2016 Talks
Spectra of Self-Similar Groups
Yuki Takahashi
October 12th, 2016 - 4:00 - 4:50pm - RH 340P
Abstract
If a group action is given, then we can naturally define a graph called a Schreier graph. We define self-similar groups and consider the spectra of Schreier graphs associated with them. This talk is based on the material that I studied recently for a conference talk, and is accessible to every math grad student.
About the Speaker
Yuki is a 5th year Ph.D. student. He received a Bachelor's degree in Mathematics from the University of Tokyo, and now he is working on Dynamical Systems and Mathematical Physics under the guidance of Professor A. Gorodetski. In his free time, Yuki enjoys juggling, swimming, and practicing yoga. Also, he recently started to learn salsa dancing!
Advisor and Collaborators
Yuki's advisor is Anton Gorodetski.
Deforming Hypersurfaces via Mean Curvature Flow with Surgery
Alex Mramor
June 1st, 2016 - 4:00 - 4:50pm - RH 440R
Abstract
The mean curvature flow provides a way to generate paths in the space of hypersurfaces so can be used to prove connectedness theorems, but it has its limitations. In this talk we explain how an extension of the mean curvature flow, mean curvature flow with surgery, can be used to partially overcome these limitations and prove stronger connectedness results.
About the Speaker
Alex is a third year grad student. When he is not doing math, he likes to exercise, hike, go to the beach, and try to learn about physics.
Advisor and Collaborators
Alex's advisor is Richard Schoen.
An Interface-Fitted Mesh Generator and Virtual Element Methods for Elliptic Interface Problems
Min Wen
May 25th, 2016 - 4:00 - 4:50pm - RH 440R
Abstract
We propose virtual element methods for solving one- or multi-domain elliptic interface problems using interface-fitted meshes. The main challenge is to design an efficient and robust mesh that can capture certain properties while preserving arbitrary complex geometries of the interface. Moreover, it is a tricky problem in classical finite element methods when the domain is decomposed into tetrahedra due to the existence of slivers in three dimensions. In our mesh generation, every element in three dimensions could be any polyhedron instead of tetrahedron. Then we apply virtual element methods for solving elliptic interface problems with solution and flux jump conditions. The purpose of using virtual element methods rather than classical finite element methods is that every element could be a different shape. We use multi-grid solvers to solve the discrete system. Lastly, numerical examples demonstrating the theoretical results for linear elements are shown
About the Speaker
Min is a 3rd year graduate student supervised by Prof. Long Chen. In her free time, she likes playing cards and scouting new restaurants on Yelp.
Advisor and Collaborators
Min's adviser is Long Chen.
Sums of Two Cantor Sets and Palis Conjecture
Yuki Takahashi
May 16th, 2016 - 4:00 - 4:50pm - RH 340P
Abstract
Sums of two Cantor sets arise naturally in homoclinic bifurcations, Markov and Lagrange dynamical spectra, and the spectrum of the square Fibonacci Hamiltonian. In the 1970s Palis conjectured that for generic pairs of regular Cantor sets either the sum has zero Lebesgue measure or else it contains an interval. This problem is known to be extremely difficult, and is still open for affine Cantor sets. In this talk, we will discuss the history of sums of two Cantor sets, and also introduce my recent results about sums of two homogeneous Cantor sets.
About the Speaker
Yuki is a 4th year Ph.D. student at UC Irvine. He received a Bachelor's degree in Mathematics from the University of Tokyo, and now he is working on Dynamical Systems and Mathematical Physics under the guidance of Professor A. Gorodetski. In his free time, Yuki enjoys juggling, swimming, and practicing yoga.
Advisor and Collaborators
Yuki's advisor is Anton Gorodetski.
Effecient Adaptive Algorithms for Stochastic ODEs
Chris Rackauckas
February 23rd, 2016 - 4:00 - 4:50pm - RH 440R
Abstract
Stochastic ODEs (SODEs) and PDEs (SPDEs) have become a significant modeling framework for problems ranging from biology to mathematical finance. However, the research in numerical algorithms for simulating such models are in their infancy compared to the deterministic counterparts. In this presentation we focus on an efficient algorithm for adaptive time-stepping. Methods to address this problem have been particularly underdeveloped since, unlike in adaptive deterministic methods, naive rejection sampling changes the distribution of the underlying Brownian path. An introduction will be given to show the importance of adaptive numerical algorithms and the shortcomings of the current algorithms. This will be followed by a detailed �chalkboard� derivation of a new general adaptive time-stepping algorithm. The purpose of this style will be to both further understand the correctness of the algorithm and display it in a way that will be intuitive for other practitioners to implement the algorithm in their own works. The talk is aimed at a general audience and will be mostly self-contained with no background in stochastic numerics required.
About the Speaker
Chris is a 3rd year MCSB student (2nd year Math) interested in stochastic analysis/computation and its applications to developmental biology. He tends to enjoy doing a diverse array of activities including the Mathletes softball, going to the beach with his beautiful pup Tara, and going on roadtrips with his girlfriend Diana.
Advisor and Collaborators
Chris' advisor is Qing Nie. Projects done in collaboration with the Schilling Lab will be briefly mentioned.
