Tuesday, September 28, 2021

2021-10-05 / 16:00 ~ 17:00

학과 세미나/콜로퀴엄 - 대수기하학: Multiplicity-free products of Schubert divisors

by ()

In my first talk I am going to speak about Schubert calculus. Let G/B be a flag variety, where G is a linear simple algebraic group, and B is a Borel subgroup. Schubert calculus studies (in classical terms) multiplication in the cohomology ring of a flag variety over the complex numbers, or (in more algebraic terms) the Chow ring of the flag variety. This ring is generated as a group by the classes of so-called Schubert varieties (or their Poincare duals, if we speak about the classical cohomology ring), i. e. of the varieties of the form BwB/B, where w is an element of the Weyl group. As a ring, it is almost generated by the classes of Schubert varieties of codimension 1, called Schubert divisors. More precisely, the subring generated by Schubert divisors is a subgroup of finite index. These two facts lead to the following general question: how to decompose a product of Schubert divisors into a linear combination of Schubert varieties. In my talk, I am going to address (and answer if I have time) two more particular versions of this question: If G is of type A, D, or E, when does a coefficient in such a linear combination equal 0? When does it equal 1?

2021-10-01 / 10:00 ~ 11:00

SAARC 세미나 - SAARC 세미나:

by 예종철(KAIST 바이오및뇌공학과)

Recently, deep learning approaches have become the main research frontier for image reconstruction and enhancement problems thanks to their high performance, along with their ultra-fast inference times. However, due to the difficulty of obtaining matched reference data for supervised learning, there has been increasing interest in unsupervised learning approaches that do not need paired reference data. In particular, self-supervised learning and generative models have been successfully used for various inverse problem applications. In this talk, we overview these approaches from a coherent perspective in the context of classical inverse problems and discuss their various applications. In particular, the cycleGAN approach and a recent Noise2Score approach for unsupervised learning will be explained in detail using optimal transport theory and Tweedie’s formula with score matching.

2021-09-30 / 10:00 ~ 12:30

학과 세미나/콜로퀴엄 - 기타: Small and Big mapping Class Groups

by Mladen Bestvina(The University of UTAH)

(KAIX Distinguished Lectures Series)

2021-10-05 / 16:30 ~ 17:30

IBS-KAIST 세미나 - 이산수학: Feedback Vertex Set on Geometric Intersection Graphs

by 오은진(포스텍)

I am going to present an algorithm for computing a feedback vertex set of a unit disk graph of size k, if it exists, which runs in time $2^{O(\sqrt{k})}(n + m)$, where $n$ and $m$ denote the numbers of vertices and edges, respectively. This improves the $2^{O(\sqrt{k}\log k)}(n + m)$-time algorithm for this problem on unit disk graphs by Fomin et al. [ICALP 2017].

2021-10-01 / 16:00 ~ 17:00

학과 세미나/콜로퀴엄 - PDE 세미나: Global dynamics around 2-solitons for the nonlinear damped Klein-Gordon equation

by Kenji Nakanishi(Kyoto University)

This is joint work with Kenjiro Ishizuka (Kyoto). We study global behavior of solutions to the nonlinear Klein-Gordon equation with a damping and a focusing nonlinearity on the Euclidean space. Recently, Cote, Martel and Yuan proved the soliton resolution conjecture completely in the one-dimensional case: every global solution in the energy space is asymptotic to a superposition of solitons getting away from each other as time tends to infinity. The next question is to see which initial data evolve into each of the asymptotic forms. The asymptotic decomposition is very sensitive to initial perturbation because all the solitons are unstable. We consider the simplest non-trivial setting in general space dimensions: the global behavior of solutions starting near a superposition of two ground states. Cote, Martel, Yuan and Zhao proved that the solutions asymptotic to 2-solitons form a codimension-2 manifold in the energy space. Our question is what happens for the other initial data in the neighborhood. As an answer, we give a complete classification of those solutions into 5 types of global behavior. Two of them are asymptotic to the positive ground state and the negative one respectively. They form two codimension-1 manifolds that are joined at their boundary by the Cote-Martel-Yuan-Zhao manifold of 2-solitons. The connected union of those three manifolds separates the remainder of the neighborhood into the open set of global decaying solutions and that of blow-up. The main difficulty to prove it is in controlling the direction of instability in two dimensions attached to the two soliton components, because the soliton interactions are not integrable in time, breaking the simple superposition of the linearized approximation around each soliton. It is resolved by showing that the non-integrable interactions do not essentially affect the direction of instability, using the reflection symmetry of the equation and the 2-solitons. I will also explain the difficulty for the 3-solitons due to a more dramatic phenomenon, which may be called soliton merger.

2021-09-28 / 16:30 ~ 17:30

IBS-KAIST 세미나 - 이산수학: Extremal functions for sparse minors

by Kevin Hendrey(IBS 이산수학그룹)

The extremal function $c(H)$ of a graph $H$ is the supremum of densities of graphs not containing $H$ as a minor, where the density of a graph is the ratio of the number of edges to the number of vertices. Myers and Thomason (2005), Norin, Reed, Thomason and Wood (2020), and Thomason and Wales (2019) determined the asymptotic behaviour of $c(H)$ for all polynomially dense graphs $H$, as well as almost all graphs of constant density. We explore the asymptotic behavior of the extremal function in the regime not covered by the above results, where in addition to having constant density the graph $H$ is in a graph class admitting strongly sublinear separators. We establish asymptotically tight bounds in many cases. For example, we prove that for every planar graph $H$, \[c(H) = (1+o(1))\max (v(H)/2, v(H)-\alpha(H)),\] extending recent results of Haslegrave, Kim and Liu (2020). Joint work with Sergey Norin and David R. Wood.

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