歷史演講 PAST TALKS

摘要詳如附件

20230707演講_Prof. Muthukumar P (IIT Kanpur, India)

摘要詳如附件

20230630演講_黎懷仁先生 (國立中央大學數學系)

Abstract：

The Parisi formula is a fundamental result in spin glass theory. In this talk I will present a new approach to (an enhanced Version of) Guerra's identity, from which the upper bound in the Parisi formula follows immediately. Among the techniques from stochastic analysis we will use include path space integration parts for the Wiener measure (Brownian motion), the Girsanov transform (exponential martingales), and the Feynman-Kac formula. The key observation is that the nonlinear partial differentiation equation figuring in Parisi’s variation formula becomes linear after differentiating with respect to a parameter, thus allowing the full strength of stochastic analysis based on Ito’s calculus into play. We hope that this approach will shed some lights on the much more difficult lower bound in the Parisi formula.

20230615演講_Prof. Elton P. Hsu (Northwestern University)

Abstract

For a general entropy-regularized stochastic control problem on an infinite horizon, we prove that a policy improvement algorithm (PIA) converges to an optimal relaxed control. Contrary to the standard stochastic control literature, classical Holder estimates of value functions do not ensure the convergence of the PIA, due to the added entropy-regularizing term. To circumvent this, we carry out a delicate estimation by moving back and forth between appropriate Hölder and Sobolev spaces. This requires new Sobolev estimates designed specifically for the purpose of policy improvement and a nontrivial technique to contain the entropy growth. Ultimately, we obtain a uniform H ̈older bound for the sequence of value functions generated by the PIA, thereby achieving the desired convergence result. Characterization of the optimal value function as the unique solution to an exploratory Hamilton-Jacobi-Bellman equation comes as a by-product.

20230609演講

Abstract：

Reinforcement learning (RL) is an active subarea in machine learning, and has been developed predominantly for Markov decision processes (MDPs) in discrete time with discrete state and action spaces. In this talk, I will report the recent development of RL theory in continuous time and state space with possibly continuous action space, focusing on the q-learning theory. The theory centers around various martingale conditions that lead naturally to on-policy and off-policy RL algorithms. Time permitting I will also present applications to data-driven portfolio selection.

20230608演講_Prof. Xunyu Zhou (Columbia University)

Abstract：

Images captured in low-light conditions often suffer from uneven illumination, low contrast, and poor visibility. In addition to easily resulting in unsatisfactory images for human vision, these undesirable effects also degrade the performance of algorithms in many image processing and computer vision applications. To improve the quality of such degraded images, various image enhancement methods have been studied extensively in the past few decades. In this talk, we will briefly introduce some models for low-light image enhancement.

20230518演講_謝博文教授 (國立中興大學應用數學系)

我們考慮下列一維的隨機微分方程：

對應到的起始值為正數。這方程的係數函數並不滿足利普希茨連續，所以不能用一般已知的定理得到強解的存在唯一性，但我們可以將係數函數截斷使其滿足利普希茨連續，得到截斷後方程強解的存在與唯一性，再取極限得到原方程強解的存在唯一性，而這解為一強馬氏過程。這篇論文主要想要研究解的長時間行為，而這會與解中的λ有關。利用解的比較定理，我們證明當λ大於根號2的時候，只有一個顯然的不變測度；當λ小於2的時候，存在了一個非顯然的不變測度。利用耦合的方法，我們得到藉由遍歷性可以得到全變差距離收斂。另外，我們也得到強範數為指數型收斂。

20230516演講_楊氏三妹小姐 (國立中央大學數學系)

Abstract：

Polarized morphisms are important objects in various mathematical and computational fields, and their composition is a fundamental operation. In this talk, we present a new algorithm to efficiently compute the composition relationship of two polarized morphisms using the concept of dynamical canonical height. Our approach utilizes the properties of dynamical systems and the theory of heights to represent polarized morphisms in a computationally efficient manner, while still allowing for flexibility in handling more complex cases. We also provide an analysis of the time and space complexity of our algorithm and compare its performance to existing approaches on a set of benchmarks. Our experimental results suggest that our proposed algorithm has the potential to outperform state-of-the-art methods and can handle a wide range of cases, which could make it a valuable tool for practical applications of polarized morphisms composition.

20230504演講_彭俊文博士（國家科學理論研究中心）

Abstract：

We discuss some results on multicomponent coupled KPZ equation such as analytic results from the viewpoint of singular SPDE (jointly with Masato Hoshino, JFA 2017), derivation from interacting particle systems (jointly with Cedric Bernardin and Sunder Sethuraman, AAP 2021), its decoupleability and an application of Littlewood-Paley-Stein inequality to show a better L^p-estimate.

20230502演講_舟木直久教授 (北京雁栖湖應用數學研究院研究員、東京大學名譽教授)

Abstract：

Over the past 25 years, meshfree methods have emerged as a promising class of computational techniques, with considerable success. The reproducing kernel particle method (RKPM) and moving least squares (MLS) are examples of meshfree methods that offer flexible ways to construct basis functions with higher-order continuity, arbitrary order of completeness, implicit smooth derivative, and control of locality, making them effective for solving various partial differential equations (PDEs). Due to their "meshfree" nature, these methods have proven to be robust in handling computational mechanics problems involving extreme conditions such as high strain-rates, strong hydrodynamical effects, high incompressibility, and damage/fragmentation processes. These extreme phenomena are commonly encountered in event analysis such as landslides, air blasts, tsunami, and structural failures, which are difficult or even impossible to model using conventional numerical tools like the finite element method (FEM) due to severe mesh distortion or entanglement. In this presentation, we will introduce the state-of-the-art meshfree particle formulations, including the development of robust stabilization and quadrature rules that enhance accuracy, stability, and efficiency. We will also present different extreme event studies to benchmark the effectiveness of the proposed methods. Additionally, if time permits, we will discuss the latest developments in machine learning techniques from a meshfree perspective.

20230420演講_黃琮暉教授 (國立清華大學動力機械工程學系)

Abstract：

The study of the value distribution of the Riemann zeta function ζ(s) is central to analytic number theory, and the moment estimate is one of the important topics. The second and fourth moments ζ(1/2+it) were established asymptotically by Hardy-Littlewood (1918) and Ingham (1926), respectively, and there has been folklore that sixth and higher moments are beyond current techniques. Nonetheless, a few years ago, Ng proved an asymptotic formula for the sixth moment under a conjecture for ternary additive divisor sums.  

In this talk, I will review some background knowledge and then explain how the Riemann hypothesis and a conjecture for quaternary additive divisor sums imply the conjectural asymptotic for the eighth moment of the Riemann zeta function. This is joint work with Nathan Ng and Quanli Shen, and it builds on the above-mentioned work of Ng. A key new idea is to use sharp bounds for shifted moments of the zeta function on the critical line.

20230330演講_翁鵬絜教授（國立中山大學應用數學系）

Abstract：

We consider an optimal consumption and investment problem on a finite time horizon to optimize the discounted expected exponential utility of consumption and terminal wealth. We treat a stochastic factor model that the mean returns of risky assets depend linearly on underlying economic factors formulated as the solutions of linear stochastic differential equations. The problem becomes a standard stochastic control problem. We derive the Hamilton--Jacobi--Bellman (HJB) equation and obtain an explicit solution. Using this solution, we construct the optimal strategy and obtain the optimal value explicitly.

20230323演講_畑 宏明教授（Graduate School of Business Administration, Hitotsubashi University, Japan）

Abstract：

The Nielsen realization problem was originally a question posted by Jakob Nielsen in 1932 and answered by Steven Kerckhoff in 1983. The question asks, given an oriented topological surface, whether every finite subgroup of its mapping class group can be realized as an isometry group of a hyperbolic metric on that surface or not. Equivalently, it asks whether the action of the finite subgroup on the moduli space of hyperbolic metrics, called the Teichmüller space, has a fixed point or not. In recent years, the classical Teichmüller theory has guided, in various aspects, the development of the theory of stability conditions on triangulated categories. In this talk, I will introduce a categorical analogue of the Nielsen realization problem and its state of art.

20230316演講_賴冠文博士（University of Bonn, Germany）

Abstract：

In this talk, we introduce L-functions based on examples. We consider L-functions associated to three types of objects in number theory: Galois representations, elliptic curves, and modular forms. For a given object, we have local information provided by each prime number p, namely the trace of Frobenius at p, the number of solutions modulo p, and the Satake parameters at p, respectively. The L-function associated to the object is defined by glue together these local information. On the other hand, we have global invariant associated to the object, namely the irreducibility, the rank of the group of rational points, and the periods, respectively. In the literature, we have the celebrated Artin conjecture, Birch–Swinnerton-Dyer conjecture, and Deligne conjecture which relate these global invariants to special values of L-functions. If time permits, we also introduce our recent work on Deligne conjecture for symmetric power L-functions of modular forms.

20230309演講_陳昰宇博士（Department of Mathematics, Kyoto University）

Abstract：

Minimal representations of reductive groups have been studied for long time and are of significant importance in number theory. The most famous example is the Weil representation of the metaplectic double cover of the symplectic group, which plays a pivotal role in automorphic form and theta correspondence. In this talk, we will first discuss some history of the development of representation theory of reductive groups. Then, more specifically, we will discuss non-minimal small representations arising from a nonlinear double cover of various real reductive groups. We will study some algebraic properties that characterize these small representations. These small representations are expected to be part of the building blocks of unitary representations. The talk is based on some recent results appearing in my papers and joint work with Dan Barbasch.

20230221演講_蔡宛育教授 (中原大學應用數學系)