
6月演講

Colloquium, 中央大學數學系 魏福村教授
Wednesday, June 6, 16:10—17:00 數學系3174
Title: Kronecker Limit Formula in Mixed Characteristic Abstract: The celebrated Kronecker limit formula expresses the “second term” of the real analytic Eisenstein series in terms of the modular discriminants of elliptic curves. One application is to give a “period interpretation” of the logarithmic derivatives of the Dedekind zeta function associated to quadratic number fields. Consequently, this provides significant evidences for the Colmez conjecture on the “Faltings height” of “CM” abelian varieties. In this talk, I will review the classical story, and then discuss this phenomenon in the “mixedcharacteristic” setting.

Colloquium, 清華大學數學系 蔡孟傑教授
Thursday, June 7, 16:10—17:00 數學系3174
Title: Inner and Outer Automorphisms in Lie Theory Abstract: For many algebraic and geometric objects, their automorphism groups are topological spaces in natural ways. We say that an automorphism is inner if its lies in the identity component, and is outer otherwise. This concept provides a stronger condition than mere automorphisms. We explore several issues in Lie theory that arise from it.

Colloquium, University of North Carolina at Greensboro 鍾佑民教授
Wednesday, June 13, 16:10—17:00 數學系3174
Title: Computational Topology with Applications in the Imaging Sciences Abstract: Computational topology is a relatively young field in algebraic topology. Tools from computational topology have proven successful in many scientific disciplines, such as fluid dynamics, biology, material science, climatology, etc. In this talk, we will give a brief introduction to computational topology, focusing primarily on persistent homology. Applications to various datasets from cell biology and climatology will be presented to illustrate the methods.

Colloquium, 中央大學數學系 李明憶教授
Thursday, June 14, 16:10—17:00 數學系3174
Title: Boundedness of MongeAmp\`ere Singular Integral Operators on Besov Spaces Abstract: Let be a strictly convex and smooth function, and \mu= \text{det}\,D^2 \phi be the MongeAmp\`ere measure generated by \phi. For x\in \Bbb R^n and t>0, let S(x,t):=\{y\in \Bbb R^n: \phi(y)<\phi(x)+\nabla \phi(x)\cdot(yx)+t\} denote the section. If \mu satisfies the doubling property, Caffarelli and Guti\'errez (Trans. AMS 348:10751092, 1996) provided a variant of the Calder\'onZygmund decomposition and a JohnNirenbergtype inequality associated with sections. Under a stronger uniform continuity condition on \mu, they also (Amer. J. Math. 119:423465, 1997) proved an invariant Harnack's inequality for nonnegative solutions of the MongeAmp\`ere equations with respect to sections. The purpose of this talk is to establish a theory of Besov spaces associated with sections under only the doubling condition on \mu and prove that MongeAmp\`ere singular integral operators are bounded on these spaces

Colloquium, 中央研究院數學研究所 鄭日新研究員
Thursday, June 21, 16:10—17:00 數學系3174
Title: Complex/CR duality (?)—Volume Renormalization and Invariant Geometric Equations Abstract: Inspired by submanifold observables in AdS/CFT correspondence, we consider volume renormalization of surfaces or hypersurfaces with boundary curves or surfaces in possible complex/CR duality. Precisely I will talk about biholomorphically invariant curves and surfaces on the boundary of a strongly pseudoconvex domain in C^2. A distinguished class of such invariant curves satisfies a system of 2nd order ODEs, called chains in CR geometry. We interpret chains as geodesics of a Kropina metric in Finsler geometry. The associated energy functional of a curve on the boundary can be recovered as the log term coefficient in a weighted renormalized area expansion of a minimal surface that it bounds inside the domain. For surfaces on the boundary, we express two CR invariant surface area elements in terms of quantities in pseudohermitian geometry. We deduce the EulerLagrange equations of the associated energy functionals. In relation to the singular CR Yamabe problem, we show that one of the energy functionals appears as the coefficient (up to a constant multiple) of the log term in the associated volume renormalization. We ask how these “CR Willmore” surfaces are related to geometric quantities inside the domain.





