It is known that “if a transformation is mixing, then it is weakly mixing; if it is weakly mixing, then it is ergodic.” This generally does not hold true in reverse, and as an example of this, we introduce Chacon's transformation and Induced transformation of Dyadic odometer on Lebesgue Measure Spaces. In particular, we provided a unique proof that the induced transformation of Dyadic odometer is not mixing.

The Hilbert cube is the countable product of the unit interval, where each interval factor is endowed with the standard topology and the normalized Lebesgue measure. We construct a measure-preserving transformation (a continuous map resp.) on the Hilbert cube whose measure-theoretic entropy (topological entropy resp.) is positive and finite.

Our construction is based on the following two results. First, the measure-theoretic entropy (the topological entropy, resp.) of a product map on the Hilbert cube is equal to the sum of the entropies of the factor maps. Second, every positive real number is realized as the measure-theoretic entropy of a piecewise monotone map on the unit interval, and as the topological entropy of a continuous piecewise monotone map on the unit interval.

The realization of measure-theoretic entropy is obtained by explicitly constructing a piecewise monotone map and computing its entropy via symbolic dynamics. The realization of topological entropy follows from a theorem due to Misiurewicz and Szlenk and the result was explicitly stated by Ruette in 2019. Here we focus on the realization problem of topological entropy for real numbers that arise as the largest eigenvalues of certain 0-1 matrices and take a different approach from that of Ruette. We explicitly construct a continuous piecewise monotone map and compute its topological entropy using the corresponding topological Markov shift.

``Sturmian lattice'' is a grid-like structure on the Euclidian plane, which is composed of infinitely many straight lines in three directions. This is almost like a trigonal lattice, but three lines intersect in different ways. In usual trigonal lattices, they meet at a single point; on the other hand, in our Sturmian lattices, they form a tiny regular triangle. See the figures in our preprint [1] on arXiv.

Giving the axiom and finishing classification of Sturmian lattices are of our fundamental interest; due to the apperance of tiny triangles, they possess non-trivial structures. In particular, almost all of them are ``quasi-periodic'', lacking translational symmetry. In this talk, we focus on such a class and give an explicit description for quasi-periodic Sturmian lattices. They can be represented using ``Sturmian words'', a certain class of bi-infinite words on alphabet {0, 1} that is strongly related to irrational rotation on a circle. We also observe their self-similarity, which is described by continued fraction expansion.

This is a joint work with Shigeki Akiyama and Tadahisa Hamada (Univ. of Tsukuba).

[1] S. Akiyama, T. Hamada, and K. Ito, Sturmian lattices and Aperiodic tile sets, https://arxiv.org/abs/2506.19362

• Presentation slides (PDF)

Class field theory, which describes abelian extensions of local fields and global fields, is discussed. The main objective is to introduce the fundamental theorems of local class field theory and global class field theory. In both the local and global cases, the homomorphism $\rho_K$​ is derived by considering the structure of the Brauer group.

In the case of local fields, the structure of the Brauer group is determined using the theory of cyclic algebras. In the case of global fields, the Brauer group is studied via the Hasse reciprocity law.

A major feature of global fields is the use of idèles. Idèles are obtained from the multiplicative groups of local fields at each place. Similarly in class field theory, global class field theory is obtained from the results of local class field theory. Finally, by considering several concrete examples derived from these fundamental theorems, we deepen the understanding of class field theory.

This thesis aims to organize the theory of divisors and the Riemann-Roch theorem in algebraic function fields of one variable, starting from valuation theory, and introduces the proof of the rationality of congruence zeta functions associated with non-singular complete curves over finite fields. The discussion is developed through classical methods without delving into the generalities of scheme theory or modern algebraic geometry. Chapters 1 and 2 follow Kenkichi Iwasawa’s Algebraic Function Theory, while Chapter 3 is based on Dino Lorenzini’s An Invitation to Arithmetic Geometry.  In the first half, we establish valuation theory to organize the concepts of places and divisors in an algebraic function field $K/k$. Places defined through valuations provide a framework for locally describing the zeros and poles of functions. For any divisor $A$, we define the linear space $L(A) = \{x \in K \mid (x) \ge -A\}$, which is finite-dimensional over $k$. By introducing the ring of adeles and differentials by Weil, we define the differential divisor $W$ and present the proof of the Riemann-Roch theorem: $l(A) - l(W - A) = \deg(A) - g + 1$. In the second half, we consider the congruence zeta function $Z(X/\mathbb{F}_q, T)$ associated with a non-singular complete curve $X/\mathbb{F}_q$. We confirm that the zeta function can be expressed using the number of rational points $N_n = \#X(\mathbb{F}_{q^n}) $ as $Z(X/\mathbb{F}_q, T) = \exp(\sum_{n=1}^{\infty} \frac{N_n T^n}{n})$. Finally, we demonstrate that for a curve of genus $g$, the zeta function is a rational function of the form: $Z(X/\mathbb{F}_q, T) = \frac{f(T)}{(1-T)(1-qT)} $.

Let $\fr{g} = \fr{g}(A)$ be the Kac-Moody algebra associated with a symmetrizable and irreducible generalized Cartan matrix $A$. Let $\Delta$ to be the root system of $\fr{g}$ and $\Delta_{\mathrm{im}}$ to be the set of imaginary roots of $\fr{g}$. For $\alpha\in \bar{\Delta} := \Delta \cup \{0\}$ and $\beta\in \Delta$, $S_{\alpha}(\beta)$ denotes the maximal interval of $\Z$ including $0$ such that $\alpha + p\beta \in \bar{\Delta}$ for every $p\in S_{\alpha}(\beta)$. The set $R_{\alpha}(\beta) := \{\alpha + p\beta \mid p\in S_{\alpha}(\beta)\}$ is called the $\beta$-root string through $\alpha$, which is studied by Lisa Carbone at al. However, there are still many things we do not know about root multiplicities and the structure of root strings. In this thesis, building on their work, we investigate the structure of the set $\{p\in \Z \mid \alpha + p\beta \in \bar{\Delta}\}$ and the asymptotic behavior of $\dim \fr{g}_{\alpha \pm n\beta}$ as $n\to \infty$.

Let $\mathfrak{g}=\mathfrak{g}(A)$ be the Kac--Moody algebra associated with a symmetrizable Cartan matrix $A$, and let $\mathfrak{h}$ be its Cartan subalgebra. For $\lambda\in\mathfrak{h}^*$, define the root space $\mathfrak{g}_{\lambda} = \{ x\in\mathfrak{g} \mid [h,x] = \lambda(h)x \,(h \in \mathfrak{h}) \}$. Then $\mathfrak{g}$ admits the root space decomposition $\mathfrak{g} = \bigoplus_{\lambda \in \mathfrak{h}^{*}}\mathfrak{g}_{\lambda}$, where each $\mathfrak{g}_\lambda$ is finite-dimensional. The dimension $\dim\mathfrak{g}_\lambda$ is called the root multiplicity, which is the main object of study in this paper. When the Cartan matrix $A$ is of finite or affine type, root multiplicities are well understood. In contrast, for indefinite type, much less is known. Although formulas for root multiplicities are known in the indefinite case, they involve complicated partition functions, which motivates the study of effective upper bounds. A classical upper bound is given by Frenkel's conjecture, which is known to fail in general. On the other hand, in the rank two hyperbolic case, Kang--Lee--Lee obtained explicit formulas and sharper upper bounds using combinatorial methods. Chen--Luo--Sun proposed a combinatorial formula for root multiplicities in the rank three acyclic case; however, their proof contains gaps and the formula does not hold in general. In this paper, we describe an inductive method for computing root multiplicities and present explicit counterexamples to their main result. We also examine their proof in detail and point out several problematic arguments. Finally, based on their approach, we aim to obtain upper bounds for root multiplicities.

Let $\mathfrak{g}=\mathfrak{g}(A)$ be the Kac-Moody Lie algebra associated with a Cartan matrix $A$. Let $\Delta$ denote the set of all roots of $\mathfrak{g}$ and $\Delta_{\mathrm{re}}$ the set of all real roots. A subset $\Sigma \subseteq \Delta$ is called a $\pi$-system if $\alpha - \beta \notin \Delta$ for all $\alpha, \beta \in \Sigma$. $\pi$-systems are fundamental in the study of Kac-Moody Lie algebras as they naturally appear in embedding problems. The concept of a $\pi$-system generalizes the properties of a simple root system and was introduced by Eugene B. Dynkin during the classification of regular subalgebras of finite-dimensional semisimple Lie algebras. Subsequently, this concept was extended to all symmetrizable Kac-Moody Lie algebras by Jun Morita and Satoshi Naito. Recently, Irfan Habib and Chaithra Pilakkat completely classified the $\pi$-systems contained in $\Delta_{\mathrm{re}}$ for Kac-Moody Lie algebras associated with rank $2$ hyperbolic Cartan matrices. Furthermore, for the specific family of Cartan matrices $A_{a}$ ($a \ge 3$), they constructed $\pi$-systems $\Sigma_{1}$ and $\Sigma_{2}$ containing imaginary roots. However, the sizes of these sets, $|\Sigma_{1}|$ and $|\Sigma_{2}|$, depend on the entry $a$ of the matrix. In this presentation, building on their work, we investigate the existence of a $\pi$-system $\Sigma$ containing imaginary roots such that $| \Sigma | = n $ for any $n \in \mathbb{Z}_{\ge 1}$. In particular, we focus on the case $a=3$, where the sets $\Sigma_{1}$ and $\Sigma_{2}$ are empty.

Two-dimensional supersymmetric quantum field theories(SQFT) with minimal $\mathcal{N}=(0,1)$ supersymmetry induced some interests recently due to their close relationship with topological modular forms(TMF). Elliptic genera, which may be seen as either a rationalized version of TMF or a 2d generalization of Witten indices, play an important role for both mathematical and physical consideration on (0,1) SQFTs. 

In this work, I will explain our recent work on a multi-residue-type formula for elliptic genera of (0,1) Gauged Linear Sigma Models(GLSM). This generalizes earlier results  [1308.4896] of Benini, Eager, Hori and Tachikawa for (0,2) GLSMs. We also provide some validity check by applying it to the family of (0,1) GLSMs proposed by Gukov, Pei and Putrov in [1910.13455]. 

This presentation is based on the work [2508.06865] in collaboration with Jiakang Bao and Masahito Yamazaki.