# Seminars & Colloquia

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Let $X$ be an abelian variety of dimension $g$ over a field $k$. In general, the group $\textrm{Aut}_k(X)$ of automorphisms of $X$ over $k$ is not finite. But if we fix a polarization $\mathcal{L}$ on $X$, then the group $\textrm{Aut}_k(X,\mathcal{L})$ of automorphisms of the polarized abelian variety $(X,\mathcal{L})$ over $k$ is known to be finite. Then it is natural to ask which finite groups can be realized as the full automorphism group of a polarized abelian variety over some field $k.$ In one of earlier works, a complete classification of such finite groups was given for the case when $k$ is a finite field, $g$ is an odd prime, and $X$ is simple. One interesting thing is that the maximal such a finite group was a cyclic group of order $4g+2$ only when $g$ is a Sophie Germain prime. Another notable thing is the fact that the abelian variety that was constructed to achieve the maximal cyclic group splits over $\overline{k}$, an algebraic closure of $k.$ \\
In this talk, we provide a construction of an absolutely simple abelian variety of dimension $g$ ($g$ being a Sophie Germain prime) over a finite field $k$, which attains the maximal automorphism group. This can be regarded as the counterpart for the previous construction. Also, we briefly describe the asymptotic behavior of the characteristic of the base field $k$ for which we can give such a construction. Finally, if time permits, we take a closer look at the case of $g=5$ by introducing the Newton polygon of an abelian variety of dimension $5.$
(If you want to join the seminar, please contact Bo-Hae Im to get the zoom link.)

This talk will be presented online. Zoom link: 709 120 4849 (pw: 1234)
Abstract: Life science has been a prosperous subject for a long time, and is still developing with high speed now. One of its major aims is to study the mechanisms of various biological processes on the basis of biological big-data. Many statistics-based methods have been proposed to catch the essence by mining those data, including the popular category classification, variables regression, group clustering, statistical comparison, dimensionality reduction, and component analysis, which, however, mainly elucidate static features or steady behavior of living organisms due to lack of temporal data. But, a biological system is inherently dynamic, and with increasingly accumulated time-series data, dynamics-based approaches based on physical and biological laws are demanded to reveal dynamic features or complex behavior of biological systems. In this talk, I will present a new concept “dynamics-based data science” and the approaches for studying dynamical bio-processes, including dynamical network biomarkers (DNB), autoreservoir neural networks (ARNN) and partical cross-mapping. These methods are all data-driven or model-free approaches but based on the theoretical frameworks of nonlinear dynamics. We show the principles and advantages of dynamics-based data-driven approaches as explicable, quantifiable, and generalizable. In particular, dynamics-based data science approaches exploit the essential features of dynamical systems in terms of data, e.g. strong fluctuations near a bifurcation point, low-dimensionality of a center manifold or an attractor, and phase-space reconstruction from a single variable by delay embedding theorem, and thus are able to provide different or additional information to the traditional approaches, i.e. statistics-based data science approaches. The dynamical-based data science approaches will further play an important role in the systematical research of biology and medicine in future.

Elie Cartan's celebrated paper ˝Pfaffian systems in 5 variables˝ in 1910 studied the equivalence problem for general Pfaffian systems of rank 3 in 5 variables as the curved version of the Pfaffian system with G_2 symmetry. The G_2 case admits a beautiful geometric correspondence with Engel's PDE system. We give a historical overview of Cartan's paper and discuss recent works extending the correspondence to curved cases, which is based on ideas from geometric control theory and algebraic geometry.

In this talk, I will present a result on the existence of 2-dimensional subsonic steady
compressible flows around a finite thin profile with a vortex line at the trailing edge, which is a
special case in the celebrated lifting line theory by Prandtl. Such a flow pattern is governed the
two-dimensional steady compressible Euler equations. The vortex line attached to the trailing
edge is a free interface corresponding to a contact discontinuity. Such a flow pattern is obtained
as a consequence of structural stability of a uniform contact discontinuity. The problem is
formulated and solved by an application of the implicit function theorem in a suitable weighted
space. The main difficulties are the possible singularities at the fitting of the profile and the
vortex line and the subtle instability of the vortex line. Some ideas of the analysis will be
presented. This talk is based on joint works with Jun Chen and Aibin Zang at Yichun University.
The research is supported in part by Hong Kong Earmarked Research Grants CUHK 14305315,
CUHK 14302819, CUHK 14300917, and CUHK 14302917.

https://kaist.zoom.us/j/3098650340

https://kaist.zoom.us/j/3098650340

This is part IV of the series of lectures on the foundations of infinity-categories.
n the first half, we'll cover the precise definition of infinity-categories based on quasi-categories. Some relationship between infinity-categories and model categories will be presented to help better understand the theory of infinity-categories in the remaining half.

Zoom ID: 352 730 6970, Password: 9999

Zoom ID: 352 730 6970, Password: 9999

This talk will be presented online. Zoom link: 709 120 4849 (pw: 1234)
Age brings the benefit of experience and looking back at my job as a professor, there are a couple of things that fall into the category “I wish someone had told me that earlier”. In this seminar, I would like to share some of the things I learned and which, I hope, will be useful for younger scientists.
The questions I will touch upon include
What is productivity, for a scientist?
What are qualities of successful people?
How can one create motivation and success?
How to organize myself? (project management; getting things done)
How to communicate effectively?
Seeking fulfillment
The seminar is targeted at PhD students, postdocs, and junior group leaders.

In this talk, we present an algebraic and graph theoretic (data-based) image inpainting algorithm. The algorithm is designed to reconstruct area or volume data from one and two dimensional slice data. More precisely, given one or two dimensional slice data, our algorithm begins with a simple algebraic pre-smoothing of the data, constructs low dimensional representation of pre-smoothed data via Dynamic Mode Decomposition, performs initial area or volume reconstruction via interpolation, and finishes with smoothing the outcome using a constraint bilateral smoothing. Numerical experiments including MRI of a three year old and a CT scan of a Covid-19 patient, are presented to demonstrate the superiority of the proposed techniques in comparisons with other commercial and published methods. Some further applications we are currently doing will also be presented.
This work is jointly done with Gwanghyun Jo and Ivan Ojeda-Ruiz.

Oscillatory signals are ubiquitously observed in many different intracellular systems such as immune systems and DNA repair processes. While we know how oscillatory signals are created, we do not fully understand what a critical role they play to regulate signal pathway systems in cells. Recently by using a stochastic nucleosome system, we found that a special signal (NFkB signal) in an immune cell can enhance the variability of the immune response to inflammatory stimulations when the signal is oscillatory. Hence in this talk, we discuss the roles of oscillatory and non-oscillatory NFkB signals in an inflammatory system of immune cells as the main example for revealing the role of oscillatory signals. And then we will talk about how this finding can be generalized for other intra- or extra-cellular systems to study why cells use oscillations.

A notion of sublinear expander has played a central role in the resolutions of a couple of long-standing conjectures in embedding problems in graph theory, including e.g. the odd cycle problem of Erdos and Hajnal that the harmonic sum of odd cycle length in a graph diverges with its chromatic number. I will survey some of these developments.

The standard machine learning paradigm optimizing average-case performance performs poorly under distributional shift. For example, image classifiers have low accuracy on underrepresented demographic groups, and their performance degrades significantly on domains that are different from what the model was trained on. We develop and analyze a distributionally robust stochastic optimization (DRO) framework over shifts in the data-generating distribution. Our procedure efficiently optimizes the worst-case performance, and guarantees a uniform level of performance over subpopulations. We characterize the trade-off between distributional robustness and sample complexity, and prove that our procedure achieves this optimal trade-off. Empirically, our procedure improves tail performance, and maintains good performance on subpopulations even over time.

Morel and Voevodsky constructed the A^1-motivic homotopy category SH(k).
One purpose of motivic homotopy theory is to incorporate various cohomology theories into a single framework so that one can discuss relations between cohomology theories more efficiently.
However, there are still lots of non A^1-invariant cohomology theories, e.g. Hodge cohomology and topological cyclic homology.
There is no way to represent these in SH(k).
In the first talk, I will explain the construction of logDM^{eff}(k) for a perfect field k, which is strictly larger than Voevodsky's triangulated categories of motives DM^{eff}(k) because Hodge cohomology is representable in logDM^{eff}(k).
This work is joint with Federico Binda and Paul Arne Østvær.

Zoom connection details will be announced a few days before the talk. The date here is subject to change. Check it often.

Zoom connection details will be announced a few days before the talk. The date here is subject to change. Check it often.

Let G be a finite group. The minimal ramification problem famously asks about the minimal number µ(G) of ramified primes in any Galois extension of Q with group G. A conjecture due to Boston and Markin predicts the value of µ(G). I will report on recent progress on this problem, as well as several other problems which may be described as
minimal ramification problems in a wider sense, notably: what is the smallest number k = k(G) such that there exists a G-extension of Q with discriminant not divisible by any (k + 1)-th power?, and: what is the smallest number m = m(G) such that there exists
a G-extension of Q with all ramification indices dividing m? Apart from partial results over Q, I will present function field analogs.
(If you would like to join the seminar, contact Bo-Hae Im to get the zoom link.)

The logarithmic analog of SH(k) is logSH(k).
In logSH(k), topological cyclic homology is representable.
Furthermore, the cyclotomic trace map from K to TC is representable too when k is a perfect field with resolution of singularities.
In the second talk, I will explain the construction of logSH(k) and how we can achieve these results.
This work is joint with Federico Binda and Paul Arne Østvær.

Zoom connection details will be provided a few days before the talk. The date and the time may change if the circumstances of the speaker require it. Please check it often.

Zoom connection details will be provided a few days before the talk. The date and the time may change if the circumstances of the speaker require it. Please check it often.

The rapid development of high-throughput sequencing technology in recent years is providing unprecedented opportunities to profile microbial communities from a variety of environments, but analysis of such multivariate taxon count data remains challenging. I present two flexible Bayesian methods to analyze complex count data with application to microbiome study. The first project is to develop a Bayesian sparse multivariate regression method that model the relationship between microbe abundance and environmental factors. We extend conventional nonlocal priors, and construct asymmetric non-local priors for regression coefficients to efficiently identify relevant covariates and their effect directions. The developed Bayesian sparse regression model is applied to analyze an ocean microbiome dataset collected over time to study the association of harmful algal bloom conditions with microbial communities. For the second project, we develop a Bayesian nonparametric regression model for count data with excess zeros. The approach provides straightforward community-level insights into how characteristics of microbial communities such as taxa richness and diversity are related to covariates. The baseline counts of taxa in samples are carefully constructed to obtain improved estimates of differential abundance. We apply the model to a chronic wound microbiome dataset, comparing the microbial communities present in chronic wounds versus in healthy skin.

Co-authors; Kurtis Shuler (Sandia National Lab), Marilou Sison-Mangus (Ocean Sciences, UCSC), Irene A. Chen (Chemistry and Biochemistry, UCLA), Samuel Verbanic (Chemistry and Biochemistry, UCLA)

Co-authors; Kurtis Shuler (Sandia National Lab), Marilou Sison-Mangus (Ocean Sciences, UCSC), Irene A. Chen (Chemistry and Biochemistry, UCLA), Samuel Verbanic (Chemistry and Biochemistry, UCLA)

Ordered Ramsey numbers were introduced in 2014 by Conlon, Fox, Lee, and Sudakov. Their results included upper bounds for general graphs and lower bounds showing separation from classical Ramsey numbers. We show the first nontrivial results for ordered Ramsey numbers of specific small graphs. In particular we prove upper bounds for orderings of graphs on four vertices, and determine some numbers exactly using SAT solvers for lower bounds. These results are in the spirit of exact calculations for classical Ramsey numbers and use only elementary combinatorial arguments.
This is joint work with Jeremy Alm, Kayla Coffey, and Carolyn Langhoff.

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Meeyoung Cha (IBS, KAIST)
[SAARC Colloquium] Theoretical Challenges in Data Science: Customs Fraud Detection and Poverty Mapping

At Data Science Group, we try to offer computational models for challenging real-world problems.
This talk will introduce two such problems that can benefit from collaboration with mathematicians and theorists. One is customs fraud detection, where the goal is to determine a small set of fraudulent transactions that will maximize the tax revenue when caught. We had previously shown
a state-of-the-art deep learning model in collaboration with the World Customs Organization [KDD2020].
The next challenge is to consider semi-supervised (i.e., using very few labels) and unsupervised (i.e., no label information) settings that better suit developing countries' conditions. Another research problem is poverty mapping, where the goal is to infer economic index from high-dimensional visual features learned from satellite images. Several innovative algorithms have been proposed for this task [Science2016, AAAI2020, KDD2020]. I will introduce how we approach this problem under extreme conditions with little validation data, as in North Korea.

We will discuss about “Highly accurate fluorogenic DNA sequencing with information theory–based error correction”, Chen et al., Nature Biotechnology (2017)
Eliminating errors in next-generation DNA sequencing has proved challenging. Here we present error-correction code (ECC) sequencing, a method to greatly improve sequencing accuracy by combining fluorogenic sequencing-by-synthesis (SBS) with an information theory–based error-correction algorithm. ECC embeds redundancy in sequencing reads by creating three orthogonal degenerate sequences, generated by alternate dual-base reactions. This is similar to encoding and decoding strategies that have proved effective in detecting and correcting errors in information communication and storage. We show that, when combined with a fluorogenic SBS chemistry with raw accuracy of 98.1%, ECC sequencing provides single-end, error-free sequences up to 200 bp. ECC approaches should enable accurate identification of extremely rare genomic variations in various applications in biology and medicine.

This is the part 3 of the lectures on infinity-categories:
This talk will be focused on introducing quasi-categories as our model for infinity-categories. After reviewing some background material needed to define quasi-categories, we'll see how the definition works.

Zoom ID: 352 730 6970, Password: 9999

Zoom ID: 352 730 6970, Password: 9999

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ZOOM 875 0445 5572, 산업경영학동(E2) Room 222
Colloquium
Insuk Seo (Seoul National Univerisy)
Reduction of stochastic systems via resolvent equations

ZOOM 875 0445 5572, 산업경영학동(E2) Room 222

Colloquium

In this talk, we consider stochastic systems with several stable sets. Typical examples are low-temperature physical systems and stochastic optimization algorithms. The macroscopic description of such systems is usually carried out via a so-called model reduction. We explain a necessary and sufficient condition for model reduction in terms of solutions of certain form of partial differential equations.

pw: 2021math

pw: 2021math

Topological defect structure is one of the most interesting topics in natural sciences. Especially, the topological defect transition was highlighted by Nobel prize in 2016. But this topic is hard to understand and realize in the practical condition because the size and time-scale are huge in cosmos or so tiny in skyrmion system. So, we proposed to use liquid crystal (LC) materials to directly show this interesting topic, phase transition of topological defect.

화학과&수리과학과 공동 주관 세미나

화학과&수리과학과 공동 주관 세미나

Thurston classified mappings from a given surface to itself. By iterating the surface mappings, one can view this as a dynamical system. Most of those surface mappings are so-called pseudo-Anosov. We briefly explain how we should understand these pseudo-Anosov maps and their physical meaning.

화학과&수리과학과 공동 주관 세미나

화학과&수리과학과 공동 주관 세미나

Overparametrized neural networks have infinitely many solutions that achieve zero training loss, but gradient-based optimization methods succeed in finding solutions that generalize well. It is conjectured that the optimization algorithm and the network architecture induce an implicit bias towards favorable solutions, and understanding such a bias has become a popular topic. We study the implicit bias of gradient flow (i.e., gradient descent with infinitesimal step size) applied on linear neural network training. We consider separable classification and underdetermined linear regression problems where there exist many solutions that achieve zero training error, and characterize how the network architecture and initialization affects the final solution found by gradient flow. Our results apply to a general tensor formulation of neural networks that includes linear fully-connected networks and linear convolutional networks as special cases, while removing convergence assumptions required by prior research. We also provide experiments that corroborate our theoretical analysis.

This is part II of the lecture series in infinity-categories.
I'll continue to talk about higher categories and some difficulty in defining them. In the end, a few models for infinity-categories will be introduced very roughly.

Zoom ID: 352 730 6970, password: 9999

Zoom ID: 352 730 6970, password: 9999

Bose-Einstein condensation (BEC) is one of the most famous phenomena, which cannot be explained by classical mechanics. Here, we discuss the time evolution of BEC in the mean-field limit. First, we review quantum mechanics briefly, and we understand the problem in a mathematically rigorous way. Then, we taste the idea of proof by using coherent state and the Fock space. Finally, some recent developments will be provided.

Introduction: In this lecture series, we'll discuss algebro-geometric study on fundamental problems concerning tensors via higher secant varieties.
We start by recalling definition of tensors, basic properties and small examples and proceed to discussion on tensor rank, decomposition, and X-rank
for any nondegenerate variety $X$ in a projective space. Higher secant varieties of Segre (resp. Veronese) embeddings will be regarded as a natural
parameter space of general (resp. symmetric) tensors in the lectures. We also review known results on dimensions of secants of Segre and Veronese,
and consider various techniques to provide equations on the secants.
In the end, we'll finish the lectures by introducing some open problems related to the theme such as syzygy structures and singularities of higher secant varieties.

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ZOOM 875 0445 5572, E2동 2222호
Colloquium
Ernest K. Ryu (Seoul National Univerisy)
WGAN with an Infinitely Wide Generator Has No Spurious Stationary Points

ZOOM 875 0445 5572, E2동 2222호

Colloquium

Generative adversarial networks (GAN) are a widely used class of deep generative models, but their minimax training dynamics are not understood very well. In this work, we show that GANs with a 2-layer infinite-width generator and a 2-layer finite-width discriminator trained with stochastic gradient ascent-descent have no spurious stationary points. We then show that when the width of the generator is finite but wide, there are no spurious stationary points within a ball whose radius becomes arbitrarily large (to cover the entire parameter space) as the width goes to infinity.