# 세미나 및 콜로퀴엄

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Room B232, IBS(기초과학연구원)
Discrete Math
Cory T. Palmer (University of Montana, Missoula, MT)
A survey of Turán-type subgraph counting problems

Let F and H be graphs. The subgraph counting function ex(n,H,F) is defined as the maximum possible number of subgraphs H in an n-vertex F-free graph. This function is a direct generalization of the Turán function as ex(n,F)=ex(n,K2,F). The systematic study of ex(n,H,F) was initiated by Alon and Shikhelman in 2016 who generalized several classical results in extremal graph theory to the subgraph counting setting. Prior to their paper, a number of individual cases were investigated; a well-known example is the question to determine the maximum number of pentagons in a triangle-free graph. In this talk we will survey results on the function ex(n,H,F) including a number of recent papers. We will also discuss this function’s connection to hypergraph Turán problems.

Stochastic heat equations usually refer to heat equations perturbed by noise. Depending on noise, stochastic heat equations have similar properties as heat equations such as strict positivity or properties which cannot be seen from heat equations such as intermittency. We consider various properties of stochastic heat equations in this talk. (This talk will be a survey talk and should be accessible to all graduate students.)

It is a common theme in algebraic geometry that many constructions have only been done for schemes and morphisms of finite type. However, in arithmetic geometry one would also like to work with infinite objects, as for example infinite level modular curve. In my talk I motivate and define schemes and morphism satisfying a weaker finiteness property, which contain many examples from arithmetic geometry. The aim of this talk is to extend the definition of cohomology with compact support to them; in fact, we even obtain Grothendieck's six operations for this class of morphism.

Synchronization phenomenon is ubiquitous in an ensemble of coupled oscillators, e.g., hand clapping in opera and musical halls, flashing of fireflies and heart beating of pacemaker cells, etc. In the last forty years, the Kuramoto model served as a prototype model for describing such synchronization phenomena. In particular, we will consider the Kuramoto model under the stochastic noise. As the number of oscillators tends to infinity, we can derive the kinetic

equation for the Kuramoto model by using the standard BBGKY hierarchy. In this talk, we will consider the asymptotic behavior for the kinetic Kuramoto models under the stochastic noise, and talk about their large time behaviors.

Given a graph GG, we define exc(G)exc(G) to be the minimum value of tt for which there exists a constant N(t,G)N(t,G) such that every tt-connected graph with at least N(t,G)N(t,G) vertices contains GG as a minor. The value of exc(G)exc(G) is known to be tied to the vertex cover number τ(G)τ(G), and in fact τ(G)≤exc(G)≤312(τ(G)+1)τ(G)≤exc(G)≤312(τ(G)+1). We give the precise value of exc(G)exc(G) when GG is a forest. In particular we find that exc(G)≤τ(G)+2exc(G)≤τ(G)+2 in this setting, which is tight for infinitely many forests.

In this talk we consider a reaction-diffusion model for the spreading of farmers in Europe, which was occupied by hunter-gatherers; this process is known as the Neolithic agricultural revolution. The spreading of farmers is modelled by a nonlinear porous medium type diffusion equation which coincides with the singular limit of another model for the dispersal of farmers as a small parameter tends to zero. From the ecological viewpoint, the nonlinear diffusion takes into account the population density pressure of the farmers on their dispersal. The interaction between farmers and hunter-gatherers is of the Lotka-Volterra prey-predator type. We show the existence and uniqueness of a global in time solution and study its asymptotic behaviour as time tends to infinity.

After the discovery of algebraic $K$-theory of schemes by Quillen, it was anticipated that algebraic cycles would play a fundamental role in computing the $K$-groups of schemes. These groups are usually very hard to compute even as they contain a lot of information about the underlying scheme. For smooth schemes, the problem of describing $K$-theory by algebraic cycles was satisfactorily settled by works of several mathematicians, spanned over many years.

However, solving this problem under the presence of singularities is still a very challenging task.

The theory of additive Chow groups and the Chow groups with modulus were invented to address this problem.

Even if it is not yet clear if these Chow groups will finally solve the problem, I shall present some positive results in this direction in my talk. In particular, I shall show that the additive Chow groups do completely describe the $K$-theory of infinitesimal neighborhoods of the origin in the affine line. This was the initial motivation of the discovery of additive Chow groups by Bloch and Esnault. This talk is based on a joint work with Rahul Gupta.

After the discovery of algebraic $K$-theory of schemes by Quillen, it was anticipated that algebraic cycles would play a fundamental role in computing the $K$-groups of schemes. These groups are usually very hard to compute even as they contain a lot of information about the underlying scheme. For smooth schemes, the problem of describing $K$-theory by algebraic cycles was satisfactorily settled by works of several mathematicians, spanned over many years.

However, solving this problem under the presence of singularities is still a very challenging task.

The theory of additive Chow groups and the Chow groups with modulus were invented to address this problem.

Even if it is not yet clear if these Chow groups will finally solve the problem, I shall present some positive

results in this direction in my talk. In particular, I shall show that the additive Chow groups do completely describe the $K$-theory of infinitesimal neighborhoods of the origin in the affine line. This was the initial motivation of the discovery of additive Chow groups by Bloch and Esnault. This talk is based on a joint work with Rahul Gupta.

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Room B232, IBS (기초과학연구원)
Discrete Math
강미현 (TU Graz)
The genus of a random graph and the fragile genus property

In this talk we shall discuss how quickly the genus of the Erdős-Rényi random graph grows as the number of edges increases and how dramatically a small number of random edges can increase the genus of a randomly perturbed graph. (Joint work with Chris Dowden and Michael Krivelevich)

(This is a reading seminar for graduate students.) Algebraic $K$-theory of schemes has a sequence of $K$-groups for a closed immersion which is exact except only one place, namely $K_0$ of the open immersion of the complement, as the proto-localization theorem shows. Following the idea of Hyman Bass, we define a non-connective spectrum of a scheme whose non-negative part coincides with the usual algebraic $K$-theory defined by perfect complexes. This non-connective $K$-theory spectrum in particular gives a long exact sequence for a closed immersion. Our aim is to construct it, which is done by descending induction and requires the projective bundle formula that is also important regardless of our situation.

We introduce the mechanical model designed by W. Malkus and L. Howard for the Lorenz system. Some mechanical equation explaining this will be derived. Based on this mechanical equations, the Fourier modes and velocity of the system will be presented as a complete system following Lorenz's model. Some key factors, including Rayleigh number, will be introduced to compare Malkus's model and the fluid convection, key interest of Lorenz that gave birth to his model.

Abstract: Linkage is a classical topic in algebraic geometry and commutative algebra. Fix an affine space A. We say two subschemes X, Y of A are linked if their union is a complete intersection in A and X and Y do not have a common component. Two linked subschemes share several properties in common. Linkage has been studied by various people, Artin-Nagata, Peskine-Szprio, Huneke-Ulrich, to name a few.

In 2014, Niu showed that if Y is a generic link of a variety X, then LCT (A, X) <= LCT (A, Y), where LCT stands for log canonical threshold. In this talk, we show that if Y is a generic link of a determinantal variety X, then X and Y have the same log canonical threshold. This is joint work with Lance E. Miller and Wenbo Niu.

The models of directed polymers are based on Gibbs measures on paths with the reference measure usually describing a process with independent increments where the energy of the interaction between the path and the environment is given by a space-time random potential accumulated along the path. One of the intriguing phenomena that these models exhibit is the localization/delocalization transition between high/low temperature regimes. In order to study directed polymers in Euclidean space, we will first review a new metrization of the Mukherjee-Varadhan topology, introduced as a translation-invariant compactification of the space of probability measures on Euclidean spaces. This new metrization allows us to prove that the asymptotic clusterization (a natural continuous analogue of the asymptotic pure atomicity property) holds in the low temperature regime and that the endpoint distribution is geometrically localized with positive density if and only if the system is in the low temperature regime.

In the first part of the talk, I will present my recent working paper on continuous time game between a suspect and a defender. Economically, this is a sequential game model with asymmetric information, imperfect observation, and optimal stopping. Mathematically, search of a Markov equilibrium boils down to finding a solution of the system of equations (a HJB equation from the suspect’s optimal control, variational inequality from the defender’s optimal stopping, and a SDE from the filtering equation).

In the second part of the talk, I will roughly overview subjects in mathematical finance, and place where my past, current, and future research topics lie.