In this talk, we prove a generalization of the del Pezzo-Bertini classification of varieties of minimal degree to higher secant varieties of minimal degree. It states that higher secant varieties of minimal degree are mostly divided into two classes: scroll type and Veronese type. Its proof is based on methods of gluing some 1-generic matrices. We also present some simple examples to explain our result. This is a joint work with Prof. Sijong Kwak.

Free-by -cyclic groups have been studied as algebraic counterparts of cusped hyperbolic mapping torus groups. Free-by-cyclic groups and cusped hyperbolic mapping torus groups share many algebraic properties. Nonetheless, free-by-cyclic groups are more complicated because not every free-by-cyclic group is realized as a cusped hyperbolic mapping torus group. In this talk, I explain some basic concepts and summarize some previous results related to free-by-cyclic groups. Also, I discuss some problems about free-by-cyclic groups.

The Yau-Zaslow formula describes the number of rational curves in a linear system on a smooth projective K3 surface in terms of a modular form. In this talk, I will review the Yau-Zaslow formula with some examples and then discuss an equivariant version of the formula for K3/abelian surfaces. When the K3/abelian surface admits a finite group G-action, we can consider a linear system with the induced action. It turns out that the equivariant version of the formula will count G-rational curves and it will also provide interesting modular forms.

심사위원장 : 김동환, 심사위원 : 강완모, 이창옥, 임미경, 윤세영(AI 대학원)

심사위원장 : 강완모, 심사위원 : 황강욱, 김동환, 윤세영(AI대학원), 전광성(Assistant Professor, Department of Computer Science, University of Arizona)

심사위원장 : 한상근, 심사위원 : 곽도영(명예교수), 황강욱, 강완모, 조현숙(이사장, 코드게이트 보안포럼)

We give some natural sufficient conditions for balls in a metric space to have small intersection. Roughly speaking, this happens when the metric space is (i) expanding and (ii) well-spread, and (iii) certain random variable on the boundary of a ball has a small tail. As applications, we show that the volume of intersection of balls in Hamming space and symmetric groups decays exponentially as their centers drift apart. To verify condition (iii), we prove some deviation inequalities `on the slice’ for functions with Lipschitz conditions. We then use these estimates on intersection volumes to obtain a sharp lower bound on list-decodability of random q-ary codes, confirming a conjecture of Li and Wootters [IEEE Trans. Inf. Theory 2021]; and improve sphere-covering bound from the 70s on constant weight codes by a factor linear in dimension, resolving a problem raised by Jiang and Vardy [IEEE Trans. Inf. Theory 2004]. Our probabilistic point of view also offers a unified framework to obtain improvements on other sphere-covering bounds, giving conceptually simple and calculation-free proofs for q-ary codes, permutation codes, and spherical codes. This is joint work with Jaehoon Kim and Hong Liu.

심사위원장 : 김용정, 심사위원 : 강문진, 김재경, 임미경, 안인경(고려대 세종캠퍼스 데이터계산학과)

심사위원장 : 배성한, 심사위원 : 김완수, 임보해, 임수봉(성균관대 수학교육학과), 최도훈(고려대 수학과)

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.

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(KAIST 입시일정과 겹쳐 1주 연기합니다)

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.

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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.)

The independent domination number of a graph $G$, denoted $i(G)$, is the minimum size of an independent dominating set of $G$. In this talk, we prove a series of results regarding independent domination of graphs with bounded maximum degree. Let $G$ be a graph with maximum degree at most $k$ where $k \ge 1$. We prove that if $k = 4$, then $i(G) \le \frac{5}{9}|V(G)|$, which is tight. Generalizing this result and a result by Akbari et al., we suggest a conjecture on the upper bound of $i(G)$ for $k \ge 1$, which is tight if true. Let $G'$ be a connected $k$-regular graph that is not $K_{k, k}$ where $k\geq 3$. We prove that $i(G')\le \frac{k-1}{2k-1}|V(G')|$, which is tight for $k \in \{3, 4\}$, generalizing a result by Lam, Shiu, and Sun. This result also answers a question by Goddard et al. in the affirmative. In addition, we show that $\frac{i(G')}{\gamma(G')} \le \frac{k^3-3k^2+2}{2k^2-6k+2}$, strengthening upon a result of Knor, \v Skrekovski, and Tepeh, where $\gamma(G')$ is the domination number of $G'$. Moreover, if we restrict $G'$ to be a cubic graph without $4$-cycles, then we prove that $i(G') \le \frac{4}{11}|V(G')|$, which improves a result by Abrishami and Henning. This talk is based on joint work with Ilkyoo Choi, Hyemin Kwon, and Boram Park.

심사위원장 : 폴정, 심사위원 : 이지운, 남경식, 양홍석(전산학부), Greg Markowsky(Senior Lecturer, School of Mathematics, Monash University)

In astrophysical fluid dynamics, stars are considered as isolated fluid masses subject to self-gravity. A classical model to describe the dynamics of Newtonian stars is given by the gravitational Euler-Poisson system, which admits a wide range of star solutions that are in equilibrium or expand for all time or collapse in a finite time or rotate. In particular, using numerics, the Euler-Poisson system in the super-critical regime has been widely used inastrophysics literature todescribe the gravitational collapse, but its rigorous proof has been established only recently. The main challenge comes from thepressure, which actsagainstgravitational force. In this talk, I will discuss some recent progress on Newtonian dust-like collapse and self-similar collapse.

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.

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.

What is the largest number $f(d)$ where every graph with average degree at least $d$ contains a subdivision of $K_{f(d)}$? Mader asked this question in 1967 and $f(d) = \Theta(\sqrt{d})$ was proved by Bollob\'as and Thomason and independently by Koml\'os and Szemer\'edi. This is best possible by considering a disjoint union of $K_{d,d}$. However, this example contains a much smaller subgraph with the almost same average degree, for example, one copy of $K_{d,d}$. In 2017, Liu and Montgomery proposed the study on the parameter $c_{\varepsilon}(G)$ which is the order of the smallest subgraph of $G$ with average degree at least $\varepsilon d(G)$. In fact, they conjectured that for small enough $\varepsilon>0$, every graph $G$ of average degree $d$ contains a clique subdivision of size $\Omega(\min\{d, \sqrt{\frac{c_{\varepsilon}(G)}{\log c_{\varepsilon}(G)}}\})$. We prove that this conjecture holds up to a multiplicative $\min\{(\log\log d)^6,(\log \log c_{\varepsilon}(G))^6\}$-term. As a corollary, for every graph $F$, we determine the minimum size of the largest clique subdivision in $F$-free graphs with average degree $d$ up to multiplicative polylog$(d)$-term. This is joint work with Jaehoon Kim, Youngjin Kim, and Hong Liu.

심사위원장: 백형렬, 심사위원: 최서영, 박정환, 김상현(겸직교수), 이상진(건국대 수학과)

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.

The proof of the central limit theorem (CLT) is often deferred to a graduate course in probability because the notion of characteristic functions is sometimes considered too advanced. I’ll start the talk by reviewing the past efforts to provide an elementary proof of the CLT which is not based on characteristic functions. Then I will explain a new proof of the CLT that derives it from the de Moivre-Laplace theorem, which is the CLT for Bernoulli random variables. The de Moivre-Laplace theorem is the first instance of the CLT in the history, and can be proved directly by computation.

Let $G$ be a graph and let $g, f$ be nonnegative integer-valued functions defined on $V(G)$ such that $g(v) \le f(v)$ and $g(v) \equiv f(v) \pmod{2}$ for all $v \in V(G)$. A $(g,f)$-parity factor of $G$ is a spanning subgraph $H$ such that for each vertex $v \in V(G)$, $g(v) \le d_H(v) \le f(v)$ and $f(v)\equiv d_H(v) \pmod{2}$. In this paper, we prove sharp upper bounds for certain eigenvalues in an $h$-edge-connected graph $G$ with given minimum degree to guarantee the existence of a $(g,f)$-parity factor; we provide graphs showing that the bounds are optimal. This is a joint work with Suil O.

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.

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

Simple mathematical models have had remarkable successes in biology, framing how we understand a host of mechanisms and processes. However, with the advent of a host of new experimental technologies, the last ten years has seen an explosion in the amount and types of quantitative data now being generated. This sets a new challenge for the field – to develop, calibrate and analyse new models to interpret these data. In this talk I will use examples relating to intracellular transport and cell motility to showcase how quantitative comparisons between models and data can help tease apart subtle details of biological mechanisms.

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This talk will be presented online. Zoom link: 709 120 4849 (pw: 1234)

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.

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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.

The poset Ramsey number $R(Q_{m},Q_{n})$ is the smallest integer $N$ such that any blue-red coloring of the elements of the Boolean lattice $Q_{N}$ has a blue induced copy of~$Q_{m}$ or a red induced copy of $Q_{n}$. Axenovich and Walzer showed that $n+2\le R(Q_{2},Q_{n})\le2n+2$. Recently, Lu and Thompson improved the upper bound to $\frac{5}{3}n+2$. In this paper, we solve this problem asymptotically by showing that $R(Q_{2},Q_{n})=n+O(n/\log n)$. Joint work with Dániel Grósz and Abhishek Methuku.

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.

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.

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.

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.

TBA

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This talk will be presented online. Zoom link: 709 120 4849 (pw: 1234)

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.

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.

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

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.

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.

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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

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

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일정에 변동이 생겨 부득이하게 11.12.(금)으로 변경되었음을 알려드립니다.

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.

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.

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.

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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.

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.

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This talk will be presented online. Zoom link: 709 120 4849 (pw: 1234)

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.

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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

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.

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.

A well-known theorem of Whitney states that a 3-connected planar graph admits an essentially unique embedding into the 2-sphere. We prove a 3-dimensional analogue: a simply-connected 2-complex every link graph of which is 3-connected admits an essentially unique locally flat embedding into the 3-sphere, if it admits one at all. This can be thought of as a generalisation of the 3-dimensional Schoenflies theorem. This is joint work with Agelos Georgakopoulos.

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.

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11.03.(수) ~ 11.05.(금)

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.

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11.03.(수) ~ 11.05.(금)

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.

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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11.03.(수) ~ 11.05.(금)

A family F of subsets of {1,2,…,n} is called maximal k-wise intersecting if every collection of at most k members from F has a common element, and moreover, no set can be added to F while preserving this property. In 1974, Erdős and Kleitman asked for the smallest possible size of a maximal k-wise intersecting family, for k≥3. We resolve this problem for k=3 and n even and sufficiently large. This is joint work with Kevin Hendrey, Casey Tompkins, and Tuan Tran.

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.

심사위원장 최건호, 심사위원 곽도영, 권순식, 김동석(경영공학부), 김완수

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.