# 세미나 및 콜로퀴엄

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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.