DOmath 2022 ran from **May 16** until **July 8, 2022**. The 2022 program featured 6 projects and 21 undergraduate student researchers.

*One of these projects has been featured on the Duke research blog! See here for the article by Robin Smith: Modeling the COVID-19 Roller Coaster.*

### Projects for DOmath 2022

**Spatial and temporal epidemic prediction by neural networks**

**Project leader: **Professor **Xiuyuan Cheng**

**Project manager: **Yixuan Tan

**Team members: **Brian Lee, Flora Shi, Nick Talati

Understanding the spread of disease is extremely important for public health and policymaking. An epidemic is just one example of a spatial temporal phenomenon–meaning that over time, a disease spreads across space. We simulate the spread of an epidemic using a stochastic SEIR model, which describes how people transition from being susceptible to a disease, to exposed, to infected, and to recovered. We also adapt this model for populations represented by graphs by simulating movement between nearby regions (nodes) over the course of the simulated epidemic. We analyze the predictive performance of several different artificial intelligence models (all neural networks) on simulated epidemics. Importantly, we modify standard recurrent neural networks to allow them to preserve information about the graph on which the epidemic simulation took place. We also demonstrate the predictive performance of these models on real world data. *Read more*

**Structure and stability for Brascamp–Lieb inequalities**

**Project leader:** Professor **Nicholas Cook**

**Project manager: **Haotian Gu

**Team members: **Harry Chen, Jean-Luc Rabideau, Alden Wang

Finner's inequality is a large family of inequalities with many applications in different branches of mathematics. It relates the integrals of products of functions with the product of integrals of functions. Some special cases of Finner's inequality include the Cauchy-Schwarz Inequality, which relates the length of two vectors with their dot product. Another application is the Loomis-Whitney Inequality, which relates the volume of an object with the areas of its shadows. The types of functions where equality holds in Finner's Inequality are already known. Our project instead looked at when equality is close to holding. If there is almost equality in the expression proposed by Finner, then are the functions in question necessarily close to these known types of functions where equality does hold? This is what it means for an inequality to be stable. Our project investigated the stability of Finner's inequality. *Read more*

**Mathematical questions arising from the COVID epidemic**

**Project leader:** Professor **Rick Durrett**

**Project manager: **Hwai-Ray Tung

**Team members:** Laura Boyle, Sofia Hletko, Jenny Huang, June Lee, Gaurav Pallod

The aim of this project was to use mathematics to understand the observed patterns of covid variant evolution. *Read more*

**Exploring minimal surfaces modulo p**

**Project leader:** Professor **Demetre Kazaras**

**Project manager:** Kai Xu

**Team members: **Hiba Benjeddou, Ben Goldstein, Khiyali Pillalamarri, Nico Salazar

A minimal surface is a surface in 3-space which locally minimizes surface area. These structures are found in nature as membranes experiencing equal pressure from opposing sides, such as soap films spanning a wireframe. In our project we investigated and found new examples of 'minimal mod-3 surfaces'. These objects are made up of pieces of minimal surfaces meeting each other 3 at a time along 'singular curves' so that the angles between colliding faces are always 120 degrees. Not much is known about these minimal mod-3 surfaces -- our goal was to find new examples and investigate what properties the singular curves might have. *Read more*

**Modeling the dynamics of filter fouling**

**Project leader:** Professor **Thomas Witelski**

**Project manager: **Yuqing Dai

**Team members: **Dominic Jeong, Juliet Jiang, Ada Zhang

Filter membranes have many applications in industrial fields and practices, such as filter masks and fabrics. However, materials composed of filter membranes may experience particle contamination and clogging at a microscopic scale. Over time, the fluid evaporates, leaving particles that deposit on the pore channel walls. Previous models made far more restrictive simplifications. With this in mind, we are able to take a couple of different approaches in investigating the behavior of the system. By using either a probabilistic model or a differential equation, we are able to compare and validate these different models to investigate what holistically happens in the system. In the probabilistic model, we make use of different models to predict particle behavior at the boundaries; the particle can either reflect off of the wall or be absorbed. Furthermore, we can systematically analyze how changing the relative strengths of different effects, such as particle deposition rate and evaporation rate, change both the concentration and the radius of different parts of the pore. We can also model the system as a droplet shrinking over time with a fixed contact radius, for which we develop similar probabilistic and differential equation models. *Read more*

**Smoothness of subspace-valued maps**

**Project leader:** Professor **Ruda Zhang**

**Project manager:** Yixin Tan

**Team members:** Noah Harris, Matt Robbins, Marie-Hélène Tomé

Our project revolved around approximating subspace-valued functions, i.e., rules assigning some parameter to a set of vectors forming a basis for a subspace. The applications of this include reduced order modeling where subspaces are used to model complex systems in a more computationally efficient manner by reducing the dimension of the matrices used to mathematically represent the system. Instead of recomputing the subspaces needed through different kinds of model reduction methods, approximating the subspace-valued map directly from the parameter to the subspace cuts out all of this computation. Given some data about the mapping, we approximate the function from the parameter to the subspace so that we can get a good prediction of the function value, i.e., the bases needed for model reduction, at a new value of the parameter for which we have not computed the function value. This is especially useful in modeling second order systems where multiple model order reduction methods are used to create the final reduced order model. Our work helps to give theoretical guidance as to when GPS will work well, i.e., when you come up with a new combination of model order reduction methods, is GPS suitable to approximate this new variant? *Read more*