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Information Center for Mathematical Science

### 세미나

Information Center for Mathematical Science

#### 세미나

##### KAIST Discrete Math 세미나

A celebrated conjecture of Sidorenko and Erdős–Simonovits states that, for all bipartite graphs H, quasirandom graphs contain asymptotically the minimum number of copies of H taken over all graphs with the same order and edge density. This conjecture has attracted considerable interest over the last decade and is now known to hold for a broad range of bipartite graphs, with the overall trend saying that a graph satisfies the conjecture if it can be built from simple building blocks such as trees in a certain recursive fashion. Our contribution here, which goes beyond this paradigm, is to show that the conjecture holds for any bipartite graph H with bipartition A∪B where the number of vertices in B of degree k satisfies a certain divisibility condition for each k. As a corollary, we have that for every bipartite graph H with bipartition A∪B, there is a positive integer p such that the blow-up H_A^p formed by taking p vertex-disjoint copies of H and gluing all copies of A along corresponding vertices satisfies the conjecture. Joint work with David Conlon.

##### 2018 KAIST Physics Seminar

Sparse modeling is a powerful framework for data analysis and processing such as image denoising and super-resolution. Sparse modeling assumes that a signal to be processed can be efficiently represented by a sparse linear combination of some basis vectors (e.g., wavelets for image). In this talk, we introduce our “sparse-modeling” methods for solving many-body problems. Our framework relies on a recently developed generic compact representation of imaginary-time (Matsubara) Green’s functions [1]. The basis functions of this “intermediate representation” (IR) are defined by the singular value decomposition of the kernel of the Lehmann representation of Green’s functions [1].We show that the data size of any single/two-particle Green’s function increases only logarithmically with inverse temperature [2,3]. We present applications of this natural compact representation. First, we introduce our new method to extract a spectral function from noisy quantum Monte Carlo data (analytic continuation) [4]. Furthermore, we show some unpublished data on efficient quantum many-body simulations accelerated by the use of the IR basis. Open-source softwares for analytic continuation [5] and the IR basis [6] are available online.

##### 2018 서울대학교 수리과학부 세미나

(Affine) W-algebras are a family of vertex algebras defined by Drinfeld-Sokolov reductions. We introduce free field realizations of W-algebras by using Wakimoto representations of affine Lie algebras, which we call Wakimoto representations of W-algebras. Then W-algebras may be described as the intersections of kernels of screening operators. As applications, parabolic inductions for W-algebras are obtained. This is motivated by results of Premet and Losev on finite W-algebras. In type A, this becomes a chiralization of coproducts by Brundan-Kleshchev. In type BCD, we also have analogs of the coproducts in special cases.

##### 2018 서울대학교 수리과학부 세미나

I'll talk about Cheeger-Gromov L^2 rho-invariant of 3-manifolds. Cheeger and Gromov analytically defined L^2 rho-invariant to Riemannian manifolds and showed L^2 rho-invariant has a universal bound by using deep analytic argument. Chang and Weinberger extended the definition to topological manifolds. Cha proved existence of universal bound for L^2 rho-invariant of topological manifolds and found an explicit bound in terms of a complexity of given 3-manifold. To be specific, L^2 rho-invariant of a 3-manifold can be linearly bounded by a number of 2-handles of a 4-manifold which has a boundary of given 3-manifold. We will discuss about the topological definition and proof about Cheeger-Gromov L^2 rho-invariant of 3-manifolds.

##### 2018 서울대학교 수리과학부 세미나

In this talk, I present a canonical decomposition of operator-valued strong L^2-functions by the aid of the Beurling-Lax-Halmos Theorem which characterizes the shift-invariant subspaces of vector-valued Hardy space. I also introduce a notion of the "Beurling degree" for inner functions by employing a canonical decomposition of strong L^2-functions induced by the given inner functions. Eventually, we establish a deep connection between the Beurling degree of the given inner function and the spectral multiplicity of the model operator on the corresponding model space.

##### 한국고등과학원 세미나

Although a Higgs boson was discovered at the LHC, the Nature of the Higgs boson is still unknown. So far, two big paradigms have been discussed to clarify the Nature of the Higgs boson, namely, supersymmetry and compositeness, where Higgs physics is described by weak and strong dynamics, respectively. Both of these scenarios predict a 2-Higgs doublet model (2HDM) as a low energy effective theory. In this talk, we discuss how we can determine the true dynamics at a TeV scale by focusing on differences in various properties of the 2HDM.

##### 한국고등과학원 세미나

We explain equidistribution theorems for a family of holomorphic Siegel cusp forms of GSp_4 in the level and weight aspects. A main tool is Arthur's invariant trace formula. While Shin-Templier used Euler-Poincare functions at the infinity in the formula, we use pseudo-coefficients of holomorphic discrete series to extract only holomorphic Siegel cusp forms. Then the non-semisimple contributions arise from the geometric side, and this provides new second main terms which have not been studied, and which correspond to endoscopic cuspidal representations with large discrete series at the infinity. We give several applications, including the vertical Sato-Tate theorem and low-lying zeros for degree 4 spinor L-functions and degree 5 standard L-functions of holomorphic Siegel cusp forms. This is a joint work with Satoshi Wakatsuki and Takuya Yamauchi. If time permits, we explain work in progress to the generalization to Sp_{2n}.

##### 한국고등과학원 세미나

Let L be a finite extension of $\mathbb{Q}_l$ for some odd prime l and let $L_{\infty}$ be a $\mathbb{Z}_p$-extension of L for some odd prime p. For an elliptic curve E over L, we completely classify all the cases when E($L_{\infty}$) [p^{\infty}] is finite. In the case when this group is infinite, we study its $\Lambda$-module structure and prove the pseudo-cyclicity.

##### 한국고등과학원 세미나

The contact process describes an elementary epidemic model by a continuous-time Markov chain on a given graph. In the process, each vertex is either infected or healthy, and an infected vertex gets healed at rate $1$ while it passes its disease to each of its neighbors at rate $\lambda$, where all the recoveries and infections are independent. In this talk, we discuss the phase diagram of the contact process on Galton-Watson trees and random graphs. To be specific, we show that the infection spread is subcritical for small enough $\lambda$ if $D$, the offspring distribution (resp. degree distribution) of the Galton-Watson tree (resp. random graph), has an exponential tail. On the other hand, when $D$ is subexponential, we prove that the contact process on the random graph is always supercritical. Our result on Galton-Watson trees, in particular, establishes a stronger version of the conjecture by [Huang-Durrett ‘18]. Joint work with Shankar Bhamidi, Oanh Nguyen and Allan Sly.

##### 한국고등과학원 세미나

It was unclear until recently whether the ergodicity of the geodesic flow on a given Riemannian manifold $M$ has any significant impact on the growth of the number of nodal domains of eigenfunctions of Laplace-Beltrami operator $\Delta_M$, as the eigenvalue $\lambda \to \infty$. In this talk, I'm going to explain my recent work with Steve Zelditch, where we prove that, when $M$ is a principle $S^1$-bundle equipped with a generic Kaluza-Klein metric, the nodal counting of eigenfunctions is typically $2$, independent of the eigenvalues. Note that principle $S^1$-bundle equipped with a Kaluza-Klein metric never admits ergodic geodesic flow. This, for instance, contrasts the case when $(M,g)$ is a surface with non-empty boundary with ergodic geodesic flow (billiard flow), in which case the number of nodal domains of typical eigenfunctions tends to $+\infty$ (proven in my paper with Seung Uk Jang). I will also present an orthonormal eigenbasis of Laplacian on a flat 3-torus, where every non-constant eigenfunction has exactly two nodal domains. This provides a negative answer to the question raised by Thomas Hoffmann-Ostenhof: For any given orthonormal eigenbasis of the Laplace--Beltrami operator, can we always find a subsequence where the number of nodal domains tends to $+\infty$?''

##### 한국고등과학원 세미나

By the Fourier inversion numerous operators appearing in mathematical analysis, partial differential equations, and mathematical physics are represented as Fourier multiplier operators. In most cases the associated multipliers have their singularities on certain submanifolds, and regularity properties of such operators are naturally related to the singularities of the multipliers. I will talk about several singular multiplier operators which play important roles in mathematical analysis, and aim to completely characterize mapping properties of these operators on Lebesgue spaces. Those include sharp estimates for the spherical harmonic projections, sharp resolvent estimates for the Laplacian, and Carleman estimates for second order differential operators. I will also give some applications of the estimates to partial differential equations and spectral theory. This talk is based on recent joint works with Eunhee Jeong and Sanghyuk Lee.

##### 한국고등과학원 세미나

We explain equidistribution theorems for a family of holomorphic Siegel cusp forms of GSp_4 in the level and weight aspects. A main tool is Arthur's invariant trace formula. While Shin-Templier used Euler-Poincare functions at the infinity in the formula, we use pseudo-coefficients of holomorphic discrete series to extract only holomorphic Siegel cusp forms. Then the non-semisimple contributions arise from the geometric side, and this provides new second main terms which have not been studied, and which correspond to endoscopic cuspidal representations with large discrete series at the infinity. We give several applications, including the vertical Sato-Tate theorem and low-lying zeros for degree 4 spinor L-functions and degree 5 standard L-functions of holomorphic Siegel cusp forms. This is a joint work with Satoshi Wakatsuki and Takuya Yamauchi. If time permits, we explain work in progress to the generalization to Sp_{2n}.