# 세미나 및 콜로퀴엄

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(This is a reading seminar for graduate students.) In the setting of Quillen's $K$-theory, the $K$-groups are defined for the category of (algebraic) vector bundles. However, even in SGA 6, the need of replacing vector bundles by complexes quasi-isomorphic to bounded complexes of vector bundles were noticed. Such complexes are called perfect complexes and their K-theory provide better local-to-global properties, for example, Nisnevich descent or the localization sequence. In this talk, the basis notions of perfect complexes and pseudo-coherent complexes will be investigated and characterized categorically. We embrace as many technical details as possible.

Many structures in mathematics have both covariant and contravariant operations, which can often be described using categories of spans or correspondences. In some cases there are two compatible classes of covariant operations, such as the additive and multiplicative transfers (or norms) that occur in representation theory and motivic homotopy theory. I will explain how these can be encoded using higher categories of "bispans", and discuss how this can be used to describe operations on motivic spectra. This is joint work with Elden Elmanto.

In numerical computations for wave propagation problems in unbounded domains, it is essential to use an accurate domain truncation technique, which allows wave fields to propagate through artificial boundaries resulting from domain truncation with reflections as small as we wish. In this talk, we will discuss two high-order absorbing boundary conditions as domain truncation techniques, so-called, perfectly matched layers and complete radiation boundary conditions, for time-harmonic problems in waveguide geometry. Unlike exterior problems, waveguides involve three different types of modes including propagating modes, evanescent modes and cutoff modes. We will present a general idea on parameter selection of the absorbing boundary conditions to minimize reflection errors of three different types of modes.

We discuss several problems regarding non-compact complete singularities of geometric PDEs including the mean curvature flow. Depending on the equations, the linearized operator would be a self-adjoint operator. Then, the eigen-decomposition allows us to reduce the equation into a system of ODEs. By solving the ODEs under prescribed asymptotic behavior, we improve their asymptotic behavior so that we classify the singularities.

We will discuss how to use PDEs in Geometry, especially for students working in Analysis. We will begin by considering how the geometric PDEs differ from the standard PDEs. Then, we will study some particular examples where the maximum principle, the spectral analysis, and the contraction mapping have been applied in geometric analysis. Also, we would talk about the recent trend in the collaborations between Geometry and Analysis.

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Room B232, IBS (기초과학연구원)
Discrete Math
박진영 (Rutgers University)
The number of maximal independent sets in the Hamming cube

Let Qn be the n-dimensional Hamming cube (hypercube) and N=2n. We prove that the number of maximal independent sets in Qn is asymptotically 2n2N/4, as was conjectured by Ilinca and Kahn in connection with a question of Duffus, Frankl and Rödl. The value is a natural lower bound derived from a connection between maximal independent sets and induced matchings. The proof of the upper bound draws on various tools, among them “stability” results for maximal independent set counts and old and new results on isoperimetric behavior in Qn. This is joint work with Jeff Kahn.

Since the early 2000s, it has turned out that well developed mathematical tools can play crucial roles in quantum information theory. One monumental work was made by Hastings in 2009. He disproved a long standing conjecture in quantum information theory, which is called the additivity conjecture of Holevo capacities. A natural way to prove this result will be covered in this talk based on the theory of i.i.d. random unitary matrices. Another outstanding application of random matrix theory in quantun information theory is to provide a systematic way to produce PPT entangled states in high dimensional tensor product spaces. This construction comes from i.i.d. random Gaussian matrices and I will try to explain why this application is important in view of quantun information theory.

Directed polymer models are well known Gibbs measures on random walk paths. Canonically they are defined so as to tilt the path distribution towards regions of space-time where an independent random field happens to be large, and as a result the paths tend to exhibit superdiffusive Kardar-Parisi-Zhang type fluctuation exponents, somehow betraying their random walk upbringing. Constructing these models on in the discrete space-time setting with a finite time horizon is straightforward, but extending them to infinite time horizons is difficult even in the fully discrete setting. I will review some relatively recent progress in the discrete and semi-discrete setting by myself and several other authors, some previous work of myself, Khanin, and Quastel on constructing continuous space-time models in the finite time horizon setting, and some attempts in progress to connect the two.

The still open topological surgery conjecture for 4-manifolds is equivalent to the statement that all good boundary links are freely slice. In this talk, I will show that every good boundary link with a pair of derivative links on a Seifert surface satisfying a homotopically trivial plus assumption is freely slice. This subsumes all previously known methods for freely slicing good boundary links with two or more components, and provides new freely slice links. This is joint work with Jae Choon Cha and Mark Powell.

We impose a rather unknown algebraic structure called a `hyperstructure' to the underlying space of an affine algebraic group scheme. This algebraic structure generalizes the classical group structure and is canonically defined by the structure of a Hopf algebra of global sections. This paper partially generalizes the result of A.Connes and C.Consani.

Pairs of commuting nilpotent matrices have been extensively studied, especially from the view point of quivers, but the space of commuting nilpotents modulo simultaneous conjugation has not received any attention at all despite its moduli theory flavor. I will explain how a 'moduli space' can be constructed via three different methods and demonstrate many interesting properties of the space:

- It is isomorphic to an open subscheme of a punctual Hilbert scheme.

- Over the field of complex numbers, it is diffeomorphic to a direct sum of twisted tangent bundles over a projective space.

- It is isomorphic to a bundle of regular jets.

- It gives examples of affine space bundles that are not vector bundles.

This is a joint work with W. Haboush (Illinois) and G. Bérczi (Zurich).

A b-coloring of a graph G is a proper coloring of its vertices such that each color class contains a vertex that has at least one neighbor in all the other color classes. The b-Coloring problem asks whether a graph G has a b-coloring with k colors.

The b-chromatic number of a graph G, denoted by χb(G), is the maximum number k such that G admits a b-coloring with k colors. We consider the complexity of the b-Coloring problem, whenever the value of k is close to one of two upper bounds on χb(G): The maximum degree Δ(G) plus one, and the m-degree, denoted by m(G), which is defined as the maximum number i such that G has ivertices of degree at least i−1. We obtain a dichotomy result stating that for fixed k∈{Δ(G)+1−p,m(G)−p}, the problem is polynomial-time solvable whenever p∈{0,1} and, even when k=3, it is NP-complete whenever p≥2.

We furthermore consider parameterizations of the b-Coloring problem that involve the maximum degree Δ(G) of the input graph G and give two FPT-algorithms. First, we show that deciding whether a graph G has a b-coloring with m(G) colors is FPT parameterized by Δ(G). Second, we show that b-Coloring is FPT parameterized by Δ(G)+ℓk(G), where ℓk(G) denotes the number of vertices of degree at least k.

This is joint work with Paloma T. Lima.

An equitable tree-k-coloring of a graph is a vertex coloring using kdistinct colors such that every color class (i.e, the set of vertices in a common color) induces a forest and the sizes of any two color classes differ by at most one. The minimum integer k such that a graph G is equitably tree-k-colorable is the equitable vertex arboricity of G, denoted by vaeq(G). A graph that is equitably tree-k-colorable may admits no equitable tree-k′-coloring for some k′>k. For example, the complete bipartite graph K9,9 has an equitable tree-2-coloring but is not equitably tree-3-colorable. In view of this a new chromatic parameter so-called the equitable vertex arborable threshold is introduced. Precisely, it is the minimum integer k such that G has an equitable tree-k′-coloring for any integer k′≥k, and is denoted by va∗eq(G). The concepts of the equitable vertex arboricity and the equitable vertex arborable threshold were introduced by J.-L. Wu, X. Zhang and H. Li in 2013. In 2016, X. Zhang also introduced the list analogue of the equitable tree-k-coloring. There are many interesting conjectures on the equitable (list) tree-colorings, one of which, for example, conjectures that every graph with maximum degree at most Δ is equitably tree-k-colorable for any integer k≥(Δ+1)/2, i.e, va∗eq(G)≤⌈(Δ+1)/2⌉. In this talk, I review the recent progresses on the studies of the equitable tree-colorings from theoretical results to practical algorithms, and also share some interesting problems for further research.

(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.

In this talk, we study the Schrödinger-Lohe (S-L) model as a phenomenological model for describing possible quantum synchronization phenomenon. In order to reflect real-world phenomenon, we employ several dynamical properties such as general network structure and interaction frustration, and present sufficient frameworks leading to complete and practical synchronizations. First, we begin with an all-to-all network topology to show that the complete synchronization occurs for generic initial data. Moreover, uniform stability of standing wave solutions is established using the complete synchronization results. Next, we consider the general network structure to observe that several emergent behaviors arise, for instance, repulsive behavior, bi-polar synchronized state and periodic orbits. On the other hand, we employ frustration functions into our model and provide sufficient conditions leading to complete and practical synchronizations and periodic trajectories.

Let NN be the set of natural numbers. A set A⊂NA⊂N is called a Sidon set if the sums a1+a2a1+a2 , with a1,a2∈Sa1,a2∈S and a1≤a2a1≤a2 , are distinct, or equivalently, if |(x+w)−(y+z)|≥1|(x+w)−(y+z)|≥1 for every x,y,z,w∈Sx,y,z,w∈S with x<y≤z<wx<y≤z<w . We define strong Sidon sets as follows:

For a constant αα with 0≤α<10≤α<1 , a set S⊂NS⊂N is called an αα -strong Sidon set if |(x+w)−(y+z)|≥wα|(x+w)−(y+z)|≥wα for every x,y,z,w∈Sx,y,z,w∈S with x<y≤z<wx<y≤z<w .

The motivation of strong Sidon sets is that a strong Sidon set generates many Sidon sets by altering each element a bit. This infers that a dense strong Sidon set will guarantee a dense Sidon set contained in a sparse random subset of NN .

In this talk, we are interested in how dense a strong Sidon set can be. This is joint work with Yoshiharu Kohayakawa, Carlos Gustavo Moreira and Vojtěch Rödl.

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 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.

The real Ginibre ensemble consists of square real matrices whose entries are i.i.d. standard normal random variables. In the large dimension limit, the empirical distribution converges to the circular law in a disk. This is same as the complex Ginibre ensemble. However, unlike the complex version, there is a positive probability that there are real eigenvalues. The law of the largest real eigenvalue was studied by Rider, Sinclair and Poplavskyi, Tribe, Zaboronski. Building on their work, we will show that the limiting distribution of the largest real eigenvalue admits a closed form expression in terms of a distinguished solution to an inverse scattering problem for the Zakharov-Shabat system which is related to the nonlinear Schr¨odinger equation. The results of this talk are based on a recent joint with Thomas Bothner (King's College London).

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.