Analysis Seminar, Fall 2024

Schedule

During Fall 2024, the Analysis Seminar will meet on Mondays 3:30-4:20pm in Keller 413.

Contact

If you would like to give a talk, please contact me at myounsi@hawaii.edu.

Talks

• September 30 : David Ross (University of Hawaii)
  
  Title: Infinite systems of equations and inequalities in infinitely many variables
  
  Abstract: 100+ years ago, mathematicians like Helly and Banach and Riesz were interested in the question of when an infinite set of linear equations in infinitely many variables had a solution. This led (for example)to the formulation of the Hahn-Banach Extension Theorem (HBET). This talk is motivated by some relatively recent results of a similar type, framed in the following way: if every finite subset of F has a common solution of some form then F has a common solution of that form, where F is a certain set of equations or inequalities in infinitely many variables. The proofs are fairly elementary: the most sophisticated tool used is the Tychonoff Theorem from point-set topology (which incidentally is set-theoretically independent of the HBET).



• October 7 : Shubham Joshi (University of Hawaii)
  
  Title: Conformal Removability
  
  Abstract: A set in the plane is called conformally removable if every homeomorphism of the plane that is conformal off of the set can be extended to a conformal map of the entire plane. I will briefly discuss the problem of removability and examples of sets known to be removable. Characterizing removable sets has been a challenging endeavor and I will present a proposed solution.



• October 21 : Guang Yang (University of Hawaii)
  
  Title: Coupling method and the fundamental gap problem on the sphere.
  
  Abstract: Coupling is a powerful probabilistic tool in the study of harmonic functions and hypoelliptic operators. In this talk, I will provide an overview of some fundamental ideas of the coupling method and stochastic differential geometry. We will then focus on applying these ideas to the study of the fundamental gap conjecture on the sphere. This talk is based on a recent joint work with Gunhee Cho and Guofang Wei.





• November 4 : Katheryn Menssen (University of Hawaii)
  
  Title: The Variation of Area and Logarithmic Capacity under Holomorphic Motions.
  
  Abstract: Holomorphic motions are, in essence, functions in two complex variables which are holomorphic in one variable and injective in the other. The variation of Area and Logarithmic capacity (along with other notions of capacity and dimensions) under these functions has only recently been studied, yet has already achieved a number of interesting results. In this talk, I will cover a small subset of these results--originating in research performed by Dr. Younsi, Thomas Ransford, Aiden Fuhrer, and Wen-Hui Ai--along with a few new results from my own research with Dr. Younsi.





• November 25 : Donald Lee (University of Hawaii)
  
  Title: Local fields, Iterated Extensions, and Julia Sets
  
  Abstract: p-adic dynamics involves the study of properties of orbits of self maps under a nonarchimedean local field setting. Although a lot of the theory are modeled after it's complex dynamics counterpart, the nonarchimedean absolute values of p-adic fields give rise to striking differences between the p-adic and complex theories. In this talk, I will describe the Berkovich Julia sets of unicritical polynomials and also the field extensions generated from their iterations. This talk is based on joint work with Michelle Manes and Nha Truong.