The 14th SNU-HU Symposium on Mathematics
November 13, 2020
Online (zoom links will be sent to participants)

09:00 - 09:10 Opening

Analysis session (See below for the Number theory session)

(PDE and Probability related topics)

09:10 - 09:40 Ki-Ahm Lee (SNU)
Title: HIGHER ORDER CONVERGENCE IN HOMOGENIZATIONS
Abstract: In this talk, we are going to higher order convergence in Homogenization process through higher order interior and boundary correctors in various Nonlinear equations. Similar issues arises in vanishing viscous Hamiltonian Equations and Lower order convergence in Parabolic problems with various space and time scales. We also discuss how this idea can be applied to the other problem where some parameters approaches to zero: for example diffusive limit in Kinetic theory, asymptotic limit in parabolic flows, and so on.

9:40 - 10:10 Hirotoshi Kuroda (HU)
Title: The behavior of the Laplacian with Neumann boundary condition on thin domains
Abstract: We consider the convergent problems of Dirichlet forms associated with the Laplace operators on thin domains. In this talk we study that the sequence of Dirichlet forms, which associate with the Laplace operators with Neumann boundary conditions and some suitable potential function, on thin domains Mosco converges to the form associated with the Laplace operator with the delta-type boundary conditions on the graph in the sense of Gromov-Hausdorff topology. From this results we can make use of many results about the convergence of the semigroups and resolvents generated by the infinitesimal generators associated with the Dirichlet forms.

Break

10:20 - 10:50 Jaehoon Lee (SNU, postdoc)
Title: Law of iterated logarithm for general Markov processes
Abstract: In this talk, we will present sharp sufficient conditions for limsup and liminf LILs which can be applicable for various Markov processes containing both diffusion processes and jump processes. In particular, we distinguish between sufficient conditions for LILs at time 0 and infinity.

10:50 - 11:20 Jun Masamune (HU)
Title: H-compactness of elliptic operators on weighted Riemannian manifolds
Abstract: We study the asymptotic behavior of second-order uniformly elliptic operators on weighted Riemannian manifolds. They naturally emerge when studying spectral properties of the Laplace?Beltrami operator on families of manifolds with rapidly oscillating metrics. We appeal to the notion of H-convergence introduced by Murat and Tartar. In our main result we establish an H-compactness result that applies to elliptic operators with measurable, uniformly elliptic coefficients on weighted Riemannian manifolds. We further discuss the special case of ``locally periodic'' coefficients and study the asymptotic spectral behavior of compact submanifolds of ?^n with rapidly oscillating geometry. Major results in this talk was obtained in a collaborative effort with H. Hoppe and S. Neukamm.

Break

11:30 - 12:00 Soobin Cho (SNU, student)
Title: Estimates on transition densities of subordinators with jumping density decaying in mixed polynomial orders
Abstract: In this talk, we discuss estimates on transition densities for subordinators, which are global in time. We establish the sharp two-sided estimates on the transition densities for subordinators whose Levy measures are absolutely continuous and decaying in mixed polynomial orders. Under a weaker assumption on Levy measures, we also obtain a precise asymptotic behaviors of the transition densities at infinity. Our results cover stable subordinators, geometric stable subordinators, Gamma subordinators and much more.

12:00 - 12:30 Panki Kim (SNU)
Title: Green function estimates and Boundary Harnack principles for non-local operators whose kernels degenerate at the boundary
Abstract: In this talk, we discuss the potential theory of Markov processes with jump kernels decaying at the boundary of the half space. The boundary part of kernel is comparable to the product of three terms with parameters appearing as exponents in these terms. The constant c in the killing term can be written as a function of a parameter p which is strictly increasing in p. We establish sharp two-sided estimates on the Green functions of these processes for all admissible values of p and parameters in the boundary part of kernel. Depending on the regions where parameters and p belong, the estimates on the Green functions are different. In fact, the estimates have three different forms depending on the regions the parameters belong to. As applications, we completely determine the region of the parameters where the boundary Harnack principle holds or not.

Lunch Break

(Operator algebra related topics)

13:30 - 12:00 Keisuke Yoshida (HU, student)
Title: Simplicity of C$^*$-algebras associated to some non-Hausdorff groupoids
Abstract: Non-Hausdorff topological groupoids are not rare. For example, we can construct them as groupoids of germs from some self-similar groups which are subgroups of the group of homeomorphisms on the Cantor space. It is more difficult to show the simplicity of C$^*$-algebras from non-Hausdorff groupoids than from Hausdorff ones. We see an algorithm to show the simplicity of C$^*$-algebras associated to non-Hausdorff groupoids of germs from some special self-similar groups.

14:00 - 14:30 Sang-jun Park (SNU, student)
Title: Twisted Fourier analysis: generalized Weyl-Wigner representations and symmetry groups.
Abstract: Weyl unitaries, formulating canonical commutation relation(CCR) in quantum kinematics, play a central role in continuous-variable quantum information theory and quantum optics. From Weyl unitaries the defintion of characteristic function and Wigner function, which are deeply analyzed when dealing with Gaussian states in multi-mode bosonic system, are induced. By considering Weyl unitaries as a unique projective representation of the abelian group R^n, we can say that the characteristic function is actually the inverse map of the (twisted) Fourier transform in certain sense. Starting with this observation the notions can be generalized in more abstract settings. In the first half of this talk, we formulate the Weyl representation and corresponding characteristic function in the abstract phase space, namely the locally compact abelian group. With a noncommutative generalization of Fourier analysis, we can relate a quantum state and its characteristic function. In the second half, two types of symmetry concerning Weyl unitary are introduced: symplectic group and Clifford group. In addition, several important examples are to be exhibited including calculations of symmetry groups. This talk is based on joint works with Cedric Beny and Hun Hee Lee

14:30 - 15:00 Takahiro Hasebe (HU)
Title: Monotone increment processes, Markov processes and Loewner chains
Abstract: We establish bijections of the three objects: non-commutative stochastic processes with monotonically independent additive increments; a special class of real-valued Markov processes; certain decreasing Loewner chains in the upper half-plane. The Markov processes above, in the time-homogeneous case, have explicit generators.

Break

15:10 - 15:40 Yuhei Suzuki (HU)
Title: Equivariant O_2-absorption theorem for exact groups
Abstract: Study of group actions on operator algebras is a central theme in operator algebra theory. This provides interesting examples via the crossed product construction, and their classification is important in the structure theory. In this talk, I will briefly introduce backgrounds on group actions, and then explain my recent result on the title, which gives the first classification result beyond amenable groups. Based on arXiv:2004.09461

15:40 - 16:10 Sang-gyun Youn (SNU)
Title: Sobolev embedding properties on duals of discrete groups
Abstract: The fractional Lp-Lq Sobolev embedding properties of the Euclidean space R^d has the optimal order d(1/p-1/q). Here, the real dimension d is an important geometric information since the above optimal order extends naturally to any compact Lie groups. In this talk, we will focus on the natural analogue of Sobolev embedding properties on the duals of discrete groups, where the role of real dimension is replaced by the polynomial growth order.

16:10 - 16:20 Analysis session closing

Number theory session

09:10 - 10:10 Masanori Asakura (HU)
Title: p-adic Beilinson conjecture on elliptic curves over Q
Abstract: The p-adic Beilinson conjecture was formulated by Perrin-Riou, which asserts that the special values of p-adic L-functions agree with p-adic regulators up to Q^\times. However the precise statement (in general situation) is very complicated. Recently we develop a new technique of computing the p-adic regulators for K_2 of elliptic curves, which is based on the theory of rigid cohomology. In this talk, I will explain a numerical approach towards the p-adic Beilinson conjecture for K_2 of elliptic curves over Q with use of our new technique. This is a joint work with M. Chida.

10:20 - 11:20 Kento Yamamoto (HU, postdoc)
Title: Syntomic complex with modulus
Abstract: In this talk, I will explain a version of the syntomic complex for modulus pair (X, D), where X is regular semi-stable family and D is an effective Cartier divisor on X. I will also talk that the symbol map of this new object is not surjective.

11:30 - 12:30 Dohyeong Kim (SNU)
Title: Special values of L-functions and their distribution
Abstract: I will recall some non-vanishing results for modular L-functions, and how an approach based on the theory of dynamical systems can shed lights on their distribution. I will report the work in progress which generalizes this to the case of Bianchi-type L-functions.

12:30 - 12:40 Number theory session closing