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We discuss the set of critical exponents of discrete groups acting on a regular tree. If the quotient graph is finite, then the critical exponent is an algebraic number.
In general, given an arbitrary real number between 0 and the volume entropy of the regular tree, we discuss how we can construct a discrete group whose critical exponent realizes the number.
Starting with introducing a general Newtonian Boltzmann theory, we will establish global-in-time well-posedness and stability results for solutions nearby the relativistic Maxwellian to the special relativistic Boltzmann equation without angular cutoff. We assume the generic soft-potential conditions on the collision kernel in that were derived by Dudynski and Ekiel-Jezewska (Commun Math Phys 115(4):607--629, 1985). In this physical situation, the angular function in the collision kernel is not locally integrable, and the collision operator behaves like a non-isotropic fractional diffusion operator.
Right-angled Artin groups (RAAGs) are defined from finite simplicial graphs.
It is a fundamental question whether or not, given two RAAGs, there is an embedding from one group to the other.
Extension graphs are useful in solving this problem.
In this talk, I will briefly review RAAGs and extension graphs,
and show some results on solving the embeddability problem in RAAGs by using extension graphs.
Let K be a finite extension of Qp. It is believed that one can attach a smooth Fp-representation of GLn(K) (or a packet of such representations) to a continuous Galois representation Gal(Qp/K) → GLn(Fp) in a natural way, that is called mod p Langlands program for GLn(K). This conjecture is known only for GL2(Qp): one of the main difficulties is that there is no classification of such smooth representations of GLn(K) unless K = Qp and n = 2. However, for a given continuous Galois representation ρ0 : Gal(Qp/Qp) → GLn(Fp), one can define a smooth Fp-representation Π0 of GLn(Qp) by a space of mod p automorphic forms on a compact unitary group, which is believed to be a candidate on the automorphic side corresponding to ρ0 for mod p Langlands correspondence in the spirit of Emerton. The structure of Π0 is very mysterious as a representation of GLn(Qp), and it is not known that ρ0 and Π0 determine each other. In this talk, we discuss that Π0 determines ρ0 , provided that ρ0 is ordinary and generic. More precisely, we prove that the tamely ramified part of ρ0 is determined by the Serre weights attached to ρ0 , and the wildly ramified part of ρ0 is obtained in terms of refined Hecke actions on Π0. The talk is based on a joint work with Zicheng Qian.
In a constantly changing world, animals must account for fluctuations and changes in their environment when making decisions. They must make use of recent information, and appropriately discount older, irrelevant information. But to do so they need to learn the rate at which the environment changes. Recent experimental studies show that humans and other animals can indeed do so. However it it is unclear what underlying computations they use to achieve this. Developing normative models of evidence accumulation is a first step in quantifying such decision-making processes. While optimal, these algorithms are computationally intensive. To address this problem we developed an approximation of the normative inference process, and show how this approximate computation can be implemented in neural circuits. In the second part of the talk I will discuss evidence accumulation on networks where private information can be shared between neighboring nodes.
Hyperbolic dynamical systems are nowadays fairly well understood from the topological and ergodic point of view. In this talk, we discuss some recent and ongoing works on the dynamics beyond hyperbolicity. In the ﬁrst part, we will provide a characterization of robustly shadowable chain transitive sets for C1-vector ﬁelds on compact smooth manifolds. In the second part, we extend the concepts of topological stability and pseudo-orbit tracing property from homeomorphisms to Borel measures, and prove that every expansive measure with the pseudo-orbit tracing property is topologically stable. This represents a measurable version of the stability theorem by Peter Walters. The ﬁrst part is joint work with M. Reza, and the second part is joint work with C.A. Morales.
I will introduce the concept of oriented cohomology in the sense of Levine and Morel, and the work of Kostant and Kumar on algebraic construction of singular cohomology and Grothendieck group of flag varieties. Then I will introduce the formal group algebra of Calmes-Petrov-Zainoulline, which is the algebraic replacement of T-equivariant oriented cohomology of a point. I will introduce the definition of formal affine Demazure algebra and sketch the proof of the structure theorem.
I will introduce the dual of the formal affine Demazure algebra, which is the algebraic replacement of T-equivariant oriented cohomology of complete flag variety. I will also mention the proof of the generalized Borel Isomorphism. In the second half of this talk, I will define the two Hecke actions on the dual of the formal affine Demazure algebra. Then I will define the push-pull operators of the oriented cohomology and define perfect pairings on the equivariant cohomology of complete and partial flag varieties. I will also mention hyperbolic cohomology and its relation with Kazhdan-Lusztig basis.
제 7회 CMC 정오의 수학산책
일시: 11월 24일(금) 12:00 - 13:15
장소: KAIST 자연과학동 E6-1 3435호
강연자: 채동호 교수 (중앙대학교)
제목: Navier-Stokes 방정식
참가: https://goo.gl/forms/ORj7J4QD69k6vbUy1 를 통해 사전등록
One of the most canonical ways to define an invariant of a singular variety is to take a resolution of its singularities, and use invariants of the resulting smooth variety. We will begin this talk with a quick overview of Grothendieck's, and then Voevodsky's theory of motives. We will explain how to use resolutions to extend the theory to singular varieties, even when resolutions don't exist. Then, if there is time, we will discuss transportation of these techniques to differential forms leading to potential applications in birational geometry.
In this talk, we will introduce a notion of a noncommutative probability space and useful properties.
Then we will discuss various convergence results of weighted sums in a noncommutative probability space, e.g., weak law of large numbers, convergence rates and precise asymptotics, etc.
Also, we will discuss some noncommutative inequalities, e.g., Fuk-Nagaev inequalities, Bennett inequality and Rosenthal inequality, etc
One of the major developments in 20th century mathematics is homotopy theory, which studies topological spaces up to stretching and bending. Basic tools in homotopy theory, like singular homology and homotopy groups, are constructed using paths from the unit interval. Such techniques were unavailable for a long time in algebraic geometry, due to polynomials being too rigid. In this talk we will discuss ways around this developed by Morel, Suslin, Voevodsky, et al.
Observational data along with mathematical models play a crucial role in improving prediction skills in science and engineering. In this talk we focus on the recent development of Bayesian inference techniques, data assimilation and parameter estimation, for Physics-constrained problems that are often described by partial differential equations. We discuss the similarities shared by the two methods and their differences in mathematical and computational points of view and future research topics. As applications, numerical weather prediction for geophysical flows and parameter estimation of kinetic reaction rates in the hydrogen-oxygen combustion are provided. This talk aims for researchers and students in all disciplines of science and engineering and only a minimum level of undergraduate mathematics is required.
2017년 제5회 정오의 수학산책
일시: 10월 13일(금) 12:00 - 13:15
장소: KAIST 수리과학과 E6-1 3435호
강연자: 임선희 교수 (서울대)
제목: On the work of Mirzakhani : from counting geodesics to the classification of measures
(미르자카니의 결과들 : 측지선의 개수부터 측도의 분류까지)
내용: 본 강연에서는 미르자카니의 업적들에 대해 살펴볼 예정입니다. 곡면에서 단순 측지선의 갯수에 대한 문제, 곡면들의 공간 (Moduli space of Riemann surfaces)에서 측지선의 갯수 세기, 부피 변화, 그리고 특정한 군의 작용에 대한 불변 측도의 분류 문제 등에서 미르자카니가 얻은 결과들을 음미하고, 그러한 문제들이 서로 어떻게 연결되어 있는지를 쌍곡 동역학 (dynamics on hyperbolic spaces) 에서 이해해 보고자 합니다.
등록: 아래 링크를 통해 사전등록 바랍니다.
We calculate the global log canonical thresholds of log del Pezzo surfaces embedded in weighted projective spaces as codimension two. As important applications, we show that most of them are weakly exceptional and admit K\"ahler-Einstein metrics. This is a joint work with Joonyeong Won.
The spatially varying coefficient process model is a nonstationary approach to explaining spatial heterogeneity by allowing coefficients to vary across space. To accommodate geographically hierarchical data, we develop a methodology for generalizing this model. We consider two-level hierarchical structures and allow for the coefficients of both low-level and high-level units to vary over space. We assume that the spatially varying low-level coefficients follow the multivariate Gaussian process, and the spatially varying high-level coefficients follow the multivariate simultaneous autoregressive model that we develop by extending the standard simultaneous autoregressive model to incorporate multivariate data. We apply the proposed model to transaction data of houses sold in 2014 in a part of the city of Los Angeles. The results show that the proposed model predicts housing prices and fits the data effectively.