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Information Center for Mathematical Science

세미나

Information Center for Mathematical Science

세미나

한국고등과학원 세미나

한국고등과학원 세미나

In this talk, we mainly show some progress towards two "domination" conjectures made on the moduli space of Higgs bundles relating the Hitchin fibration and Hitchin section. Firstly, we show for a Hitchin representation in PSL(n,R), every equivariant minimal immersion from a hyperbolic plane into the corresponding symmetric space is distance-increasing. Secondly, consider Hitchin fibers at (q2,0,...,0), that is, the fibers containing Fuchsian locus. We show a comparison theorem on the length spectrum between surface group representations in such fibers with Fuchsian ones, as a generalization of the SL(2,C) case shown by Deroin and Tholozan.

한국고등과학원 세미나

Consider a closed surface of genus at least 2. We first recall the non-abelian Hodge theory, which relates the moduli space of Higgs bundles and the moduli space of surface group representations. Next we introduce the Hitchin fibration, which gives a fibration structure of the moduli space of Higgs bundles. As a special family in the moduli space of Higgs bundles, we study cyclic Higgs bundles and show several domination results. Lastly, based on the results we obtained for the cyclic case, we describe a conjectural picture of the whole moduli space.

한국고등과학원 세미나

We consider the pointwise convergence of the solution to the generalized Schroodinger equation. Recently, Du, Guth and Li [1] and Du and Zhang [2] obtained optimal estimates of the convergence for Schrodinger equation in two dimension and in higher dimensions, respectively. We generalize the results to the operators with more general phases, which gives the convergence for generalized Schrodinger equation. This is joint work with Hyerim Ko.(SNU)

한국고등과학원 세미나

In this talk, we study the classical result ``The L^{2}-boundedness for the Cauchy integral operator on Lipschitz curves". The theorem was first proved by Calder?n (PNAS, 1977) when the Lipschitz curve has a small Lipschitz constant. The smallness assumption on the Lipschitz constant was later removed by Coifman-McIntosh-Meyer (Ann. Math., 1982). To give a motivation for this theorem, we study the Dirichlet problem for the Poisson equation in an arbitrary bounded Lipschitz domain. There are several proofs on the theorem of Coifman-McIntosh-Meyer. Here we follow a martingale proof given by Coifman-Jones-Semmes (JAMS, 1989). Necessary materials for this method will be given in this talk.

한국고등과학원 세미나

In the theory of Teichm\"uller space of Riemann surfaces, we consider the set of Riemann surfaces which are quasiconformally equivalent. Here, we say that two Riemann surfaces are quasiconformally equivalent if there is a quasiconformal homeomorphism between them. Hence, at the first stage of the theory, we have to know a condition for Riemann surfaces to be quasiconformally equivalent. The condition is quite obvious if the Riemann surfaces are topologically finite. On the other hand, for Riemann surfaces of topologically infinite type, the situation is rather difficult. We consider geometric conditions for the quasiconformal equivalence of open Riemann surfaces. We also discuss the quasiconformal equivalence of regions which are complements of some Cantor sets, e. g., the limit sets of Schottky groups and the Julia sets of some rational functions.

한국고등과학원 세미나

Kontsevich established a formality theorem and applied it to deformation quantization of Poisson manifolds. Following Kontsevich's method, we obtained a formality theorem for dg manifolds. As application, we extended Duflo theorem in Lie theory to the context of dg manifolds. In this talk, I'll start with an introduction to L-infinity algebras, Kontsevich formality theorem, and deformation quantization problem. After that, I plan to explain formality and Kontsevich-Dulfo theorem for dg manifolds.

KAIST 수리과학과 세미나

In this talk we outline the construction of certain higher Chow cycles on Abelian surfaces. The existence of these cycles is predicted by certain conjectures on special values of L-functions in the local case and by the existence of certain modular forms in the case of the universal family over a Shimura curve - providing evidence for the conjecture described in the first talk. The construction uses beautiful 19th century constructions of Kummer and Humbert.

KAIST 수리과학과 세미나

Stochastic models of chemical reactions are of interest when considering chemical systems in which the stochastic effects are important. Examples of such systems in mathematical biology are frequent both in molecular- and population-level models (e.g., gene transcription or onset of an epidemic). In this talk I will give a brief introduction to the theory of Markovian stochastic reaction networks and describe some of the examples of its application. This talk will be understandable by undergraduate students who has a basic probability background.

KAIST 수리과학과 세미나

Katz and Sarnak proposed a conjecture that the behavior of low-lying zeros of L-functions in a family is determined by classical matrix groups such as the orthogonal matrix group. There are abundant evidences which supports the conjecture. In this talk, we compute the n-level density for the family of cubic Dirichlet characters. Furthermore, we compare the one-level density with the prediction by the Ratios conjecture.

KAIST 수리과학과 세미나

Financial markets are often driven by latent factors which traders cannot observe. Here, we address an algorithmic trading problem with collections of heterogeneous agents who aim to perform statistical arbitrage, where all agents filter the latent states of the world, and their trading actions have permanent and temporary price impact. This leads to a large stochastic game with heterogeneous agents. We solve the stochastic game by investigating its mean-field game (MFG) limit, with sub-populations of heterogenous agents, and, using a convex analysis approach, we show that the solution is characterized by a vector-valued forward-backward stochastic differential equation (FBSDE). We demonstrate that the FBSDE admits a unique solution, obtain it in closed-form, and characterize the optimal behaviour of the agents in the MFG equilibrium. Moreover, we prove the MFG equilibrium provides an -Nash equilibrium for the finite player game. We conclude by illustrating the behaviour of agents using the optimal MFG strategy through simulated examples.

KAIST 해외 석학 특별 강연 시리즈

One of the most fundamental questions in representation theory asks for a description of the simple representations. I will give an introduction to this problem with an emphasis on the representation theory of Lie groups and algebras. A recurring theme is the appearance of geometric techniques in seemingly algebraic problems. I will also emphasise the important role played by invariant forms and signature.

KAIST 해외 석학 특별 강연 시리즈

One of the most fundamental questions in representation theory asks for a description of the simple representations. I will give an introduction to this problem with an emphasis on the representation theory of Lie groups and algebras. A recurring theme is the appearance of geometric techniques in seemingly algebraic problems. I will also emphasise the important role played by invariant forms and signature.

KAIST Discrete Math 세미나

Any structure whose language is finite has a model of graph theory which is bi-interpretable with it. From this idea, Mekler further developed a way of interpreting a model into a group. This Mekler's construction preserves various model-theoretic properties such as stability, simplicity, and NTP2, thus helps us find new group examples in model theory. In this talk, I will introduce to you what Mekler's construction is and briefly show that this preserves NTP1.

KAIST Discrete Math 세미나

Graphs are mathematical structures used to model pairwise relations between objects. Graph decomposition problems ask to partition the edges of large/dense graphs into small/sparse graphs. In this talk, we introduce several famous graph decomposition problems, related puzzles and known results on the problems.

KAIST Discrete Math 세미나

We say a subgraph H of an edge-colored graph is rainbow if all edges in H has distinct colors. The concept of rainbow subgraphs generalizes the concept of transversals in latin squares. In this talk, we discuss how these concepts are related and we introduce a result regarding approximate decompositions of graphs into rainbow subgraphs. This has implications on transversals in latin square. It is based on a joint work with K"uhn, Kupavskii and Osthus.

2018 KAIST 학부 콜로퀴움

2018 KAIST 학부 콜로퀴움

2018 KAIST 학부 콜로퀴움

2018 KAIST 학부 콜로퀴움