# 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.)

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.

This is joint work with Sławomir Kołodziej. We show that the complex m-Hessian

operator of a Holder continuous m-subharmonic function is well dominated by the corresponding capacity. As consequence we obtain the Holder continuous subsolution theorem for the complex m-Hessian equation.

The Boussinesq abcd system was originally derived by Bona, Chen and Saut [J. Nonlinear. Sci. (2002)] as a rst order 2-wave approximations of the incompressible and irrotational, two dimensional water wave equations in the shallow water wave regime. Among many particular regimes, the Hamiltonian generic regime is characterized by the setting b = d > 0 and a; c < 0. It is known that the system in this regime is globally well-posed for small data in the energy space H1 H1 by Bona, Chen and Saut [Nonlinearity (2004)]. In this talk, we are going to discuss about the decay of small solutions to abcd system in three directions: First, for a weakly dispersive abcd systems (characterized only in terms of parameters a; b and c), all small solutions must decay to zero, locally strongly in the energy space, in proper subset of the light cone jxj jtj. Second, for every ray x = vt, jvj < 1 inside the light cone, small solutions to suciently dispersive system (smallness and dispersion are characterized by v) decay to zero, in proper subset along the ray. Last, small solutions decay to zero in exterior regions jxj jtj under suitable conditions of parameters (a; b; c). All results rule out, among other things, the existence of zero or nonzero speed or super-luminical small solitary waves in each regime where decay is present.

This is joint work with Claudio Munoz.

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.