Wednesday, November 10, 2021

2021-11-17 / 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.

2021-11-12 / 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.

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

학과 세미나/콜로퀴엄 - 대수기하학:

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Let  C⊂P^r be a nondegenerate projective integral curve of degree d and arithmetic genus g. A celebrated theorem of Castelnuovo gives an explicit upper bound pi_0(d,r) on g in terms of d and n. Moreover, if d ≥ 2r+1 then g=pi_0 (d,r) if and only if C is ACM and it lies on a surface of minimal degree. In 1980, Joe Harris and David Eisenbud proved that (i) C lies on a surface of minimal degree if g> pi_1 (d,r), and (ii) if g=pi_1(d,r) and C does not lie on a surface of minimal degree, then there exists a del Pezzo surface which contains C. Along this line, we will show that there exists an integer pi_1(d,r)^' < pi_1(d,r) such that  C lies on a del Pezzo surface if g> pi_1(d,r)^' This is a joint work with Wanseok Lee

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

학과 세미나/콜로퀴엄 - 대수기하학:

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In this talk we consider the Waring rank of monomials over the rational numbers. We give a new upper bound for it by establishing a way in which one can take a structured apolar set for any given monomial. This bound coincides with all the known cases for the real rank of monomials, and is sharper than any other known bounds for the real Waring rank. Since all of the constructions are still valid over the rational numbers, this provides a new result for the rational Waring rank of any monomial as well. We also apply the methods developed in the paper to the problem of finding an explicit rational Waring decomposition of any homogeneous polynomial over rational numbers, which is important in many applications, especially to the integration of a polynomial over a simplex. We will present examples and computational implementation for potential use.

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

학과 세미나/콜로퀴엄 - PDE 세미나:

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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.

2021-11-10 / 13:00 ~ 14:00

학과 세미나/콜로퀴엄 - PDE 세미나:

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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.

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

IBS-KAIST 세미나 - 수리생물학:

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Abstract: From fertilization to birth, successful mammalian reproduction relies on interactions of elastic structures with a fluid environment. Sperm flagella must move through cervical mucus to the uterus and into the oviduct, where fertilization occurs. In fact, some sperm may adhere to oviductal epithelia, and must change their pattern of oscillation to escape. In addition, coordinated beating of oviductal cilia also drive the flow. Sperm-egg penetration, transport of the fertilized ovum from the oviduct to its implantation in the uterus and, indeed, birth itself are rich examples of elasto-hydrodynamic coupling. We will discuss successes and challenges in the mathematical and computational modeling of the biofluids of reproduction.

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

학과 세미나/콜로퀴엄 - PDE 세미나:

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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.

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

학과 세미나/콜로퀴엄 - 정수론: On Euler systems for the multiplicative group over general number fields

by Prof. Soogil Seo(Yonsei University)

We formulate, and provide strong evidence for, a natural generalization of a conjecture of Robert Coleman concerning higher rank Euler systems for the multiplicative group over arbitrary number fields. This is a joint work with Burns, Daoud, and Sano.

Events for the 취소된 행사 포함