Kenichi Bannai, Keio University/RIKEN
Pure Mathematicians meet Machine
Learning
I have worked in the field of arithmetic geometry ever since my graduate studies. My speciality is the Bloch-Beilinson-Kato conjecture concerning the special values of Hasse-Weil L-functions associated to algebraic varieties defined over number fields. However, in 2016, on the occasion of the founding of AIP, a new research project in artificial intelligence and machine learning affiliated with RIKEN, I was suddenly asked to organize a team of pure mathematicians to conduct research in Machine Learning. In this talk, I will describe the circumstance of the formation of our team, the Mathematical Science Team, as well as some of the recent projects we are undertaking.
Lung-Chi Chen, National Chengchi University
The Critical Behavior for
Percolation and the Mean Field Behavior for Oriented Percolation in High Dimensions
In this talk, we consider percolation on the $d$-dimensional integral lattice
$\mathbb{Z}^d$, and oriented percolation on the $d$-dimensional
body-centered cubic (BCC) lattice $\mathbb{L}^d$ in high dimensions.
For percolation, we focus on long-range models, whose bond-occupation probability is proportional to
$D$. Suppose that D(x) decays as $|x|^{-d-\alpha}$ for some
$\alpha>0$. Thanks for lace expansion, we can estimate the bootstrapping functions in the
lace-expansion analysis and obtain that the following critical two-point function
$$
G_{p_c}(x):=P_{p_c}(o\leftrightarrow x)\sim\begin{cases}
C_\alpha |x|^{\alpha\wedge2-d}, &\mbox{ $d>3(\alpha\wedge 2)$ and $\alpha\neq 2$},\\
C_2 |x|^{2-d}/\log|x|,&\mbox{$d\geq6$ and $\alpha=2$},
\end{cases}
$$
where the constant $C_\alpha\in(0,\infty)$ is depending only on $\alpha$ and $L$.
For oriented percolation, we focus on nearest-neighbor oriented percolation with independent
Bernoulli bond-occupation probability on $\mathbb{L}^d$ and the set of non-negative integers
$\mathbb{Z}_+$.
Thanks to the nice structure of the BCC lattice, we obtain that the infrared bound holds on
$\mathbb{L}^d\times\mathbb{Z}_+$ in all dimensions $d\geq 9$.
As opposed to ordinary percolation, we have to deal with the complex numbers due to asymmetry
induced by time-orientation, which makes it hard to estimate the bootstrapping functions in the
lace-expansion analysis from above. It’s the joint work with Akira, Handa at Hokkaido university and
Kamijima at NCTS.
Chiun-Chuan Chen, National Taiwan University
Diffusive Lotka-Volterra
competition systems
In this talk, we consider the wave phenomenon of two and three species Lotka-Volterra competition systems and present the results collaborated with or inspired by Professor Masayasu Mimura and his collaborators during the past years. In particular, we investigate the problems related to exact solutions, 2-species singular limit problems with “latent heat” effect, and 3-species non-monotone waves.
Sijong Kwak, Korea Advanced Institute of Science and Technology
Higher secant
varieties of minimal degree and del Pezzo higher secant varieties
There are two basic objects in projective algebraic geometry: one is a variety of minimal degree and the other is a del Pezzo variety. In this talk, I'd like to introduce higher secant varieties of minimal degree and del Pezzo higher secant varieties to nonexpert with modest backgrounds. Classification and characterization of such varieties have been focused recently. I will also introduce many interesting examples explaining main results.
Chi-Kwong Li, College of William and Mary
Numerical range techniques in
quantum information science
In this talk, we will discuss how to use numerical range techniques to study problems arising in quantum information sciences such as quantum tomography, quantum error corrections. Recent results are open problems will be mentioned.
Matthew M. Lin, National Cheng Kung University
Low Rank Approximation of
Entangled Bipartite Systems
Gauging the distance between a mixed state and its nearest separable state is important but challenging in the quantum mechanical system. We, in this talk, propose a dynamical system approach to tackle low rank approximation of entangled bipartite systems, which has several advantages, including 1) A gradient dynamics in the complex space can be described in a fairly concise way; 2) The global convergence from any starting point to a local solution is guaranteed; 3) The requirement that the combination coefficients of pure states must be a probability distribution can be ensured; 4) The rank can be dynamically adjusted. The theory, algorithms, and some numerical experiments will be presented in this talk.
Mao-Pei Tsui, National Taiwan University
Data, Geometry and Randomness
Modern data science uses geometric analysis to find the structural features of data sets before further supervised or unsupervised analysis. Geometry are very natural tools for analysing massive amounts of data since geometry can be regarded as the study of distance functions. The point clouds are finite samples taken from a geometric object, perhaps with noise. We will explain how geometry and probability can be used to study in understanding the structure of data. We will also show some examples to demonstrate the success and limitation in real application.
Hau-Tieng Wu, Duke University
Some recent advances in time-frequency analysis
methods for biomedical signals
Analysis of signals with oscillatory modes with crossover instantaneous frequencies is a challenging problem in time series analysis. One way to handle this problem is considering the chirplet transform (CT) that lifts the 2-dimensional time-frequency representation to a 3-dimensional representation, called time-frequency-chirp rate (TFC) representation, by adding one extra chirp rate parameter so that crossover frequencies are disentangles in the higher dimension. However, in practice we found that CT has a strong "blurring" effect in the chirp rate axis, which limits its application in real world data. With my PhD student Dr. Ziyu Chen, we propose the synchrosqueezed chirplet transform (SCT) that enhances the TFC representation given by the CT. The resulting concentrated TFC representation has a high contrast so that one can better distinguish different modes and even crossover instantaneous frequencies. The basic idea is using the phase information in the TFC representation to determine a reassignment rule that reallocates coefficients to the "true" frequency and chirp rate of the signal. Besides providing various numerical results, we also provide theoretical guarantees of SCT based on the well-known oscillatory integral results in the harmonic analysis society.
Bo-Yin Yang, Academia Sinica
Postquantum Cryptography and the NIST
Competition
Shor's algorithm with a large-scale quantum computer will break all currently deployed public-key cryptography. In anticipation, the U.S. National Institute of Standards and Technology (NIST) is calling for new standard Key Establishment Methods (KEMs) and Digital Signature Schemes (DSSs) in an open competition to determine the next generation standards for public-key cryptography. The result will no doubt decide what the internet will be using for the next two decades. We will go over the story of postquantum cryptography and the competition, and will describe some of the NISTPQC candidates as well some mathematics behind them.
Shih-Hsien Yu, Academia Sinica
Green's function and Path integral
In this talk, one presents a way to build a Green's function of a heat equation for heterogeneous media.
Jeng-Daw Yu, National Taiwan University
Moments of Airy Functions as Ulterior
Motives
Anderson introduced the arithmetic Hodge structures and ulterior motives in order to factorize the motives of Fermat hypersurfaces into simpler, although non-classical, objects. We indicate that such structures appear as a special case of irregular Hodge structures, and in addition, show that the symmetric moments of classical Airy functions provide an interesting example. Joint work with Claude Sabbah.