# 학과 세미나 및 콜로퀴엄

Recently, mapping a signal/image into a low rank Hankel/Toeplitz matrix has become an emerging alternative to the traditional sparse regularization, due to its ability to alleviate the basis mismatch between the true support in the continuous domain and the discrete grid. In this talk, we introduce a novel structured low rank matrix framework to restore piecewise smooth functions. Inspired by the total generalized variation to use sparse higher order derivatives, we derive that the Fourier samples of higher order derivatives satisfy an annihilation relation, resulting in a low rank multi-fold Hankel matrix. We further observe that the SVD of a low rank Hankel matrix corresponds to a tight wavelet frame system which can represent the image with sparse coefficients. Based on this observation, we also propose a wavelet frame analysis approach based continuous domain regularization model for the piecewise smooth image restoration.

The degree-shifting action on the cohomology of locally symmetric spaces, which has its origins in the representation theory of real reductive groups, enjoys a surprising connection with arithmetic, as expected by the so-called motivic action conjectures of A. Venkatesh. Although these conjectures are expected to hold in great generality, there is a disparity between the algebraic and non-algebraic locally symmetric spaces. We will discuss the nature of the degree-shifting action in both cases
(For those who cannot attend the in-person seminar, we will also stream the seminar talk via Zoom. Please contact Wansu Kim for the Zoom connection details.)

In this talk, we study the behaviour of rational points on the expanding horospheres in the space of unimodular lattices. The equidistribution of these rational points is proved by Einsiedler, Mozes, Shah and Shapira (2016). Their proof uses techniques from homogeneous dynamics and relies particularly on measure-classification theorems due to Ratner. We pursue an alternative strategy based on Fourier analysis, Weil's bound for Kloosterman sums, recently proved bounds (by M. Erdélyi and Á. Tóth) for matrix Kloosterman sums, Roger's formula, and the spectral theory of automorphic functions. Our methods yield an effective estimate on the rate of convergence for a specific horospherical subgroup in any dimension.
This is a joint work with D. El-Baz, B. Huang, J. Marklof and A. Strömbergsson.

For the last decade, there have been a number of studies reporting that certain surface singularities give rise to vector bundle on their smoothing. The first result is by Hacking, who studies this correspondence for Wahl singularities. I am going to introduce a generalization of Hacking's result to singularities of class T, which is a natural extension of Wahl singularities. Also, if time permits, I will introduce a recent result of Tevelev-Urzua which generalizes this to arbitrary cyclic quotient surface singularities.

Given a space, one can study its singularities. The converse direction is called reconstruction problem: How to reconstruct spaces from given singularity information? In this talk, by introducing a notion called a semicascade we derive a bound of Picard number for toric log del Pezzo surfaces in terms of the singular points generalizing some results of Dais and Suyama, which solves the reconstruction problem with the help of computer. We also discuss Kähler-Einstein toric log del Pezzo surfaces as an application of semicascades.

A monotone symplectic manifold is a symplectic analogue of a smooth Fano variety and it provides an important classes of objects, called monotone Lagrangian tori, in view of mirror symmetry. In this talk, I will explain a way of producing monotone Lagrangian tori in a given smooth Fano variety using toric degeneration. Using this technique, we prove that there exist infinitely many monotone Lagrangian tori not Hamiltonian isotopic to each other in a full flag variety. This is based on joint work with Myungho Kim, Yoosik Kim, Jaehoon Kwon, and Euiyong Park at Center for Quantum Structures in Modules and Spaces (QSMS).

In this talk, we introduce a various methods of representations of graphs which are mathematical objects expressing a variety of non-Euclidean data such as Molecules, social networks, genes, transportation networks, citation networks of papers and so on. Graph representation as a Euclidean vector is inevitable in machine learning for classifications for graphs which is closely related to graph neural network in computer science. We would like to introduce a few literatures, Weisfeiler-lehman algortihm, random walks, graph convolution whci are commonly used techniques and explain the result of combining them with topological invarints of graphs

https://sites.google.com/view/mwagaag

https://sites.google.com/view/mwagaag

The introduction for the framework of geometric deep learning will be explained in the perspective of new methodology of A.I. and data analysis. Various applications can be discussed by utilizing geometry, algebra, topology.

https://sites.google.com/view/mwagaag

https://sites.google.com/view/mwagaag

Abstract: We discuss a new application of (a part of) the Iwasawa main conjecture to the non-triviality of Kato's Kolyvagin systems and a structural refinement of Birch and Swinnerton-Dyer conjecture. In particular, the structure of Selmer groups is completely determined by certain modular symbols for a large class of elliptic curves.
(Please contact Wansu Kim at for Zoom meeting info or any inquiry.)

(Please contact Wansu Kim at for Zoom meeting info or any inquiry.)

(Please contact Wansu Kim at for Zoom meeting info or any inquiry.)

One of the important work in graph theory is the graph minor theory developed by Robertson and Seymour in 1980-2010. This provides a complete description of the class of graphs that do not contain a fixed graph H as a minor. Later on, several generalizations of H-minor free graphs, which are sparse, have been defined and studied. Also, similar topics on dense graph classes have been deeply studied. In this talk, I will survey topics in graph minor theory, and discuss related topics in structural graph theory.

ZOOM Meeting ID: 873 7478 2790 Direct link: https://kaist.zoom.us/j/87374782790

ZOOM Meeting ID: 873 7478 2790 Direct link: https://kaist.zoom.us/j/87374782790