DOmath 2021

DOmath 2021 ran from May 17 until July 9, 2021. The 2021 program featured 6 projects and 20 undergraduate student researchers.

Projects for DOmath 2021

Holonomy of combinatorial surfaces

Project leader: Professor Robert Bryant
Project manager: Seppo Niemi-Colvin
Team members: Aram Lindroth, Alanna Manfredini, Nathan Nguyen

The concept of holonomy explores how smooth surfaces interact against each other by “rolling” one surface over another through arbitrarily closed loops around a base point. In our project, we focused on the discrete setup of this problem, in particular, we referred to triangular surfaces where we put an interaction between two surfaces by choosing two different faces of each one and matching vertices between them. This choice of two touching faces and the matching between their vertices define a position between the two surfaces. So our general goal of understanding the interaction between two surfaces is to figure out how the position changes after rolling the surfaces over each other, what are all possible positions we can get after rolling in arbitrary loops.  Read more

Characterizing emerging features in cell dynamics

Project leader: Professor Veronica Ciocanel
Project manager: Ray Tung
Team members: Maddie Dawson, Carson Dudley, Sasamon Omoma

Actin is one of the most abundant proteins in cells which assembles to form polymer filaments. When actin filaments interact with motor proteins, they can organize into ring channels. These ring channels are present in many critical biological processes, such as cellular division, development, and wound healing. The onset of ring channels and their maintenance throughout time is not well understood. Our research proposes new and improved methods of analyzing ring channel dynamics throughout time. To accomplish this, we used topological data analysis (TDA), a growing branch of mathematics, to understand the shape of data. We successfully developed and applied TDA methods to synthetic and experimental time-series contractile ring data.  Read more

Parameter interference in epidemiological models

Project leader: Professor David Dunson
Project managers: Tao Tang and Omar Melikechi
Team members: Trevor Bowman, Jenny Huang, Greg Orme, Pranay Pherwani

During an epidemic, it is important to gain an understanding of the intrinsic properties of an outbreak, such as how rapidly it spreads or how many individuals are currently infected. Accurate forecasts about the trajectory of the epidemic rely heavily on an understanding of these basic properties during the early stages of an outbreak. The early-stage outbreak also happens to be when parameters, such as the transmission rate, are the most difficult to pin down. We explore several methods that may allow for better estimation of these parameters. These methods include incorporating known information about the recovery rate of the disease in question and considering a stochastic model which may be more accurate during the early days when case counts are low. Moreover, as seen during the COVID-19 pandemic, transmission rates may change rapidly throughout the course of the epidemic with increased testing or shifts in containment strategies. Hence, we also focus on designing fast and simple methods for detecting a change in transmission rate. Our work is ultimately aimed at helping policy-makers make informed decisions during an early epidemic, such as what types of data (data on asymptomatic spread, recovery rate of the disease, contact network data etc.) would be most effective to collect to gain a good understanding of these basic properties of the disease and thus more accurate forecasts.  Read more

Excursions into the calculus of variations and notions of convexity

Project leader: Professor Tarek Elgindi
Project managers: Yupei Huang and Karim Shikh-Khalil
Team members: Jeffrey Cheng, Jeremy Savarese, Samia Zaman

Almost every physical phenomenon has behind it a minimization or maximization principle. The calculus of variations is the study of such principles and it contains powerful tools that are used throughout mathematics. We refer to a problem in which we are searching for a minimum or a maximum as an optimization problem. This summer, our group worked on one such problem which relates to an open question in mathematics known as the Morrey Conjecture (1952). The conjecture deals with two properties of functions which are known as quasiconvexity and rank-one convexity. It is currently unknown whether these two properties are equivalent in general (that is, whether each implies the other). If these notions were in fact equivalent, it would allow for the use of more powerful tools in the calculus of variations. Although it can be shown that every function satisfying the quasiconvexity property must also satisfy rank-one convexity, Morrey conjectured that the reverse-implication will not always hold. Proving this conjecture was the primary focus of our work this summer. Doing so would require us to find an example of a function which satisfies the rank-one convexity property but not the quasiconvexity property.  Read more

Topological insulators

Project leader: Professor Jianfeng Lu
Project managers: Stephen McKean and Kevin Stubbs
Team members: Santino Panzica, Will Strong, Luke Triplett

Phases of matter are characterized by the symmetries they preserve. For example, solids preserve orientation and periodicity, while liquids preserve neither. And between them, there are states of matter which preserve these symmetries to varying degrees. Thus, phase changes are characterized by these symmetries breaking or initializing. In addition to spatial symmetries such as these, we can also look at time-symmetry to get a more complete picture of how some special materials behave. Traditionally, matter is not sensitive to whether time flows forward or backward. However, the electromagnetic properties of some materials (called topological materials) do not respect this “time reversal symmetry.” Some of these are topological insulators, which have useful electrical properties: the interior is an insulator, while the exterior is a conductor. Our project studied the behavior of these topological insulators.  Read more

Mathematical clairvoyant: computational inverse problems

Project leader: Professors Hongkai Zhao and Yimin Zhong
Project manager: Mo Zhou
Team members: Arun Chakrabarty, Sarah Glomski, Nathanael Ong, Ashley Wang

With current technology, we are able to utilize X-rays to obtain images of the human body. But an X-ray cannot just obtain a ”picture” of internal objects inside the human body; how, then, do we obtain a legible image? When an X-ray beam passes through an object at a given intensity, a certain amount of the rays are absorbed or deflected by the object, which allows us to obtain information about the materials within the object. By examining that difference between the intensity of the X-rays that enter and exit, we can draw conclusions about the X-ray absorption of the materials in the body, which allows us to then reconstruct a sense of the materials that the X-rays had to pass through.  Read more