Prismatic cohomology, which is recently developed by Bhatt and Scholze, is a p-adic cohomology theory unifying etale, de Rham, and crystalline cohomology. In this series of two talks, we will discuss its central object of study called prismatic F-crystals, and some applications to studying p-adic Galois representations. The first part will be mainly devoted to explaining motivational background on the topic. Then we will discuss the relation between prismatic F-crystals and crystalline local systems on p-adic formal scheme, and talk about applications on purity of crystalline local system and on crystalline deformation ring. If time permits, we will also discuss recent work in progress on log prismatic F-crystals and semistable local systems. A part of the results is based on joint work with Du, Liu, Shimizu.

Prismatic cohomology, which is recently developed by Bhatt and Scholze, is a p-adic cohomology theory unifying etale, de Rham, and crystalline cohomology. In this series of two talks, we will discuss its central object of study called prismatic F-crystals, and some applications to studying p-adic Galois representations. The first part will be mainly devoted to explaining motivational background on the topic. Then we will discuss the relation between prismatic F-crystals and crystalline local systems on p-adic formal scheme, and talk about applications on purity of crystalline local system and on crystalline deformation ring. If time permits, we will also discuss recent work in progress on log prismatic F-crystals and semistable local systems. A part of the results is based on joint work with Du, Liu, Shimizu.

Affine Deligne-Lusztig varieties show up naturally in the study of Shimura varieties, Rapoport-Zink spaces, and moduli spaces of local shtukas. Among various questions on its geometric properties, the question on the connected components turns out to be a fairly important problem. For example, Kisin, in his proof of the Langlands-Rapoport conjecture (in a weak sense) for abelian type Shimura variety with the hyperspecial level structure, crucially used the description of the set of connected components. Since then, many authors have answered this question in various restricted cases. I will first discuss what is the conjectural description of the connected components and related previous works. Then, I will explain my new result (joint work with Ian Gleason and Yujie Xu) which finishes the question in the mixed characteristic case and, if time permits, new ingredients.

Affine Deligne-Lusztig varieties are first defined by Rapoport as the (conjectural) p-part of the so-called Langlands-Rapoport conjecture. It can be understood as a p-adic generalization of the classical Deligne-Lusztig varieties. One of the most basic questions is 'when they are nonempty'. For a certain union, the nonemptiness criterion is completely known (by the so-called Mazur's inequality or B(G,μ)). However, the question about the "individual" ones is moderately open (with no general conjecture). I will discuss old and new nonemptiness results and suggest a new conjecture, for the individual ones, in the basic case. As an application, I will briefly mention a new explicit dimension formula in the rank 2 case (for which no conjectural formula was stated before).

The converse theorem for automorphic forms has a long history beginning with the work of Hecke (1936) and a work of Weil (1967): relating the automorphy relations satisfied by classical modular forms to analytic properties of their L-functions and the L-functions twisted by Dirichlet characters. The classical converse theorems were reformulated and generalised in the setting of automorphic representations for GL(2) by Jacquet and Langlands (1970). Since then, the converse theorem has been a cornerstone of the theory of automorphic representations. Venkatesh (2002), in his thesis, gave new proof of the classical converse theorem for modular forms of level 1 in the context of Langlands’ “Beyond Endoscopy”. In this talk, we extend Venkatesh’s proof of the converse theorem to forms of arbitrary levels and characters with the gamma factors of the Selberg class type. This is joint work with Andrew R. Booker and Michael Farmer.

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심사위원장: 백형렬, 심사위원: 남경식, 최서영, Kasra Rafi(University of Toronto), Giulio Tiozzo(University of Toronto)

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심사위원장: 이창옥, 심사위원: 김동환, 임미경, 예종철(겸임교수), 한송희(삼성전자)

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심사위원장: 김재경, 심사위원: 김용정, 정연승, 황강욱, 이승규(경상대학교)

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심사위원장: 김용정, 심사위원: 권순식, 강문진, 김재경, 윤창욱(충남대학교)

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심사위원장: 안드레아스 홈슨, 심사위원: 김동수, 김재훈, 엄상일, 김민기(광주과학기술원)

This study is concerned with multivariate approximation by non-polynomial functions with internal shape parameters. The main topics of this presentation are two folds. First, interpolation by radial basis function (RBF) is considered. We especially discuss the convergence behavior of the RBF interpolants when the basis function is scaled to be increasingly flat. Moreover, we investigate the advantages of interpolation methods based on exponential polynomials. The second topic of this presentation is the approximation method based on sparse grids in $[0,1]^d \subset \RR^d$. The goal of sparse grid methods is to approximate high dimensional functions with good accuracy using as few grid points as possible. In this study, we present a new class of quasi-interpolation schemes for the approximation of multivariate functions on sparse grids. Each scheme in this class is based on shifts of kernels constructed from one-dimensional RBFs such as multiquadrics. The kernels are modified near the boundaries to prevent deterioration of the fidelity of the approximation. We show that our methods provide significantly better rates of approximation, compared to another quasi-interpolation scheme in the literature based on the Gaussian kernel using the multilevel technique. Some numerical results are presented to demonstrate the performance of the proposed schemes.

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Online: https://kaist.zoom.us/j/81807153144

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심사위원장: 엄상일, 심사위원: 안드레아스 홈슨, 김재훈, 권오정(한양대학교), Hong Liu(기초과학연구원)

Metal artifact reduction has become a challenging issue for practical X-ray CT applications since metal artifacts severely cause contrast degradation and the misinterpretation of information about the property and structure of a scanned object. In this talk, we propose a methodology to reduce metal artifacts by extending the method proposed by Jeon and Lee (2018) to a three-dimensional industrial cone beam CT system. We develop a registration technique managing the three dimensional data in order to find accurate segmentation regions needed for the proposed algorithm. Through various simulations and experiments, we verify that the proposed algorithm reduces metal artifacts successfully.

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(Online participation) Zoom Link: https://kaist.zoom.us/j/87958862292

In a region closer to the boundary compared to Prandtl layer, an inviscid disturbance can be manifested by the interaction with viscous mode via the no-slip boundary condition due to resonance. In some unstable range of parameters, this leads to instability in the transition regime from laminar flow to turbulence. This instability phenomenon was observed by physicists long time ago, such as Heisenberg, Tollmien and C.C. Lin, etc. And it was justified rigorously in mathematics by Grenier-Guo-Nguyen using the incompressible Navier-Stokes equation. In this talk, we will present some results on this phenomenon in some other physical situations in which the governing system is either MHD or compressible Navier-Stokes equation. The talk is based on some recent joint work with Chengjie Liu and Zhu Zhang.