▷ 금종해 (KIAS)

Title: Mori dream surfaces

Abstract:

The Cox ring of a variety is the total coordinate ring, i.e., the direct sum of all spaces of global sections of all divisors.

When this ring is finitely generated, the variety is called Mori dream (MD). A necessary condition for being MD is the finite generatedness of Pic(X), i.e., the vanishing of the irregularity. 

Smooth rational surfaces with big anticanonical divisor are MD. Thus all del Pezzo surfaces of any degree are.

A K3 surface or an Enriques surface is MD iff its automorphism group is finite.

In this talk I will consider the case of surfaces of general type with p_g=0, and provide several examples that are MD.

I will also provide non-minimal examples that are not MD.

This is a joint work with Kyoung-Seog Lee.

▷ 김찬호 (KIAS)

Title: How not to prove Birch and Swinnerton-Dyer conjecture

Abstract:

The speaker first apologizes for this title against the motto of HCMC. Without a doubt, Birch and Swinnerton-Dyer (BSD) conjecture is one of the central problems in number theory. It explains how the information of the rational points of elliptic curves is encoded in the behavior of their complex L-functions at the critical point. Although there has been significant progress toward this conjecture for several decades, our understanding is still extremely limited when the vanishing order of the L-function is larger than one. Recently we have developed a slightly different and more refined connection between the rational points of elliptic curves and a “discrete” analogue of complex L-functions naturally arising from zeta elements. This discrete analogue reveals a structural refinement of the BSD conjecture without any restriction on the "vanishing order” of discrete L-functions. Some explicit examples will be given. If time permits, we also discuss the case of modular forms of arbitrary weight.

▷ 김현규 (KIAS)

Title: Cluster varieties, higher Teichmüller spaces, and quantization

Abstract:

I will review the definitions of Fock-Goncharov's cluster varieties in elementary terms. As examples I will present ordinary and higher Teichmüller spaces of punctured surfaces. I will discuss the problem of finding a nice basis of the algebra of regular functions on a cluster variety, and also the problem of quantization. If time allows, I will mention a connection to mirror symmetry of log Calabi-Yau varieties.

▷ 라준현 (KIAS)

TBA

▷ 변성수 (Seoul National University)

Title: Real Eigenvalues of Asymmetric Random Matrices

Abstract:

In this talk, I will discuss how the fundamental concepts in probability theory—the law of large numbers, the central limit theorem, and the large deviation principle—are developed in the study of real eigenvalues of asymmetric random matrices.

▷ 오세욱 (KIAS)

Title: Maximal averages over submanifolds

Abstract:

In this talk, I will introduce maximal averages over submanifolds, which are generalizations of the Hardy-Littlewood maximal function.

I will discuss how the shape of submanifold affects to the Lp boundedness of maximal average. If time allows, I will also talk about multiparameter maximal averages.

▷ 이우영 (KIAS)

Title: Why so much attention to mathematical challenges and higher-order de Branges-Rovnyak spaces

Abatract: 

The first part of this talk briefly examines the question of why so much attention to mathematical challenges. In the second part, we briefly consider the motivation behind the emergence of higher-order de Branges-Rovnyak spaces in the context of the invariant subspace problem. The novel concept of higher-order de Branges-Rovnyak spaces is obtained by combining the studies of de Branges-Rovnyak spaces and hypercontractions, both topics have been studied intensively in the last several decades. In particular, we show they can be viewed as iterated de Branges-Rovnyak spaces. We also apply an abstract operator theoretic approach to the study of higher-order de Branges-Rovnyak spaces in the weighted Bergman spaces and compute their reproducing kernels.

Eventually we find finite dimensional higher-order de Branges-Rovnyak spaces in the weighted Bergman spaces.

▷ 이철희 (KIAS)

Title: Laplacian on a compact Riemann surface with many automorphisms

Abatract: 

The multiplicity of eigenvalues of the Laplacian on a compact Riemann surface is difficult to determine in general. As the eigenspace of a fixed eigenvalue is stable under the action of its automorphism group, there are more tools available to deal with this question when the surface admits large symmetries.
We will discuss some experimental ideas to decompose the eigenspaces of the Laplacian as a representation of the automorphism group.