# 학과 세미나 및 콜로퀴엄

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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.

Liouville quantum gravity (LQG) surfaces are random topological surfaces which are important in statistical mechanics and have deep connections to other mathematical objects such as Schramm–Loewner evolution and random planar maps. These random surfaces are too singular and fractal in the sense that the Hausdorff dimension, viewed as a metric space equipped with its intrinsic metric, is strictly bigger than two. I will talk about the interesting geometric structure and recent progress on LQG surfaces.

하나금융 융합기술원은 국내 금융그룹 최초의 AI 연구소로 2018년부터 지난 4년 간 다양한 금융서비스에 현행 AI 응용기술들을 접목시키고 금융사 내 기술 전파에 큰 성과를 올려왔다. 그 중에서도 융합기술원이 연구/개발하는 신용평가 기술은 업계를 선도하고 있으며 그런 선도 기술을 만들어나가는 과정을 소개하려 한다. 또한, 응용기술 뿐만 아니라 향후 다양한 분야의 원천기술 연구를 위해 국내 유수 산업/학계 인재들이 모이는 조직으로 변형해가는 노력을 소개할 예정이다.

온라인, 오프라인 동시진행

온라인, 오프라인 동시진행

In this presentation, I will present me, Daeyeol Jeon, and Chang Heon Kim's construction of certain points on $X_1(N)$ over ring class fields (and therefore construction of points on the abelian varieties associated to newforms of level $\Gamma_1(N)$). Our work generalizes Bryan Birch's Heegner points on $X_0(N)$. Then, we show that these points form Euler systems (like the Heegner points), and we improve Kolyvagin's Euler system techniques to show that for our point $P_{\tau_K/c}$ and any ring class character $\chi$ of the extended ring class field of conductor $c$ satisfying $\chi=\overline{\chi}$, if $P_{\tau_K/c}^\chi$ is non-torsion and $G_K \to \operatorname{Aut} A_f[\pi]$ is surjective, then the corank of $\Sel(A_\chi/K)$ is 1, which implies the rank of $A_f(K)^\chi$ is 1. (Please contact Bo-Hae Im if you want to join the seminar.)

In recent years, local regularity theory for weak solutions to nonlocal equations with fractional orders has been studied extensively. In this talk, we discuss on local regularity for weak solutions to nonlocal equations with nonstandard growth and differentiability. In particular, we consider nonlocal equations of a variable exponent type, a double phase type and an Orlicz type.

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https://kaist.zoom.us/j/84619675508
콜로퀴엄
이효정 (경북대학교 통계학과)
Mathematical modeling for infectious disease using epidemiological data

https://kaist.zoom.us/j/84619675508

콜로퀴엄

The new infectious disease are emerging around the world. Coronavirus disease 2019 (COVID-19) caused by a novel coronavirus has emerged and has been rapidly spreading. The World Health Organization (WHO) declared the COVID-19 outbreak a global pandemic on March 11, 2020. Mathematical modelling plays a key role in interpreting the epidemiological data on the outbreak of infectious disease. Moreover, mathematical modeling can give us an early warning about the size of the outbreak. First, we construct a mathematical model to estimate the effective reproduction numbers, which assess the effect of control interventions. Second, we forecast the COVID-19 cases according to the different effect of control interventions. Finally, the most effective intervention can be suggested in terms of modeling approach. In this talk, I’d like to briefly introduce the main results of recent research on the mathematical modeling for various infectious diseases.

ZOOM링크: https://kaist.zoom.us/j/84619675508

ZOOM링크: https://kaist.zoom.us/j/84619675508

The talk with start with an introduction to Stark’s conjectures. We will then specialise to the situation of Brumer-Stark conjecture and its various refinements. I will then sketch a proof of the conjecture. This is a joint work with Samit Dasgupta.

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

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

Geometric and functional inequalities play a crucial role in several problems arising in analysis and geometry.
Proving the validity of such inequalities, and understanding the structure of minimizers, is a classical and important question.
In these lectures I will first give an overview of this beautiful topic and discuss some recent results.

We design and analyze V‐cycle multigrid methods for problems posed in H(div) and H(curl). Due to the fact that traditional smoothers do not work well for the vector field problems, special approaches for smoothers in the multigrid methods are essential. We introduce new smoothing techniques which involve non-overlapping domain decomposition preconditioners based on substructuring. We prove uniform convergence of the V‐cycle methods on bounded convex hexahedral domains. Numerical experiments that support the theory are also presented.

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.

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

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.

Please contact Wansu Kim at for Zoom meeting info and any inquiry. For the list of Number Theory seminar talks, please visit the KAIST Number Theory seminar webpage. https://sites.google.com/site/wansukimmaths/kants-kaist-number-theory-seminar

Please contact Wansu Kim at for Zoom meeting info and any inquiry. For the list of Number Theory seminar talks, please visit the KAIST Number Theory seminar webpage. https://sites.google.com/site/wansukimmaths/kants-kaist-number-theory-seminar

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.

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.

Zoom ID: 352 730 6970, Password: 9999. You will be authorized individually by the host of the meeting.

Zoom ID: 352 730 6970, Password: 9999. You will be authorized individually by the host of the meeting.

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.

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.

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.

Please contact Wansu Kim at for Zoom meeting info and any inquiry. For the list of Number Theory seminar talks, please visit the KAIST Number Theory seminar webpage. https://sites.google.com/site/wansukimmaths/kants-kaist-number-theory-seminar

Please contact Wansu Kim at for Zoom meeting info and any inquiry. For the list of Number Theory seminar talks, please visit the KAIST Number Theory seminar webpage. https://sites.google.com/site/wansukimmaths/kants-kaist-number-theory-seminar

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

KAIX Distinguished lectures in Mathematics
Speaker : Wen-Ching Winnie Li (Distinguished Professor of Mathematics, Penn. State Univ.)
2021.11.09 (Tue) - Korean time
09:30-10:30 Colloquium talk
Primes in Number Theory and Combinatorics
10:30-10:50 Q&A
11:00-12:00(noon) Seminar Talk
Pair arithmetical equivalence for quadratic fields
ZOOM ID : 518 127 6292
(No password required)
Abstract:
1. colloquium talk
Title: Primes in number theory and combinatorics
Abstract: Prime numbers are a central topic in number theory. They have inspired the study of many subjects in mathematics. Regarding prime numbers as the building blocks of the multiplicative structure of positive integers, in this survey talk we shall interpret "primes" as the basic elements in a structure of interest arising from combinatorics and number theory, and explore their distributions of various kinds. More precisely, we shall examine primes in compact Riemann surfaces, graphs, and 2-dimensional simplicial complexes, respectively. These results are products of rich interplay between number theory and combinatorics.
2. number theory seminar talk
Title: Pair arithmetical equivalence for quadratic fields
Abstract: Given two nonisomorphic number fields K and M, and two finite order Hecke characters $\chi$ of K and $\eta$ of M respectively, we say that the pairs $(\chi, K)$ and $(\eta, M)$ are arithmetically equivalent if the associated L-functions coincide: $L(s, \chi, K) = L(s, \eta, M)$. When the characters are trivial, this reduces to the question of fields with the same Dedekind zeta function, investigated by Gassmann in 1926, who found such fields of degree 180, and by Perlis in 1977 and others, who showed that there are no arithmetically equivalent fields of degree less than 7.
In this talk we discuss arithmetically equivalent pairs where the fields are quadratic. They give rise to dihedral automorphic forms induced from characters of different quadratic fields. We characterize when a given pair is arithmetically equivalent to another pair, explicitly construct such pairs for infinitely many quadratic extensions with odd class number, and classify such characters of order 2.
This is a joint work with Zeev Rudnick.

In my next talk, I will define canonical dimension of varieties (which, roughly speaking, measures how hard it is to get a rational point in a given variety) and canonical dimension of algebraic groups (which, roughly speaking, measures how complicated the torrsors of an algebraic group can be). Then I will state several previously known facts from intersection theory and from theory of canonical dimension, and I will prove that if we know that a certain product of Schubert divisors is mutiplicity-free (which was defined in my first talk), then this fact implies an upper estimate on the canonical dimension of the group and its torsors. As a result, we will get some explicit numerical estimates on canonical dimension of simply connected simple split algebraic groups groups with simply-laced Dynkin diagrams.

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 first talk will be mainly devoted to giving brief expository accounts of some background materials needed to understand the notion of twisted derived equivalence in the setting of derived/spectral algebraic geometry; in particular, some familiarity with ordinary algebraic geometry will be enough for the talk.

We will survey recent development in subadditive thermodynamic formalism for matrix cocycles. In particular, in the setting of locally constant cocycles as well as fiber-bunched cocycles, we will discuss sufficient conditions for the norm potentials of such cocycles to have unique equilibrium states. If time permitting, we will also discuss ergodic properties of such equilibrium states as well as some applications.

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https://kaist.zoom.us/j/84619675508
콜로퀴엄
서수길 (연세대학교)
On a conjecture of Coleman concerning Euler systems

https://kaist.zoom.us/j/84619675508

콜로퀴엄

We introduce a distribution-theoretic conjecture of Roert Coleman of the 1980's and prove the conjecture in a recent joint work with Burns and Daoud. This accordingly gives an explicit description of the complete set of Euler systems for the multiplicative group over Q together with a connection to other conjectures in number theory.

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https://kaist.zoom.us/j/3098650340
PDE 세미나
김정호 (한양대학교 수학과)
Hydrodynamic limits of the Schrodinger equation with gauge fields

https://kaist.zoom.us/j/3098650340

PDE 세미나

In this talk, we present the hydrodynamic limits of the Schrodinger equation, affected by different gauge fields. Precisely, we first present the hydrodynamic limit of the Schrodinger equation with the Chern-Simons gauge fields (Chern-Simons-Schrodinger equation), toward to the Euler-Chern-Simons equation on the two-dimensional state space. Then, we consider the hydrodynamic limit of the Schrodinger equation with the Maxwell gauge fields (Maxwell-Schrodinger equation), toward to the Euler-Maxwell equation on the three-dimensional state space. Both estimate use the estimate on the modulated energy functionals.

First, I will say a few words about Galois descent in the particular case of a projective variety embedded into a projective space. Then I will recall the definintion of a torsor and will explain how to construct the quotient of a torsor of a simple simply connected split algebraic group modulo a Borel subgroup. Finally, I will prove that the Picard group of such a quotient does not change for one particular finite Galois extension of the base field, and then, if there is enough time, for any extension of the base field.