Friday, November 12, 2021

2021-11-19 / 13:00 ~ 14:00

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This series of lectures will focus on recent developments of the so-called a-contraction theory and its application to the study of discontinuous flow at high Reynolds numbers. We will first introduce the classical framework to study the stability of 1D shocks for compressible flows. Recent multi-D applications will be presented next, both in the context of compressible and incompressible flows.

2021-11-17 / 13:00 ~ 14:00

학과 세미나/콜로퀴엄 - PDE 세미나:

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2021-11-12 / 11:00 ~ 12:00

학과 세미나/콜로퀴엄 - 대수기하학:

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In this talk, we present how to glue linear matrices in order to obtain a bigger linear matrix in a certain circumstance, and as a consequence, classify higher secant varieties of minimal degree. It is worth noting that by the del Pezzo-Bertini classification, a variety of minimal degree has determinantal presentation whenever its codimension is not small, and that higher secant varieties of minimal degree generalize varieties of minimal degree. This is a joint work with Prof. Sijong Kwak.

2021-11-12 / 13:00 ~ 14:00

학과 세미나/콜로퀴엄 - PDE 세미나:

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2021-11-18 / 12:25 ~ 12:45

대학원생 세미나 - 대학원생 세미나: Chern classes of tautological sheaves on Hilbert schemes of points on surface

by 최도영(KAIST)

I will introduce some concepts of Chern classes, Hilbert schemes and tautological sheaves on Hilbert scheme of points which is associated to a line bundle on surfaces. Also, I will provide a brief description of Lehn's work which gives an algorithmic approach of the action of the Chern classes of tautological bundles on the cohomology of Hilbert schemes of points on a smooth surface. His work is based on the framework of Nakajima's oscillator algebra. At the end, I will present the computation of the top Segre classes of tautological bundles associated to line bundles on $Hilb^n$ up to $n \leq 7$, extending computations of Severi, LeBarz, Tikhomirov and Troshina.

2021-11-18 / 12:00 ~ 12:20

대학원생 세미나 - 대학원생 세미나: What is the correct diffusion equation in heterogeneous mediums

by 김호연(KAIST)

In the classical diffusion theory, the diffusivity has been regarded as an intrinsic property of particles. However, it can't explain diffusion phenomena in heterogeneous medium, one of the most famous example is Soret effect. The diffusivity can be changed along different mediums and it arises a question: how can we express heterogeneous diffusion. In this talk, I'll introduce the heterogeneous diffusion equation we found and give some experimental data verifying this work.

2021-11-15 / 16:30 ~ 17:30

학과 세미나/콜로퀴엄 - 대수기하학:

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2021-11-17 / 16:15 ~ 17:15

학과 세미나/콜로퀴엄 - 정수론: Average size of Selmer ranks in quadratic twist families of elliptic curves over global function fields

by Sun Woo Park(University of Wisconsin-Madison / NIMS)

In a recent joint work with Niudun Wang, we prove new results towards the Bhargava-Kane-Lenstra-Poonen-Rains conjectures on the first moment of Selmer groups over quadratic families of elliptic curves over global function fields. The key ingredients used in the proof are the Grothendieck-Lefschetz trace formula and zeroth homological stability of fiber bundles over configuration spaces. Both ideas form the backbone of a seminal work by Ellenberg, Venkatesh, and Westerland (2016), a rich incorporation of algebraic topological methods to arithmetic geometry. We shall give an overview of how these ideas are incorporated in analyzing the average size of Selmer groups, and examine how they can be implemented to approaching other arithmetic problems.

2021-11-19 / 10:00 ~ 11:00

SAARC 세미나 - SAARC 세미나:

by 김판기(서울대학교, 수리과학과)

In this talk, we first review some basics on stochastic processes. Then we discuss about the recent developments on Brownian-like jump processes. This talk is based on joint projects with Ante Mimica, Joohak Bae, Jaehoon Kang, Jaehun Lee.

2021-11-12 / 10:00 ~ 11:00

SAARC 세미나 - SAARC 세미나:

by 양홍석(카이스트, 전산학부)

Deep neural networks have brought remarkable progress in a wide range of applications, but a satisfactory mathematical answer on why they are so effective has yet to come. One promising direction, with a large amount of recent research activity, is to analyse neural networks in an idealised setting where the networks have infinite widths and the so-called step size becomes infinitesimal. In this idealised setting, seemingly intractable questions can be answered. For instance, it has been shown that as the widths of deep neural networks tend to infinity, the networks converge to Gaussian processes, both before and after training, if their weights are initialized with i.i.d. samples from the Gaussian distribution and normalised appropriately. Furthermore, in this setting, the training of a deep neural network is shown to achieve zero training error, and the analytic form of a fully-trained network with zero error has been identified. These results, in turn, enable the use of tools from stochastic processes and differential equations for analyzing deep neural networks in a novel way.In this talk, I will explain our efforts for extending the above analysis to a new type of neural networks that arise from recent studies on Bayesian deep neural networks, network pruning, and design of effective learning rates. In these networks, each network node is equipped with its own scala parameter that is intialised randomly and independently but is not updated during training. This scale parameter of a node determines the scale of weights of outgoing network edges from the node at initialisation, thereby introducing the dependency among the weights. Also, its square becomes the learning rate of those weights. I will show that these networks at given inputs become infinitely-divisible random variables at the infinite-width limit, and describe how this characterisation at the infinite-width limit can help us to understand the behaviour of these neural networks.This is joint work with Hoil Lee, Juho Lee, and Paul Jung at KAIST, Francois Caron at Oxford, and Fadhel Ayed at Huawei technologies

2021-11-12 / 15:00 ~ 16:00

학과 세미나/콜로퀴엄 - 대수기하학: Twisted equivalences in spectral algebraic geometry II

by 조창연(서울대 QSMS)

Derived equivalence has been an interesting subject in relation to Fourier-Mukai transform, Hochschild homology, and algebraic K-theory, just to name a few. On the other hand, the attempt to classify schemes by their derived categories twisted by elements of Brauer groups is very restrictive as we have a positive answer only for affines. I'll talk about how we can extend this result to a broader class of algebro-geometric objects in the setting of derived/spectral algebraic geometry at the expense of a stronger notion of twisted equivalences than that of ordinary twisted derived equivalences. I'll convince you that the new notion is not only reasonable, but also indispensable from this point of view. The second talk will be dedicated to studying twisted derived equivalences in the derived/spectral setting. As a consequence, a derived/spectral analogue of Rickard's theorem, which shows that derived equivalent associative rings have isomorphic centers, will be discussed. I'll try to avoid technicalities related to using the language of derived/spectral algebraic geometry.

2021-11-18 / 11:00 ~ 12:00

IBS-KAIST 세미나 - 수리생물학:

by ()

TBA

2021-11-12 / 16:00 ~ 17:00

학과 세미나/콜로퀴엄 - PDE 세미나:

by ()

In this talk I will consider the spectral gap for the linearized Boltzmann or Landau equation with soft potentials. It is known that the corresponding collision operators admit only the degenerated spectral gap. We rather prove the formation of spectral gap in the spatially inhomogeneous setting where the space domain is bounded with an inflow boundary condition. The key strategy is to introduce a new Hilbert space with an exponential weight function that involves the inner product of space and velocity variables and also has the strictly positive upper and lower bounds. The action of the transport operator on such space-velocity dependent weight function induces an extra non-degenerate relaxation dissipation in large velocity that can be employed to compensate the degenerate spectral gap and hence give the exponential decay for solutions in contrast with the sub-exponential decay in either the spatially homogeneous case or the case of torus domain. The result reveals a new insight of hypocoercivity for kinetic equations with soft potentials in the specified situation.

2021-11-19 / 16:00 ~ 17:00

학과 세미나/콜로퀴엄 - PDE 세미나:

by 지정민()

Singular perturbations occur when a small coefficient affects the highest order derivatives in a system of partial differential equations. From the physical point of view, singular perturbations generate thin layers near the boundary of a domain, called boundary layers, where many important physical phenomena occur. In fluid mechanics, the Navier-Stokes equations, which describe the behavior of viscous flows, appear as a singular perturbation of the Euler equations for inviscid flows, where the small perturbation parameter is the viscosity. In general, verifying the convergence of the Navier-Stokes solutions to the Euler solution (known as the vanishing viscosity limit problem) remains an outstanding open question in mathematical physics. Up to now, it is not known if this vanishing viscosity limit holds true or not, even in 2D for which the existence, uniqueness, and regularity of solutions for all time are known for both the Navier-Stokes and Euler. In this talk, we discuss a recent result on the boundary layer analysis for the Navier-Stokes equations under a certain symmetry where the complete structure of boundary layers, vanishing viscosity limit, and vorticity accumulation on the boundary are investigated by using the method of correctors. We also discuss how to implement effective numerical schemes for slightly viscous fluid equations where the boundary layer correctors play essential roles. This is a joint work in part with J. Kelliher, M. Lopes Filho, A. Mazzucato, and H. Nussenzveig Lopes, and with C.-Y. Jung and H. Lee.

2021-11-18 / 16:00 ~ 17:00

학과 세미나/콜로퀴엄 - 콜로퀴엄: The Monster and the universe

by 김현규(이화여자대학교)

I will give an introduction to the Monstrous moonshine conjectures of 70's-80's, which are on remarkable relations between Klein's j-invariant in number theory and the Monster sporadic simple group. I will only assume mild basic knowledge of complex analysis and group theory. I will start from a brief introduction to modular forms and Hauptmoduln, then connect it to finite simple groups. If I can manage the time, I will briefly explain a hint to a connection to the 3d gravity theory. https://kaist.zoom.us/j/84619675508

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