REU Vir(tu)al Conference 2021

Group Photo from the end of the conference. This is a compilation of screenshots, so some people appear twice.

The REU conference has now concluded. Thank you to everyone who made it possible! If you have the chance, please leave us some feedback here.

General Information

This conference is part of an annual series in which Math REU students from various colleges in the northeast, this year, Amherst College, Tufts Univeristy, UMass Amherst, UConn, and Yale University, will present work done in their summer research projects.

The conference will be partly synchronous and partly asynchronous.  Talks and posters will be available for viewing and discussion in advance, and there will be a discussion forum for each talk held using the Webex videoconferencing platform (you do not need a Webex account to participate).

Overview

The conference will take place on Wednesday and Thursday, July 28th-29th.

Wednesday, July 28th, will be the asynchronous portion of the conference. Participants will attend prerecorded talks online. We encourage questions to be asked in the comment section of each video. On this day, we hope that each attendee comment and ask a question on at least three other talks. Talks should be between 15 and 20 minutes long.

Thursday, July 29th will be the live (virtual, synchronous) portion of the conference. Each talk will have a 20 minute timeslot. This time will be used for speakers to answer questions from the comments on the videos as well as any follow-up questions. Speakers should prepare a 1-2 min recap of their talk to commence discussion. The tentative time for the live portion of the conference is 12:00 noon – 6pm EDT. There will also be a panel on graduate school at 2pm.

Graduate School Panel

The conference also includes a graduate school panel at 2pm on July 29th. You may view the panel and ask questions in advance on this Padlet.

Meeting Access

Anyone may join the discussion on Webex by following this meeting link. You can participate through your browser.

Schedule

Wednesday, July 28th

  • 11:30 – Brief welcome and social time
  • Rest of the day: View talks. Ask a question in the comment section of at least three.
Thursday, July 29th

Session Details:

  • Session on Algebra and Number Theory

    Chair: Dr. Amanda Folsom
    Back to Master Schedule

    • 12:00 noon
      Overpartition Ranks and Quantum Modular Forms
      Anna Dietrich, Keane Ng, Chloe Stewart, Shixiong (Leo) Xu.
      Amherst College. Mentor: Amanda Folsom.
      Abstract: A partition of a positive integer $n$ is any non-increasing sequence of positive integers which sum to $n$, and the partition function $p(n)$ counts the number of partitions of $n$. A classical identity due to Euler gives rise to the beautiful and unexpected fact that the generating function for $p(n)$ is essentially a modular form, a well-studied type of symmetric, complex analytic function. Here, we study other new combinatorial partition generating functions with the goal of establishing modularity properties, if any. In particular we study quantum modular properties of such combinatorial functions, where quantum modular forms are a new type of modular form defined by Zagier in 2010, and which have been shown to exhibit rich connections to diverse areas of mathematics, including combinatorics, number theory and more.
    • 12:20pm
      Metaplectic Analogues of Macdonald Polynomials
      Cory Eckert.
      UMass Amherst. Mentor: Paul Gunnels.
    • 12:40pm
      Classifying Groups with Two Generators and One Relation
      Rebecca Barry and Sarah Hayward.
      Tufts University. Mentors: Kim Ruane, Genevieve Walsh, Lorenzo Ruffoni.
    • 1:00pm
      Understanding Growth of Finitely Generated Groups
      Thomas Sachen.
      Tufts University. Mentors: Kim Ruane, Genevieve Walsh, Lorenzo Ruffoni, Mackenzie McPike.
      Abstract: A quick survey of growth types of finitely generated groups and how it relates to quasi-isometry and other geometric and algebraic properties.
  • Session on Machine Learning

    Chair: Dr. Luke Rogers
    Back to Master Schedule

    • 12:00 noon
      Graph-based Semi-supervised Learning with Applications in Protein Function Prediction
      James Swomley, Diego Griese.
      Tufts University. Mentors: Xiaozhe Hu, James Adler, and Kaiyi Wu.
    • 12:20pm
      Demystifying The Spectral Bias of Overparameterized Deep Neural Networks
      Genglin Liu.
      UMass Amherst. Mentor: Yulong Lu.
    • 12:40pm
      Group Convolutions
      Jack Champagne.
      UMass Amherst. Mentor: Wei Zhu.
      Abstract: Convolutional neural networks offer translational equivariance, helping networks classify images with translations effectively. This property is not true in general for rotations, mirroring, and scaling. Using a group-equivariant neural network, it is possible to construct convolutions that have these properties for rotation and mirroring. Group CNNs can effectively improve performance in classifying rotated datasets and generalize to unseen rotated data.
    • 1:00pm
      Neural Network SDE Solver
      Caleb Meridith.
      UMass Amherst. Mentor: Yao Li.
      Abstract: In a recent paper, UMass Amherst Professor Yao Li explored training neural networks to find the invariant probability measure of stochastic differential equations by incorporating the Fokker-Planck equation into the loss function for the neural network. For this project, I worked with Professor Li to train neural networks to find the time dependent solution of a 1D stochastic differential equation using this improved loss function.
  • Session on Combinatorics

    Chair: Dr. Emily Gunawan
    Back to Master Schedule

    • 12:00 noon
      Box-ball systems and RSK recording tableaux
      Marisa Cofie, Olivia Fugikawa, Madelyn Stewart, David Zeng.
      Yale University. Mentor: Emily Gunawan.
      Abstract: A box-ball system is a discrete dynamical system which can be thought of as a collection of time states. At each state, we have a collection of boxes with each integer from 1 to n assigned to a unique box. A permutation on n objects gives a box-ball system state by assigning its one-line notation to $n$ consecutive boxes. After a finite number of steps, a box-ball system will reach a steady state. From any steady state, we can construct a tableau (not necessarily standard) called the soliton decomposition of the box-ball system.We prove that the Robinson–Schensted recording tableau of a permutation completely determines the behavior of the corresponding box-ball system. We use this fact to prove many results about the box-ball system. First, we prove that the permutations whose BBS soliton partitions are L-shaped have steady-state time at most 1. This large class of permutations include column reading words and non-crossing involutions. Next, we study the permutations whose RS insertion tableaux and soliton decompositions coincide and refer to them as “good”. We prove that these good permutations are closed under consecutive permutation pattern. Furthermore, we conjecture that the number of RS recording tableaux which give good permutations are counted by the Motzkin numbers.
    • 12:20pm
      Juggling Poset for initial and terminal state 1^k
      Viviene Do.
      UMass Amherst. Mentor: Laura Colmenarejo.
      Abstract: We look at juggling from a combinatorial viewpoint. Juggling has been formalized in multiple ways. We consider two types of juggling: basic where at most one ball is thrown at a time, and multiplex where at most c balls are thrown at a time. For basic juggling, we look at Siteswaps, which are special sequences of numbers that represent a basic juggling pattern, and some known results for Siteswaps. Then we focus on an alternative representation of juggling sequences for multiplex juggling. Finally, we look at a generalized Poset for juggling sequences and consider the specific case where the initial and terminal state are .
    • 12:40pm
      Cyclic sieving for block factorizations of the long cycle.
      Justin Bailey.
      UMass Amherst. Mentor: Theo Douvropolous.
      Abstract: Hurwitz knew already in 1889 that there are n^{n-2} minimum length factorizations of the long cycle (123..n) of the symmetric group in transpositions; the same number that counts labeled trees on n vertices. If instead of transpositions, we consider cycles of fixed length, then there are n^{c-1} such factorizations with c factors; this is a special case of the Goulden-Jackson cactus formula. We study a cyclic action on the set of such factorizations and we count its orbits of different sizes. Our answer is in the form of a cyclic sieving phenomenon (CSP), which means that these orbit sizes are given as polynomial evaluations at roots of unity. We will give background on combinatorial factorizations, explain what a CSP is, and briefly discuss some proof techniques: monodromy of polynomials, and the weighted Bezout’s theorem.
    • 1:00pm
      Asymptotics for Algebraic Generating Functions
      Pranav Garg.
      UMass Amhers. Mentors: Mark Wilson, Steve Melczer, Tia Ruza and Torin Greenwood.
      Abstract: What are algebraic generating functions? New method to do asymptotics on algebraic generating functions in several variables.
  • Graduate School Panel
    • Meet our amazing panelists and ask questions in advance. The panel will answer your questions at the scheduled time. If there is time remaining, the panelists will also take live questions.
  • Session on Markov Chains

    Chair: Dr. Iddo Ben-Ari
    Back to Master Schedule

    • 3:00pm
      Ballistic Deposition Model
      Caitlyn Powell, Na’ama Nevo, and Connor Bass.
      UConn. Mentors: Iddo Ben-Ari, Daniel Mourad.
      Abstract: Ballistic Deposition is a stochastic process on a connected graph where, at each time step, a random vertex is chosen and its height is assigned to 1 plus the maximum height of itself and its neighbors. We define gamma as the limit as time goes to infinity of the expected maximum height over time multiplied by the number of vertices. We are examining existing bounds on gamma of the cycle graph as well as presenting new techniques to approximate this value.
    • 3:20pm
      Coalescing Particles on a Cycle Graph
      Christopher Vairogs, Daniel Zou.
      UConn. Mentors: Iddo Ben-Ari, Daniel Mourad.
      Abstract: At time 0, we place one particle at each vertex of the N-cycle. At each time t>=1, randomly sample an occupied vertex and move the particle at the randomly sampled vertex one step counter-clockwise. If there is already a particle at the new vertex, combine both particles into a single particle. We present, with proof, a formula for the expected time it takes for all particles on the N-cycle to coalesce into a single particle. We discuss variations of this problem.
    • 3:40pm
      Quasi-Stationary Distributions on the Invasion Model
      Julia Shapiro, Clay Allard, Shrikant Chand.
      UConn. Mentors: Iddo Ben-Ari, Daniel Mourad.
      Abstract: We study the invasion model on the complete bipartite graph. In this model, nodes are randomly selected to impose their opinions on one another until reaching consensus of each node having the same opinion on the graph. We look for quasi-stationary distributions, an analog for a stationary distribution of a Markov chain conditioned on not being absorbed, in the invasion model. We first evaluate some simple cases where the model is on a star graph and determine the geometric absorption rate for this model to reach consensus, given by its largest eigenvalue. We then study this rate for the general graph and use a coalescing Markov chain approach, generalizing the coalescing random walk used to study the voter model, to extract an approximation for its absorption rate. Using simulations, we conjecture the marginal and conditional distributions from the QSD corresponding to this rate. We attempt to derive explicit expressions for the QSD of the invasion model by taking the limit of one of the groups on the graph to infinity.
  • Session on Working with Limited Data

    Chair: Dr. Owen Gwilliam
    Back to Master Schedule

    • 3:00pm
      An Analysis of Novel Data Acquisition Methods in Compton Tomography
      Rachel Dennis
      Tufts University. Mentor: Todd Quinto.
    • 3:20pm
      Numerical Study of Reconstruction Artifacts from Limited Sonar Data
      Konstantinos Tsingas
      Tufts University. Mentor: Todd Quinto.
    • 3:40pm
      A Mathematical Analysis of Reconstruction Artifacts in Radar Limited Data Tomography
      Elena Martinez.
      Tufts University. Mentor: Todd Quinto.
      Abstract: Tomography is the mathematics, science, and engineering that is used to determine the internal structure of an object using indirect data. In X-ray computed tomography (CT), we can obtain this data by integrating over all the lines in all directions of an object. The data can in turn be used to reconstruct the object, creating detailed images of internal organs, bones, and blood vessels. It is often the case, however, that some data values are missing. This occurs in a wide variety of problems and is referred to as limited data tomography. When reconstructing such a data set, artifacts often present themselves in the reconstruction. We study this problem in radar applications. Given a radio transmitter in a fixed location and a receiver that is flying around the transmitter in circular paths, we secure our data by integrating over all ellipses with the transmitter and receiver as foci. We reconstruct this data and analyze any artifacts that present themselves. Our objective is to determine whether objects within and outside of the flight path can be well reconstructed.
  • Session 1 on Topology/Geometry

    Chair: Dr. Ian Adelstein
    Back to Master Schedule

    • 3:00pm
      Minimal Genus in 4-Manifolds with Circle Actions
      Audrey Rosevear.
      UMass Amherst. Mentor: Weimin Chen.
      Abstract: Our goal is to study the ways 2-dimensional surfaces can live in 4-dimensional spaces, as a means for understanding the structure of the spaces. Our techniques are gauge theoretic, coming from quantum physics. This talk is mostly an introduction to the problem and a sketch of our current approach; we hope to have results sometime next winter.
    • 3:20pm
      Towards Robust Curvature Computation in Point Clouds
      Kincaid MacDonald, Jennifer Paige, Dawson Thomas, and Sarah Zhao.
      Yale University. Mentors: Ian Adelstein, Smita Krishnaswamy, Dhananjay Bhaskar, Bastian Rieck.
      Abstract: With the influx of big data in recent decades, many methods have been developed to capture the structures of high-dimensional data sets with lower-dimensional manifolds. However, standard techniques for processing this data often discard a wealth of local geometric information. While many dimensionality reduction and unsupervised learning techniques can detect clusters, they do not preserve the shape of the clusters, and are blind to variance within the manifold’s intrinsic metric. We posit that the point-wise curvature of nodes on a graph constitutes a useful feature for understanding the underlying manifold. In this project, we incorporate tools from diffusion geometry to propose improvements to two previous methods of discrete curvature computation and present two novel approaches. We first adapt the Ollivier-Ricci curvature—initially defined for unweighted graphs—to point cloud data sets by incorporating diffusion probabilities and increase the computational speed with a modified Earth Mover’s Distance. The second method, proposed by Sritharan et al. (Jan 2021), uses quadratic polynomial regression to extrinsically calculate the Riemannian curvature tensor. We extend this method with diffusion-based neighborhood selection and the incorporation of the latest techniques for approximating tangent planes. Finally, we propose two novel methods for computing curvature inspired by Alexandrov geometry, using triangles and geodesic ray comparisons to capture the geometry.
    • 3:40pm
      Gaps Between Saddle Connections on Translation Surfaces
      Synneva Bratland, Jordan Grant, Jo O’Harrow, Laurel Turner.
      Yale University. Mentor: Taylor McAdam.
      Abstract: Motivated by a dynamical systems method for studying the gaps between rational numbers, we study the gaps between the slopes of special straight-line paths on geometric objects called translation surfaces. The distribution of slope gaps on translation surfaces is an active area of study that connects dynamical systems, geometry, and number theory, however the procedure for computing these distributions is in general quite challenging and has only been carried out for a small number of surfaces. We discuss a new tool for finding the slope gap distributions of families of translation surfaces and provide experimental evidence for the slope gap distribution of a new such family of surfaces.
  • Session on Analysis and Financial Math

    Chair: Dr. Kasso Okoudjou
    Back to Master Schedule

    • 4:30pm
      Minimizing and Dividing Tight Frames
      Lily Ahmed, Eric Chavez, Irv Bahe, Sofia Encarnacion.
      Tufts University. Mentors: Kasso Okoudjou, Martin Buck.
    • 4:50pm
      An Introduction to Optimal Transport on the Sierpinski Gasket
      Cecilia Mikat, Robyn Steveley, Giorgos Kementzidis.
      UConn. Mentor: Luke Rogers.
      Abstract: The Sierpinski Gasket is an object with interesting properties. We can ask questions about Optimal Transport on the Sierpinski Gasket. Taking advantage of distance properties and Measure Theory, we turn our Optimal Transport on the Sierpinski Gasket question into a one-dimensional problem, whose solution is already well-known.
    • 5:10pm
      Spectral Similarity and Decimation on Various Self-Similar Groups
      Melissa Wei and Jake Brown.
      UConn. Mentor: Luke Rogers.
      Abstract: We determine spectral similarity of, and compute spectra for, sequences of graphs arising in group theory that were previously investigated by Grigorchuk and collaborators. Using methods from fractal analysis, we relate the spectra of the Laplacians on Schreier graphs for certain self-similar groups to sequences of graphs for which the spectra can be computed using the technique of spectral decimation.
    • 5:30pm
      Fair Pricing and Hedging under Small Perturbations of the Numéraire
      William Buschin, Delphine Hintz, & Alexey Pozdnyakov.
      UConn. Mentors: Oleksii Mostovyi.
  • Session on Mathematics with Applications in Medicine

    Chair: Dr. Tanya Leise
    Back to Master Schedule

    • 4:30pm
      Using cs-fMRI visualization and General Linear Modeling to detect Cortical hemodynamic activity
      Dayishaa Daga.
      UMass Amherst. Mentor: Maryclare Griffin.
      Abstract: Dr. Maryclare Griffin and I are aiming to conduct data analyses through the General Linear Model based on neuro-imaging in order to see how different kinds of task related stimuli elicit different changes in the chemical composition of the brain. There has been research conducted on this subject using functional MRIs (fMRIs), but because fMRIs use 3D imaging, they do not take into account the spatial distribution and the folds in the anatomical structure of the cortex. Thus, to resolve these issues, we will be using 2D cortical surface fMRI imaging with spatial dependence taken into account to better represent the structure of the cortex and to observe the changes in hemodynamic activity in different regions of the brain in response to an external stimulus. For statistical modeling, we will be using the General Linear Model to analyze the variability in brain activity throughout the cortex.The overarching purpose of these data analyses is to see whether there are specific regions within the cortex that are uniquely responsible for or affected by certain stimuli.
    • 4:50pm
      Using non-linear ODEs to model the COVID-19 Pandemic
      Parikshita Gya (Computational Help: Michael Barmann).
      UMass Amherst. Mentors: Panayotis Kevrekidis, Qian-Yong Chen.
    • 5:10pm
      CIRCADA-I: App for Circadian Data Analysis
      Author: Cassandra Jin, Xinyu Ni.
      Amherst College. Mentor: Tanya Leise.
  • Session 2 on Topology/Geometry

    Chair: Dr. Ivan Contreras
    Back to Master Schedule

    • 4:30pm
      Cylinder Configurations on Translation Surfaces
      Juliet Aygun, Janet Barkdoll, Jenavie Lorman, Theodore Sandstrom.
      Yale University. Mentor: Aaron Calderon.
    • 4:50pm
      Quasi-Isometry in Geometric Group Theory
      Darien Farnham.
      Tufts University. Mentors: Lorenzo Ruffoni, Genevieve Walsh, Kim Ruane, Mackenzie McPike
      Abstract: Investigating the basic notions of quasi-isometry and how it relates to geometric group theory.
    • 5:10pm
      Spinors and Graph Theory
      Beata Casiday, Thomas Meyer, Sabrina Mi, Ethan Spingarn
      Yale University. Mentor: Ivan Contreras.
      Abstract: The evolution of quantum mechanical systems can be described by the Schrödinger equation, and the Dirac equation for spinors (particles with spin). In this talk, we study the behavior of spinors on a graph, as predicted by the Dirac equations, using analytic and algebraic methods. Analytically, we investigate the evolution of quantum states using graph-theoretic versions of the Schrödinger and Dirac equations, and we study discrete models of the Laplacian and Dirac operators. Algebraically, we study spinors as linear representations of the spin group using graph Clifford algebras. We also give gluing results on the centers of graph Clifford algebras.
    • 5:30pm
      The Generalized Pitman-Stanley Polytope and Flow Polytopes
      Maura Hegarty.
      UMass Amherst. Mentors: Alejandro Morales, William Dugan.
      Abstract: In 1999, Pitman and Stanley introduced the Pitman-Stanley polytope along with its volume formula and lattice point formula. The Pitman-Stanley polytope is well studied due to its connections to probability, parking functions, the permutahedra, and flow polytopes. Its lattice points correspond to plane partitions with entries 0 and 1. Pitman and Stanley remarked that their polytope can be generalized so that lattice points correspond to plane partitions with entries $0,1, . . . , m$. This generalization has since been untouched. We study this generalization, its lattice points, and its volume. We also realize the generalized Pitman-Stanley Polytope as a flow polytope which opens the door to new tools for studying this polytope.

Please send any questions to daniel.mourad@uconn.edu.