This biweekly seminar series is jointly organized by Academia Sinica and National Taiwan University.
The idea of the series is to present an introduction to certain basic mathematical concepts for nonexperts and graduate students.
The aim is not for the audience to become experts on the subject, but rather to develop some working knowledge of the concepts presented.

2020/06/15 謝銘倫 MingLun Hsieh (AS)
Ideal class groups, elliptic curves and zeta functions
Finding integral or rational solutions of a polynomial equation with integral coefficients is of fundamental interest in number theory.
I will begin with famous equations from Fermat’s last theorem and the congruent number problem and explain their connection with ideal class groups, elliptic curves and zeta functions.

2020/10/12 蔡政江 ChengChiang Tsai (AS)
Some aspects of Langlands program
In this presentation, we aim to introduce the Langlands program, still largely conjectural but said to unify number theory with some algebraic geometry and representation theory.
We will begin with elementary examples like integral solutions to $x^2+y^2=n$ and to number of mod p solutions to $y^2y=x^3x^2$ for all primes $p$ altogether.
We show that the former questions are surprisingly related to some mysterious (modular) holomorphic functions, and explain how the relation fits into a unified scheme given by the Langlands program.
If time permits, we also hope to mention some explicit result of the program when we focus on a single prime $p$.

2020/10/26 夏俊雄 ChunHsiung Hsia (NTU)
On the development of synchronization theory
Synchronization is a pervasive phenomena which has been observed in biological, chemical, physical and social systems. The first reported observation of synchronization dates back to the 17th century;
a Dutch scientist, Christiaan Huygens has discovered in 1665 that two pendulum clocks hanging on the wall have always ended up swinging in exactly the opposite direction from each other.
Since then, various synchronization phenomena have been reported. These include circadian rhythms, chirping crickets, flashing fireflies, croaking frogs, electrical generators,
Josephson junction arrays, intestinal muscles and menstrual cycles. In this talk, we shall introduce the development of the mathematical theory and some scientific applications of synchronization theory.

2020/11/9 賴俊儒 ChunJu Lai (AS)
What is categorification?
The term ``categorification'' originated in the 1994 work of Crane and Frenkel in the study of topological quantum field theory.
Vaguely speaking, it is the process of realizing certain algebraic structures as shadows of richer higher ones.
To people's surprise, the notion of categorification becomes a rather universal mathematical phenomenon in knot theory and representation theory.
In this talk, I will focus on examples and applications of categorification in representation theory.

2020/11/23 余正道 JengDaw Yu (NTU)
Exponential motives and irregular Hodge theory
Both theories in the title start from the investigations of the cohomological properties of varieties equipped with potential functions. I give some motivations and indicate current status of the theories.

2020/12/14 鄭日新 JihHsin Cheng (AS)
The Dirac equation  spinor, mass and mean curvature
I would like to give a brief introduction to the Dirac equation and its applications
to a proof of positive mass theorem in general relativity and several complex variables, and a proof
of Alexandrov's theorem in the geometry of constant mean curvature.

2020/12/28 吳浩榳 Hautieng Wu (Duke)
Browse data science with differential geometry and random matrix theory
Manifold learning and random matrix theory have been actively developed in past decades.
I will overview some topics in data science, particularly the manifold learning, through the lens of manifold learning and random matrix theory.
Some open mathematical problems toward statistical inference and some current progress, like L∞ spectral convergence,
local law and rigidity of eigenvalue distribution of graph Laplacian under the manifold setup, will be discussed.
The backbone of the talk will be a clinical challenge, analyzing modern longterm physiological signals.

2021/2/22 蕭欽玉 ChinYu Hsiao (AS)
Szegő kernels in complex and CR geometry
The study of Szegő kernels on CR manifolds has a profound impact in many research areas:
several complex variables, geometric quantization, Kähler geometry.
In this talk, I will introduce some basic notions in CR geometry,
some classical results about Szegő kernels and will mention some applications of Szegő kernels in complex geometry and geometric quantization.
I will also mention some of my results about Szegő kernels asymptotic expansions.

2021/3/15 佐藤信夫 Nobuo Sato (NTU)
An introduction to the theory of multiple zeta values
Multiple zeta values are multiplesum generalizations of Riemann zeta function evaluated at positive integers, which form a fundamental class of numbers.
These numbers satisfy various linear/algebraic relations, and their entire structure is not completely understood and has a deep connection to the Grothendieck–Teichmuller theory.
I will introduce both the combinatorial and theoretical aspects of multiple zeta values and talk about recent developments of the theory.

2021/3/29 李元斌 YuanPin Lee (AS) (postponed to 2021 Fall)
Moduli and Invariants, from the perspective of GromovWitten theory
This lecture will review the concept of moduli spaces and the “modern” technique of employing moduli spaces to define invariants for the purpose of classification.
After that, the theory of GromovWitten invariants will be discussed from this viewpoint. This is a repeat talk from my 2020 TMS lecture.

2021/4/12 陳逸昆 IKun Chen (NTU)
A revisit of the velocity averaging lemma: On the regularity of stationary linearized Boltzmann equation
We adopt the idea of velocity averaging lemma to establish regularity for stationary linearized Boltzmann equations in a bounded convex domain.
Considering the incoming data, with three iterations, we establish regularity in fractional Sobolev space in space variable up to order 1.
This take is based on a collaboration with PingHan Chuang, ChunHsiung Hsia, and JheKuan Su.

2021/4/26 林正洪 ChingHung Lam (AS)
Monster and Moonshine
I will first explain the two terms (Monster and Moonshine) in the title and give some historical remarks.
Then I will explain how the theory of vertex operator algebras can be used to understand these two objects.

2021/5/10 蔡忠潤 ChungJun Tsai (NTU)
Mean curvature flow
This is an introductory talk to the mean curvature flow, which is the gradient flow of the volume functional.
The focus will be on some stories in the hypersurface case.

2021/5/24 程之寧 Miranda Cheng (AS/Amsterdam)

2021/6/07 沈俊嚴 ChunYen Shen (NTU)