# 세미나 및 콜로퀴엄

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Recently, the classification of isoparametric hypersurfaces in spheres has been completed. Therefrom, various new research projects in geometry have been initiated. The study of minimal lagrangian submanifolds via isoparametric hypersurfaces is one of the most active projects a la mode. In this talk, we have an introduction to the study of Isoparametric hypersurfaces and minimal Lagrangian submanfolds, and discuss the relationship between them.

The geometry and cohomology of modular curves play an important role in constructing the Langlands correspondence for GL(2,\mathbb{Q}). Similarly, the tower of algebraic varieties called Shimura varieties plays an important role in the Langlands programme for reductive groups over number fields. It is then a natural question to ask if there is a ‘p-adic local analogue’ of modular curves and Shimura varieties, which is now commonly referred to as 'local Shimura varieties' or ‘Rapoport—Zink spaces'.

In this survey talk, I will try to motivate the concept of local Shimura varieties starting from the review of modular curves and introducing classical examples of local Shimura varieties (that have been known even before the general concept of local Shimura varieties emerged). If time permits, I will introduce my contribution and work in progress on constructing more general classes of local Shimura varieties.

This talk is about the spectra of non-Hermitian random matrix models. Their asymptotic analysis reveals remarkable high dimensional phenomena, which are, among other aspects, related to entropy maximization and free probability theory. We will present some of the general phenomena and methodologies, and we will focus mostly on the circular law phenomenon. We will present several works, notably in collaboration with Charles Bordenave and Pietro Caputo. We will also present briefly some few open problems.

Describing the eigenvalue distribution of the sum of two general Hermitian matrices is basic question going back to Weyl. If the matrices have high dimensionality and are in general position in the sense that one of them is conjugated by a random Haar unitary matrix, the eigenvalue distribution of the sum is given by the free additive convolution of the respective spectral distributions. This result was obtained by Voiculescu on the macroscopic scale. In this talk, I show that it holds on the microscopic scale all the way down to the eigenvalue spacing in the bulk and at the regular edges. This shows a remarkable rigidity phenomenon for the eigenvalues. Joint work with Z.G. Bao and L. Erdos.

S. Kamada introduced chart diagrams to describe two-dimensional braids in four-dimensional space, which (roughly speaking) are to classical braids what Cerf diagrams are to Morse functions. In this talk we recall chart diagrams, and discuss their application in defining Vassiliev invariants and approaching problems in linkhomotopy of 2-spheres in the 4-sphere.

Magombedze and Mulder (2013) studied the gene regulatory system of Mycobacterium Tuberculosis (Mtb) by partitioning this into three subsystems based on putative gene function and role in dormancy/latency development. Each subsystem, in the form of S-system, is represented by an embedded chemical reaction network (CRN), defined by a species subset and a reaction subset induced by the set of digraph vertices of the subsystem. Based on the network decomposition theory initiated by Feinberg in 1987, we have introduced the concept of incidence-independent and developed the theory of C- and C*-decompositions including their structure theorems in terms of linkage classes. With the S-system CRN N of Magombedze and Mulder's Mtb model, its reaction set partition induced decomposition of subnetworks that are not CRNs of S-system but constitute independent decomposition of N. We have also constructed a new S-system CRN N for which the embedded networks are C*-decomposition. We have shown that subnetworks of N and the embedded networks (subnetworks of N*) are digraph homomorphisms. Lastly, we attempted to explore modularity in the context of CRN.

Chemical reaction network theory (CRNT) is an area of applied mathematics that attempts to model the behavior of real world chemical systems. CRNT has become a tool to study complex biology independent of rate parameters, that is, certain behaviors of networks are examined by analyzing their structures only. In this talk, preliminary CRNT concepts will be presented. We focus on the existence of complex balanced equilibria for weakly reversible reaction networks with power law kinetics elaborating on the so called “Weak Reversibility Theorems”. We also discuss some particular applications of our theoretical results.

We survey some results in random matrix theory and their universal nature. For instance, consider the largest eigenvalue of a randomly chosen Hermitian matrix. This random variable converges to a certain distribution as the dimension becomes large. It was proved by many different researchers over the last twenty years that this distribution also describes many different models in probability which do not have an apparent connection to matrices. The examples include Coulomb gas, random tilings of a hexagon, random growth models, and directed polymers among others. We will discuss this fascinating university aspect of random matrix theory through several examples.

(This is a reading seminar for graduate students.) We defined Quillen's higher algebraic $K$-theory and examined its basic properties in previous talks. By the localization theorem and the dévissage theorem, the codimension filtration on $\operatorname{Coh}(X)$ for a finite dimensional noetherian scheme $X$ gives the Brown-Gersten-Quillen spectral sequence from page 1. If $X$ is a regular algebraic scheme, then the second page of this spectral sequence is given by $E_2^{p,-q}=H^p_{Zar}(X, G_{q})$ and $E_2^{p,-p}=CH^p(X)$, where $G_{q}$ denotes the (Zariski) sheafification of $U\mapsto G_p(U)$. To prove this, we employ Quillen's geometric presentation lemma. This is the third and last part of Quillen's algebraic $K$-theory.

Let K be a field. The monodromy group of a rational function $r(X) = f(X)/g(X) in K(X)$, i.e., the Galois group of $f(X) − tg(X)$ over $K(t)$, is an important object of study in problems from number theory, geometry, arithmetic dynamics, etc.

Classifying which finite groups occur as monodromy groups has been of great interest, since this knowledge helps reducing many arithmetic problems to pure group theory. The celebrated Guralnick-Thompson conjecture (1990; eventually proved by Frohardt and Magaard) asserts that apart from alternating and cyclic groups, only finitely many simple groups occur as composition factors of monodromy groups of rational functions over C (so-called "geometric" monodromy groups). In the case of functionally indecomposable $r(X)$, later work by Neftin, Zieve and others classified not only the "exceptional" groups, but actually the rational functions with exceptional monodromy group, assuming sufficiently large degree. In joint work in progress with Mueller, Neftin and Zieve, we reach a similar result for "arithmetic" monodromy groups. That is, we extend the above classification to arbitrary fields of characteristic zero. As a consequence, we also prove a generalization of the Guralnick-Thompson conjecture for arbitrary fields.

The current focus of our research is to reveal fundamental design principles of the biological clock mechanism and pathogenesis of circadian disorders. We develop unique mouse models to simulate human diseases that can cause circadian disorders directly and indirectly, and unravel how the diseases compromise circadian rhythms including wake-sleep cycles, the most salient feature of circadian rhythms in animals, at the molecular level. In the first half of my talk, I will discuss how cytoplasmic congestion, normally associated with metabolic diseases and aging, can disrupt the clock mechanism and how sleep disorders in patients with these diseases can be treated at the core clock level. In the second half of my talk, I will discuss arguably the most upstream event in rhythm generation. We recently found that phosphorylation of the circadian pacemaker protein, PER is regulated by a strong Hill-type reaction, the basis of robustness in many signaling pathways. Phosphorylation kinetics of PER increases proportionally to concentration of PER: the more PER is generated, the faster PER phosphorylates, similar to O2 binding to hemoglobin. Since PER phosphorylation is considered the time-generating step, we believe this is the most upstream event and simple mutations in Per genes (SNPs) may cause sleep disorders in many humans.

The circadian clock is an autonomous molecular mechanism that controls biochemical, physiological, and behavioral processes with a periodicity of 24 h in living organisms and can be entrained by environmental cues. The clock is sustained by a coordinated interplay of positive and negative transcriptional-translational feedback loops driven by circadian factors, a core group of proteins that either possess intrinsic transcriptional activity or modulate gene expression. We previously reported that the circadian factor PERIOD 2 (PER2) forms a stable complex with the tumor suppressor and checkpoint protein p53. The PER2:p53 complex undergoes time-of-day–dependent nuclear-cytoplasmic shuttling, thus generating an asymmetric distribution of each protein in different cellular compartments. In unstressed cells, PER2 mediates p53’s stability by binding to its C-terminal domain and preventing p53 from being ubiquitylated at sites targeted by the RING finger–containing E3 ligase and oncoprotein mouse double minute 2 homolog (MDM2). We found that PER2, p53, and MDM2 co-exist as a trimeric and stable complex in the nuclear compartment, although p53 is released from the complex to become transcriptionally active after cells experience genotoxic stimuli. More recently, we found that PER2 could also act as a bona fide substrate for MDM2 in the absence of p53. Indeed, PER2 was efficiently ubiquitylated in vitro and in cells at numerous sites by MDM2 in a process that was independent of PER2 phosphorylation. Accordingly, PER2’s half-life is critically influenced by the abundance and enzymatic activity of MDM2, as shown in cells in which MDM2 expression was either enhanced or silenced and its catalytic activity was pharmacologically inhibited. As a consequence, direct manipulation of MDM2 expression influenced period length by reducing PER2 stability. Our results uncover previously unknown regulatory players that likely impact our view of how other mechanisms crosstalk and modulate the clock itself. Furthermore, it exposes an uncharacterized druggable node that is often found to be deregulated during tumorigenesis.

(This is a reading seminar for graduate students.)

Grothendieck's $K_0$-group has an exact sequence $K_0(Z)\to K_0(X)\to K_0(X-Z)\to 0$ for a closed immersion $Z\to X$ of regular noetherian schemes, whose kernel of the leftmost map is usually nontrivial. To prolong this sequence functorially, Quillen defined higher algebraic $K$-groups as the higher homotopy groups of some mysterious category associated to an exact category. For this, we introduce simplicial sets, $Q$-construction of an exact category, higher $K$-groups of schemes, and their remarkable theorems, part of which extends the previously mentioned exact sequence for $K_0$ and relates higher K-theory with Chow groups. This is the second half of Quillen's algebraic $K$-theory.

In X-ray computed tomography (CT), deep learning techniques have shown great potential for reduction of various artifacts (e.g., noise arising from low-dose, streaking artifacts due to sparse view). Most of these approaches learn the relationship between artifact images and artifact-free (ground-truth) images. However, paired training data are not generally available in clinical practice. In this talk, I will introduce simulation-based and unpaired learning methods, which can be used to circumvent such issue.

A Boolean function is a function from the set Q of binary vectors of length n (i.e., the binary n-dimensional hypercube) to F2={0,1}. It has several applications to complexity theory, digital circuits, coding theory, and cryptography.

In this talk we give a connection between Boolean functions and Artificial Neural Network. We describe how to represent Boolean functions by Artificial Neural Network including linear and polynomial threshold units and sigmoid units. For example, even though a linear threshold function cannot realize XOR, a polynomial threshold function can do it. We also give currently open problems related to the number of (Boolean) linear threshold functions and polynomial threshold functions.

Many modern applications such as machine learning require solving large-dimensional optimization problems. First-order methods are widely used to solve such problems, since their computational cost per iteration mildly depends on the problem dimension. However, they suffer from relatively slow convergence rates, and this talk will discuss recent progress on the acceleration of first-order methods, particularly using the close relationship between convex optimization methods and maximally monotone operators.

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 $k.$

In this talk, we give a classification of such finite groups for the case when $k$ is a finite field and $g$ is a prime number. If $g=2,$ then we need a notion of maximality in a certain sense, and for $g geq 3,$ we achieve a rather complete list without conveying maximality.

Finite element discretization of solutions with respect to simplicial/cubical meshes has been studied for decades, resulting in a clear understanding of both the relevant mathematics and computational engineering challenges. Recently, there has been both a desire and need for an equivalent body of research regarding discretization with respect to generic polygonal/polytopal meshes. General meshes offer a very convenient framework for mesh generation, mesh deformations, fracture problems, composite materials, topology optimizations, mesh refinements and coarsening; for instance, to handle hanging nodes, different cell shapes within the same mesh and non-matching interfaces. Such a flexibility represents a powerful tool towards the efficient solution of problems with complex inclusions as in geophysical applications or posed on very complicated or possibly deformable geometries as encountered in basin and reservoir simulations, in fluid-structure interaction, crack propagation or contact problems.

In this talk, a new computational paradigm for discretizing PDEs is presented via staggered Galerkin approach on general meshes. First, a class of locally conservative, lowest order staggered discontinuous Galerkin method on general quadrilateral/polygonal meshes for elliptic problems are proposed. The method can be flexibly applied to rough grids such as highly distorted meshes. Next, adaptive mesh refinement is an attractive tool for general meshes due to their flexibility and simplicity in handling hanging nodes.

Goldman parametrizes the $mathrm{PSL}_3(mathbb{R})$-Hitchin component of a closed oriented hyperbolic surface of genus $g$ by $16g-16$ parameters. Among them, $10g- 10$ coordinates are canonical. We prove that the $mathrm{PSL}_3(mathbb{R})$-Hitchin component equipped with the Atiyah-Bott-Goldman symplectic form admits a global Darboux coordinate system such that the half of its coordinates are canonical Goldman coordinates. To this end, we establish a version of the action-angle principle and show that the Hitchin component can be decomposed into a product of smaller Hitchin components.

(This is a reading seminar for graduate students.)

Grothendieck's $K_0$-group has an exact sequence $K_0(Z)\to K_0(X)\to K_0(X-Z)\to 0$ for a closed immersion $Z\to X$ of regular noetherian schemes, whose kernel of the leftmost map is usually nontrivial. To prolong this sequence functorially, Quillen defined higher algebraic $K$-groups as the higher homotopy groups of some mysterious category associated to an exact category. For this, we introduce simplicial sets, $Q$-construction of an exact category, higher $K$-groups of schemes, and their remarkable theorems, part of which extends the previously mentioned exact sequence for $K_0$ and relates higher K-theory with Chow groups. This is the first half of Quillen's algebraic $K$-theory.