Title: Eigencurve over the Boundary of the Weight Space

Speaker: Ruochuan Liu（刘若川）

Time: Sunday, Apr. 26, 14:00—14:45

Venue: Lecture Hall, Jia Yi Bing Building, 82 Jing Chun Yuan, BICMR

Abstract:
We will prove a folklore conjecture concerning the geometry of the boundary of eigencurves in the case of definite quaternion algebras over Q. Precisely, we prove that: (a) over the boundary annuli of the weight space, the eigencurve is a disjoint union of (countably) infinitely many connected components each finite and flat over the weight annuli, (b) the Up-slopes of points on each fixed connected component are proportional to the p-adic valuations of the parameter on the weight space, and (c) the sequence of the slope ratios form a union of finitely many arithmetic progressions with the same common difference. Joint work with Daqing Wan and Liang Xiao.

Title: Fluctuating-Rate Model and Stochastic Phenotype Transition of a Single Cell

Spearker: Hao Ge（葛颢）

Time: Sunday, Apr. 26, 14:45—15:30

Venue: Lecture Hall, Jia Yi Bing Building, 82 Jing Chun Yuan, BICMR

Abstract:
We proposed a fluctuating-rate model for the stochastic biochemical dynamics in a single cell, which is indeed stochastic coupled Ordinary Differential Equations. We also found that the fluctuating-rate model yields a nonequilibrium landscape function, which, similar to the energy function for equilibrium fluctuation,provides the leading orders of fluctuations around each phenotypic state, as well as the transition ratesbetween the two phenotypic states. The rigorous proof needs to integrate the well-known Donsker-Varadhan theory and Feidlin-Wentzell theory in such an averaging case. We further apply this model to Lac operon, and show that the stochastic gene-state switching can significantly broaden the environmental parameter ranges for the existence of bistability induced by positive feedback, which can be beneficial dealing with unpredictable environmental changes. We also demonstrate that the transition rates between different phenotypic states achieve the maximal value at the intermediate region of gene-state switching, and the barrier term in the rate formula can help to distinguish two categories of bistability.

Title: Mixed Finite Element Methods for Elasticity Problems

Speaker: Jun Hu（胡俊）

Time: Sunday, Apr. 26,15:30—16:15

Venue: Lecture Hall, Jia Yi Bing Building, 82 Jing Chun Yuan, BICMR

Abstract:
Designing stable and optimal mixed finite elements for elasticity problems is a challenging and surprisingly hard problem, which has been open since sixties last century and was studied by a lot of prestigious computational Mathematician. This talk presents simple and optimal mixed finite elements which solved finally the aforementioned problem.

Title: Demailly's strong openness conjecture and related problems

Speaker: Qi’an Guan（关启安）

Time: Sunday, Apr. 26,16:15—17:00

Venue: Lecture Hall, Jia Yi Bing Building, 82 Jing Chun Yuan, BICMR

Abstract:
Multiplier ideal sheaf is an important objection in complex geometry and algebraic geometry. In this talk, we will survey some recent progress of Demailly's strong openness conjecture and related problems on multiplier ideal sheaves.