Department of Mathematical Sciences, Ritsumeikan University Web Page

Date: 5 December (Thu)

Place: West Wing, 6th floor, Colloquium Room and on the Web (zoom)

Time: 16:30–18:40 

Speaker 1: Jorge Ignacio González Cázares (Universidad Nacional Autónoma de México),16:30–17:30 

Title: Markov Chain Monte Carlo: how and why? II

Abstract 1: We will review classical and widely used Markov Chain Monte Carlo (MCMC) methods. We will begin by reviewing Markov Chain theory and posing the MCMC problem. We motivate the problem from the point of view of stochastic optimisation problems arising in statistics and machine learning. We introduce Metropolis Hastings and Gibbs samplers and show its simplicity using code. We will also hint at its connections with the popular SGD algorithm.

Speaker 2: Andrea Macrina (University College London), 17:40–18:40

Title: Continuous-Time Quantile Processes with Applications in Finance and Insurance

Abstract 2: We develop a novel approach for the construction of quantile processes governing the stochastic dynamics of quantiles in continuous time. Two classes of quantile diffusions are identified: the first, which we largely focus on, features a dynamic random quantile level that allows for direct interpretation of the resulting quantile process characteristics such as location, scale, skewness, and kurtosis, in terms of the model parameters. The second type are function--valued quantile diffusions and are driven by stochastic parameter processes, which determine the entire quantile function at each point in time. By the proposed construction method, quantile processes are obtained by transforming the marginals of a diffusion process under a composite map consisting of a distribution and a quantile function. Sub-classes of quantile diffusions are explored, with emphasis placed on the Tukey family of models whereby skewness and kurtosis are directly parameterised and thus the composite map is explicable with respect to such statistical behaviours. As an example of an application of quantile diffusions, we show how probability measure distortions, a form of dynamic tilting, can be induced. Though particularly useful in financial mathematics and actuarial science, examples of which are given in this work, measure distortions feature across multiple research areas. For instance, dynamic distributional approximations (statistics), non-parametric and asymptotic analysis (mathematical statistics), dynamic risk measures (econometrics), behavioural economics, decision making (operations research), signal processing (information theory), and not least in general risk theory including applications thereof.

日時 ：2024年11月21日（木）16:30 〜 18:40

場所 ：立命館大学BKCウエストウイング6階数理科学科談話会室＆ZOOM

講演者1 : 長谷川武博 先生（滋賀大学）16:30-17:30（質疑応答込み）

講演題目: 「Transcendence of special values of log-type hypergeometric functions attached to Carlitz modules」

講演要旨:
講演者は 2022 年に Carlitz 加群に付随する対数型超幾何関数を定義し、2024年にその関数について代数点での特殊値の代数性・超越性を調べた。 関数を motivic化することで、2024年の結果が進展したので、本講演ではそのことをご紹介します。

 
講演者2 : 篠原雅史 先生（滋賀大学）17:40-18:40（質疑応答込み）

講演題目: 「Erdős distinct distances problem に関係する固有値問題と格子予想」

講演要旨: 
平面上の n 点集合に現れる距離の個数の最小値に関する Erdős(1946) の予想がある。
本講演ではこの予想に関係する格子予想を紹介し、小さいところでの考察を行う。また、この予想に関する幾つかのアプローチについて提案する。

日時 ：2024年11月7日（木）16:30 〜 18:40

場所 ：立命館大学BKCウエストウイング6階数理科学科談話会室＆ZOOM
 
講演者1 : 富田昌 先生（明治安田生命）16:30-17:30（質疑応答込み）

講演題目: 「破産理論におけるCramér-Lundberg modelの一般化とその数学的性質」

講演要旨:
破産理論は損害保険数理の一分野で、保険会社の純資産（サープラス）の変動を確率過程としてモデル化し、そのリスクを分析する。本講演では発生頻度の不確実性と保険料の改定に着目して、古典的なCramér-Lundbergモデルの一般化を提案し、その数学的性質について述べる。提案するモデルは、クレーム発生は混合ポアソン過程に従い、保険料率はベイズ推定により設定され、被保険者は複数である。このモデルにおいて、破産確率がクレーム強度の分布や被保険者数に依存しないことを示し、またクレーム強度で条件づけた場合の破産確率の単調性について述べる。本講演の内容は高岡浩一郎氏（中央大）、石坂元一氏（中央大）との共同研究に基づく。
 

講演者2: 小澤知己 先生（東北大学） 17:40-18:40（質疑応答込み）

講演題目: 「トポロジカル量子ウォーク：平均カイラル変位を例として」

講演要旨:
物性物理学でトポロジカル絶縁体という物質が発見された。これは、電子状態が適切な意味でトポロジカルに非自明になっているような物質であり、物質の表面・端において観測可能な現象につながることが知られている。物質の電子状態はハミルトニアンと呼ばれる自己共役演算子の固有ベクトルの性質と密接に関係する。つまり、適切な意味でハミルトニアンのトポロジーが非自明であるような物質がトポロジカル絶縁体である。電子状態の時間発展はハミルトニアンを含む微分方程式であるシュレディンガー方程式で記述されるが、電子状態の時間発展を量子ウォークととらえるとき、トポロジカルに非自明なハミルトニアンに従う時間発展に対応する量子ウォークがトポロジカル量子ウォークである。本講演ではトポロジカル絶縁体の説明から始めてトポロジカル量子ウォークの基礎的な部分を紹介し、トポロジカル量子ウォーク由来の非自明な現象の例として平均カイラル変位（mean chiral displacement）を解説することを目標とする。講演者は物理学者であり、トポロジカル絶縁体の専門家ではあるが量子ウォークの、特に数学的な側面は専門としていない。講演を通じて異なる分野の科学者による有意義な交流ができることを期待している。

Date: 31 October (Thu)

Place: West Wing, 6th floor, Colloquium Room and on the Web (zoom)

Time: 16:30–18:00

Speaker: Taiho Wang (Baruch College) 

Title: Concentrated Liquidity Provision Market Making

Abstract: 
In this talk, we first review the concentrated liquidity provision amendment to the constant product market making (CPMM), a commonly employed trading protocol populated in automated market making. Unlike the plain CPMM, the feature of concentrated liquidity provision enables liquidity providers (LPs) to specify customized price ranges within which they will provide liquidty for the purpose of maximizing the transaction fee payments collected from the liquidity takers. Should the pool price leave the prespecified price range, transaction fee payments to the LP are suspended until the pool price returns to the price range, if the LP has not withdrawn his liquidity. This new feature introduces a set of new coordinates, the price-liquidity coordinates, under which liquidty is regarded as a function of pool price. This function is referred to as the liqudity profile or the liquidity distribution. We exposit the evolutions of the following quantities: the liquidity profile, the pool reserves, the pool price, as well as payments of transaction fees under continuous liquidity provisions and tradings in this framework. We touch upon the problem of optimal liquidity provision in these circumstances. The talk is based on joint works in progress with Jimmy Risk, Shen-Ning Tung, and Wei-Cheng Wang.

Date: 3 October (Thu)

Place: West Wing, 6th floor, Colloquium Room and on the Web (zoom)

Time: 16:30–18:00

Speaker: Taiho Wang (Baruch College) 

Title: Relative entropy-regularized robust optimal order execution under transient impact

Abstract: 
In this talk, we cast optimal liquidation under linear temporary and transient price impact as a relative entropy-regularized robust optimal control problem. The problem is formulated as to maximize a reward-risk functional associated with the order execution agent’s profit-and-loss of trading and the execution risk taking into account market’s liquidity and uncertainty over a class of absolutely continuous strategies. The problem is made into an entropy-regularized stochastic differential game and is solved by adopting the principle of dynamic programming, yielding that the value function of the differential game satisfies an entropy-regularized Hamilton-Jacobi-Isaacs (rHJI) equation. Under the assumption of aggregate exponential transient impact and Gaussian prior, the rHJI equation reduces to a matrix Riccati differential equation. Further imposing constancy of the corresponding coefficients, the matrix Riccati differential equation can be linearized, resulting in analytical expressions for optimal strategy and trajectory as well as the posterior distribution of market activity. The talk is based on a joint work with Xue Cheng and Meng Wang.

Date: 26th September (Thu)

Place: West Wing, 6th floor, Colloquium Room and on the Web (zoom)

Time: 16:30–18:40

Speaker 1: Sho Kubota（Aichi University of Education）

Title: Grover walk の周期性に関する最近の研究

Abstract: 
量子ウォークはランダムウォークの量子版として導入された数理モデルであり、確率論の枠を超えてグラフ理論、関数解析学、量子情報理論など幅広い分野と密接に関連して研究されている。周期性は量子状態がいくらかの時刻を経て初期状態に戻るという現象であり、特定のグラフにおいてのみ発生する珍しい現象である。本講演では、まず量子について物理的な背景を簡単に説明したうえで離散時間の1次元量子ウォークを導入する。次に、グラフ上の量子ウォークでは定番の Grover walk と、その周期性に関する基本的な結果を紹介する。最後に Grover walk の周期性に関する最近の研究成果や未解決問題を紹介する。


Speaker 2: Sennosuke Watanabe（University of Fukuchiyama）

Date: 19th September (Thu)
Place: West Wing, 6th floor, Colloquium Room and on the Web (zoom)
Time: 16:30–18:00

Speaker: Taiho Wang (Baruch College) 

Title:  Growth rate of wealth in G3Ms 

Abstract:
Geometric mean market makers (G3Ms), such as Uniswap and Balancer, represent a widely used class of automated market makers (AMMs). These G3Ms are characterized by the following rule: the reserves of the AMM must maintain the same (weighted) geometric mean before and after each trade. In this talk, we investigate the effects of trading fees on liquidity providers’ (LP) profitability in a G3M, as well as the adverse selection faced by LPs due to arbitrage activities involving a reference market. Our work expands previous models to G3Ms, integrating both transaction fees and continuous-time arbitrage into the analysis. Within this context, we analyze G3M dynamics, characterized by stochastic storage processes, and calculate the growth rate of LP wealth. We extend earlier results on constant product market maker, commonly referred to as Uniswap v2, as a special case. The talk is based on a joint work with Cheuk Yin Lee and Shen-Ning Tung.

Time: 24 September 2024 (Tue) 16:30–17:30
Place: Colloquium Room, West-Wing 6th floor, BKC
Speaker: John Parker (Durham Univ.)
Title: Fenchel-Nielsen coordinates for SL(3,C) representations of surface groups
Abstract:
In the 1940’s Fenchel and Nielsen gave geometrical coordinates on the Teichmueller space of a closed surface of genus at least two, or equivalently on discrete, faithful, totally loxodromic representations of its fundamental group to PSL(2,R). These coordinates have been generalised to quasi-Fuchsian representations in PSL(2,C), by Kourouniotis and Tan; to convex real projective representations to SL(3,R), by Goldman and later Zhang; and to complex hyperbolic representations to SU(2,1) by Platis and me. In this talk we describe how to give Fenchel-Nielsen coordinates to SL(3,C) representations of surface groups to SL(3,C) and we show how this includes all the previous constructions as special cases, thereby unifying them. This is work from the PhD thesis of my student Rodrigo Davila.