Friday, December 13, 2024

2024-12-19 / 16:00 ~ 17:00

편미분방정식 통합연구실 세미나 - 편미분방정식: Scattering problem for the generalized Korteweg-de Vries equation

by ()

In this talk, we study the scattering problem for the initial value problem of the generalized Korteweg-de Vries (gKdV) equation. The purpose of this talk is to achieve two primary goals. Firstly, we show small data scattering for (gKdV) in the weighted Sobolev space, ensuring the initial and the asymptotic states belong to the same class. Secondly, we introduce two equivalent characterizations of scattering in the weighted Sobolev space. In particular, this involves the so-called conditional scattering in the weighted Sobolev space. This talk is based on a joint work with Satoshi Masaki (Hokkaido University)

2024-12-13 / 16:00 ~ 17:00

학과 세미나/콜로퀴엄 - 박사논문심사: 시지지, 카스텔누오보-멈포드 정칙성, 그리고 텐서 계수에 대한 몇 가지 연구

by 한종인()

2024-12-13 / 11:00 ~ 12:00

학과 세미나/콜로퀴엄 - 응용 및 계산수학 세미나:

by 이영규()

We present HINTS, a Hybrid, Iterative, Numerical, and Transferable Solver that combines Deep Operator Networks (DeepONet) with classical numerical methods to efficiently solve partial differential equations (PDEs). By leveraging the complementary strengths of DeepONet’s spectral bias for representing low-frequency components and relaxation or Krylov methods’ efficiency at resolving high-frequency modes, HINTS balances convergence rates across eigenmodes. The HINTS is highly flexible, supporting large-scale, multidimensional systems with arbitrary discretizations, computational domains, and boundary conditions, and can also serve as a preconditioner for Krylov methods. To demonstrate the effectiveness of HINTS, we present numerical experiments on parametric PDEs in both two and three dimensions.

2024-12-13 / 16:30 ~ 17:30

IBS-KAIST 세미나 - 이산수학: Phase transition of degenerate Turán problems in p-norms

by Jun Gao(IBS 극단 조합 및 확률 그룹)

For a positive real number $p$, the $p$-norm $\|G\|_p$ of a graph $G$ is the sum of the $p$-th powers of all vertex degrees. We study the maximum $p$-norm $\mathrm{ex}_{p}(n,F)$ of $F$-free graphs on $n$ vertices, focusing on the case where $F$ is a bipartite graph. It is natural to conjecture that for every bipartite graph $F$, there exists a threshold $p_F$ such that for $p< p_{F}$, the order of $\mathrm{ex}_{p}(n,F)$ is governed by pseudorandom constructions, while for $p > p_{F}$, it is governed by star-like constructions. We determine the exact value of $p_{F}$, under a mild assumption on the growth rate of $\mathrm{ex}(n,F)$. Our results extend to $r$-uniform hypergraphs as well. We also prove a general upper bound that is tight up to a $\log n$ factor for $\mathrm{ex}_{p}(n,F)$ when $p = p_{F}$. We conjecture that this $\log n$ factor is unnecessary and prove this conjecture for several classes of well-studied bipartite graphs, including one-side degree-bounded graphs and families of short even cycles. This is a joint work with Xizhi Liu, Jie Ma and Oleg Pikhurko.

2024-12-17 / 16:30 ~ 17:30

IBS-KAIST 세미나 - 이산수학: Counting homomorphisms in antiferromagnetic graphs via Lorentzian polynomials

by 이준경(연세대)

An edge-weighted graph $G$, possibly with loops, is said to be antiferromagnetic if it has nonnegative weights and at most one positive eigenvalue, counting multiplicities. The number of graph homomorphisms from a graph $H$ to an antiferromagnetic graph $G$ generalises various important parameters in graph theory, including the number of independent sets and proper vertex colourings. We obtain a number of new homomorphism inequalities for antiferromagnetic target graphs $G$. In particular, we prove that, for any antiferromagnetic $G$, $|\mathrm{Hom}(K_d, G)|^{1/d} ≤ |\mathrm{Hom}(K_{d,d} \setminus M, G)|^{1/(2d)}$ holds, where $K_{d,d} \setminus M$ denotes the complete bipartite graph $K_{d,d}$ minus a perfect matching $M$. This confirms a conjecture of Sah, Sawhney, Stoner and Zhao for complete graphs $K_d$. Our method uses the emerging theory of Lorentzian polynomials due to Brändén and Huh, which may be of independent interest. Joint work with Jaeseong Oh and Jaehyeon Seo.

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