# DOmath 2021 DOmath 2021 runs from May 17 until July 9, 2021. The 2021 program features 6 projects and 20 undergraduate student researchers.

### Projects for DOmath 2021

1. Holonomy of combinatorial surfaces, led by Professor Robert Bryant

This is a project about what might be called "discrete (or combinatorial) rolling behavior".  A simple example is the following:  Start with a chessboard with squares of side length one unit and place a cube with side length one unit on one of the chessboard squares.  A valid "move" is to roll the cube over to an adjacent square keeping the common edge fixed.  Thus, if the square is not a boundary square, there are four possible valid moves.  By a succession of such moves, we can roll the cube to any other square on the chessboard, but the "path" we choose to do so will determine the final orientation of the cube at the end (and there are 24 possible orientations).  One can ask whether one can realize each of the possible orientations by a suitable choice of rolling path.  This is a special case of what is known as the "combinatorial holonomy problem".

Similar questions can be asked about rolling a regular tetrahedron, octahedron, or icosahedron over a plane tessellated by equilateral triangles of the appropriate size. Even more generally, one can consider rolling one triangulated surface over another, asking when any two configurations can be joined by a rolling path.  The idea is to consider two triangulated surfaces $S$ and $S'$ and consider the act of "rolling" $S$ over $S'$ in the following way:  Let $f$ (resp. $f'$) be a (triangular) face of $S$ (resp. $S'$) and start with an "initial position" which is an "identification" of $f$ with $f'$, i.e., matching their respective vertices.  (There are 6 ways to do this.)  Now consider moving to a another configuration $(f_1,f_1')$ by choosing an edge $e = e'$ of $f$ and moving to the new configuration $(f_1, f_1')$ where $f_1$ shares $e$ with $f$ and $f_1'$ shares $e'$ with $f'$ and the new identification $f_1$ with $f_1'$ continues to match vertices of the common edge $e = e'$.  Since there are three choices of the edge $e$, there are three ways to move to a new configuration.

The fundamental question is: Starting from a given configuration $(f,f')$ on given triangulated surfaces $S$ and $S'$, what are the possible configurations that can be reached by rolling?  For example, when can any two configurations be joined by a "rolling path"?  This defines an equivalence relation on configurations $(f,f')$. How many equivalence classes can there be for a given pair of triangulated surfaces? This is a discrete analog of the differential geometric problem of rolling one surface over the other without slipping or twisting, and this has a natural formulation in terms of holonomy of a fiber bundle.

It doesn't take much background to understand the concept of triangulations of surfaces.  The analysis of this problem and related problems will involve some graph theory, some group theory, and some topology.  It could also involve some computer programming (finding the best data structure to model this) and some geometry of random paths. We'll start by developing the tools needed to solve the chessboard and cube problem and also the cases involving the other Platonic solids, and then go on to explore the geometry, topology, and combinatorics of the general problem.  If there is time and interest, we can even explore the continuous version of this problem, which involves interesting techniques from ordinary differential equations and some differential geometry.

If you are interested in learning more about the origins of these questions, you might enjoy the following video that introduces the holonomy problem and spends a little time talking about the combinatorial version: https://www.youtube.com/watch?v=9qMv2exawsA  You might also like the following news article about some recent related breakthroughs: The Crooked Geometry of Round Trips.

2. Characterizing emerging features in cell dynamics, led by Professor Veronica Ciocanel

In cell and developmental biology, the organization of proteins into structures is often captured using time-series data of protein locations. Depending on their functions, with time proteins may accumulate into clusters, align into bundles, or organize into more complex structures such as ring channels (you can Google "contractile ring in wound healing").

Motivated by this dynamic organization which is important in cell function, we are interested in studying the shape of data generated using simple to more complex models. In this project, we will investigate how topological features emerge over time in synthetic data by drawing on tools from topological data analysis. In the process, we will also explore how to develop appropriate null models that allow us to assess whether the protein clusters or rings in the data are significant or the result of noisy data observations. This project will build on basic background in linear algebra and probability. Since we will primarily use and build on computational tools, some experience in programming will be useful, although students will learn about specific packages of interest during the program.

3. Mathematical & statistical modeling of COVID-19: SIR models and beyond, led by Professor David Dunson

In modeling of infectious disease transmission and spread, one of the most common mathematical models is the Susceptible, Infectious, Removed (SIR) model, which is a type of compartment model that can be expressed as an ODE.  In fitting this ODE to real world data from the COVID-19 pandemic, one can combine likelihood-based statistical methods with differential equation solvers to find parameters that maximize likelihood functions.  However, little is known about the properties of such maximum likelihood estimators (MLEs) and also such models are overly-simple in not accounting for the multiple peaks that have been observed in the COVID-19 pandemic.

Research outline:

Phase 1: Starting with a simple SIR model, fit the model to public data using methods recently developed by James Johndrow and co-authors.  Think critically about the model and the algorithms being used to fit the model, while also presenting the results in an interpretable manner.

Phase 2: Think carefully about why the models considered in phase 1 may be fitting the data poorly and consider model elaborations that are as simple as possible while being able to explain important factors such as the multiple epidemic peaks problem.  Develop and implement algorithms for fitting these new models.

Phase 3: Think mathematically about stability and identifiability of the SIR model parameter estimates, and use this theory to make recommendations informing practice in considering model elaborations and algorithms.

4. Excursions into the calculus of variations and notions of convexity, led by Professor Tarek Elgindi

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. The goal of the project is to introduce students to the basics of the calculus of variations and notions of convexity. At the beginning of the summer, students will attend and participate in a series of lectures on the subject, in order to understand the basic tools. Students will then explore, analytically and numerically, some more advanced research problems. One possible project is to explore the relationship between various notions of convexity in specific examples, as an approach to an open problem called the Morrey Conjecture.

It is desirable that students in this project have taken at least one course in real analysis. It is also desirable that students have coding experience. Being very strong in one could make up for lacking the other.

5. Mathematics and computation of topological insulators, led by Professor Jianfeng Lu

Topological materials are fascinating as they often exhibit intriguing behaviors on the boundary of the materials, for example, conductance along the edge while the bulk materials is insulating. Such edge behavior is also topologically protected, such that they persist even in the presence of large disorder and defects of the materials. Mathematically, the characterization of such materials is through index such as Chern number. In this project, we will investigate topological materials in the presence of large disorder both theoretically and numerically by studying toy model systems.

6. Mathematical clairvoyant: computational inverse problems, led by Professors Hongkai Zhao and Yimin Zhong

Computational inverse problems refer to a wide class of problems which are inferring important quantities computationally from physical measurements. Many of them find their important applications in the real world. One of the good examples is the CT scan, which uses X-ray to scan the body which could produce the image of underlying structure.  In this project, we start from the mathematical model of X-ray and try to formulate the mechanism of the CT-scan by the famous Radon transform, then study the theory about the Radon transform and learn when CT-scan is useful and not useful, in the next we learn to reconstruct the images from the CT-scan data by different computational methods and compare them. Besides the CT-scan, the X-ray related inverse problems have many other applications, such as X-ray crystallography to probe the structure of molecules.

An ideal student for this project would have solid background in multivariable calculus, linear algebra, ODE, and some programming knowledge.