Applied Math Collective

Applied Math Collective was initiated by my advisors and Fernando Guevara Vasquez. The aim is to provide an informal platform where the speaker discusses general-interest "SIAM review"-style applied math papers, led by either faculty or graduate student. We meet Thursdays at 4pm in LCB 222, when the Department Colloquium does not have a speaker. Please contact me if you would like to attend or give a talk so that I can add you to the mailing list.

➜ Fall 2018

August 30
Speaker: Zach Boyd
Title: Stochastic Block Models are a Discrete Surface Tension
Abstract: The accompanying paper can be found here: [arXiv].

September 6
Speaker: Franco Rota
Title: Lebesgue's integration, integrals of limits and limits of integrals
Abstract: We will introduce and motivate Lebesgue's theory of integration. One of the key results is a theorem of completeness. We will assume it and use it to prove theorems about passing the limit under the integral sign. Then, we will see these theorems in action on a variety of examples.

September 13
Speaker: Adam Brown
Title: Computing stratifications of finite topological space
Abstract: In this talk I will discuss a computational approach to the study of stratified topological spaces. We will see how techniques originating in sheaf theory and the proof of the topological invariance of intersection homology can be used to develop algorithms for computing stratifications of finite topological spaces. Motivated by the applicability of geometric and topological techniques in data science, we will conclude with an overview of current work (joint with Bei Wang) which aims to combine sheaf theory, topological data analysis, and statistics to study finite point sets sampled from stratified topological spaces.

September 20
Speaker: Yiming Xu
Title: Bernstein's inequality and Johnson-Lindenstrauss random projections
Abstract: In this talk I will introduce how to use an old technique in large deviation theory to derive the famous Bernstein’s inequality. Based on that, we will see how the renowned Johnson-Lindenstrauss random projection lemma in data compression comes into display. I will explain how this lemma can also be seen from the perspective of geometric functional analysis as well as the possible extension to a more general setting. [Notes]

October 18
Speaker: Qing Xia
Title: A Domain Decomposition Approach based on Difference Potentials Method for Chemotaxis Models in 3D
Abstract: In this talk, I will present a domain decomposition approach based on Difference Potentials Method (DPM) for approximating the solution to the classical Patlak-Keller-Segel chemotaxis models in 3D. We employ DPM and uniform Cartesian meshes to handle sub-domains of complex geometric shapes, without loss of accuracy near the irregular boundaries of the sub-domains. As a result of using uniform meshes, fast Poisson solver based on FFT is employed for better efficiency of our numerical algorithms. In addition, our domain decomposition approach is capable of mesh adaptivity and is suitable for parallel computing, which further boosts the efficiency. Numerical results from 3D simulations will be given to demonstrate the significantly improved efficiency and similar accuracy of the domain decomposition approach, in comparison to the single domain approach. This is joint work with Y. Epshteyn.

October 25
Speaker: Hyunjoong Kim
Title: What is the optimal gating strategy to maximize the survival probability?
Abstract: If we only open the gate for a limited time, then when should we open the gate to keep diffusing particles in a domain as many as possible? This problem is motivated by the stochastic gating problem of diffusing particles to explain the fluttering phase of insect respiration. The result shows that the fluttering period maximizes the oxygen uptake from outside with assuming the bang-bang control. One following question is "what is the best way not to lose particles under the limited closing time without assuming bang-bang?" We introduce some similar problems and its method. We also discuss the difficulties (like non-linearity) and our approaches to resolving the issue. This project is jointly working with Braxton Osting and Dong Wang.

November 1
Speaker: Chee Han Tan
Title: Direct Method of the Calculus of Variations and Mountain Pass Lemma
Abstract: Variational methods find solutions of equations by considering a solution as a critical point of an appropriately chosen function. Criteria for the existence of local extrema are well-known, but those for mountain passes or saddle points are less well-known. For the first half of the talk, we prove the conceptionally simple Direct Method of the Calculus of Variations that is widely used in establishing existence of minimisers. We then consider the problem of minimising an integral functional over a reflexive Banach space and give conditions on the integrand that guarantee the existence of minimisers. For the second half, we will explore the general problem of detecting critical points using geometrical intuition from finite-dimensional examples. Following James Brisgard's SIAM review article "Mountain Passes and Saddle Points", we state the Mountain Pass Lemma that guarantees existence of critical points in the finite-dimensional case and sketch its proof.

November 15
Speaker: Ryan Viertel
Title: Coarse Quad Layouts Through Robust Simplification of Cross Field Separatrix Partitions
Abstract: Streamline-based quad meshing algorithms use smooth cross fields to partition surfaces into quadrilateral regions by tracing separatrices of the cross field. In practice, reentrant corners and misalignment of singularities lead to small regions and limit cycles, negating some of the benefits a quad layout can provide in quad meshing. In this paper we implement a pipeline for coarse quad partition generation on CAD geometries. We introduce three novel methods in the pipeline that improve on previous research. First, we extend the MBO method for cross field design from Viertel et al (2017). This results in an efficient method to compute high quality cross fields on curved surfaces. Next, we introduce a method for accurately computing the trajectory of streamlines through singular triangles that prevents tangential crossings. Finally, we introduce a robust method to produce coarse quad layouts by simplifying the partitions obtained via naive separatrix tracing. Our pipeline is tested on a database of 100 objects and results are analyzed.

November 29
Speaker: Ryleigh Moore
Title: Percolation Theory and Applications
Abstract: In mathematics and statistical physics, percolation theory describes the behavior of connected clusters in a random graph. Percolation theory has been used to model various phenomena including forest fires, the spread of viruses, and, in my own research, Arctic melt pond evolution. In this talk, I will formulate and describe percolation theory in a two dimensional square lattice and motivate its usefulness in research. Specifically, I will explain what it means to percolate, discuss different types of percolation, explain the percolation threshold, and define critical exponents. I will also demonstrate how I use percolation theory to study melt pond evolution.