Summer Research Program 2024



Organizers: Alex Iosevich and Azita Mayeli

Dates: July 29 - August 9, 2024

Structure of the program: The program is going to consist of supervised research projects and series of lectures designed to help gain the necessary background as you are working on your projects. The exact topics of the lecture series will be determined in the coming weeks. The exact topics for the research projects will be selected based on your interests and preferences.

This program is partly a culmination of undergraduate research activities that have transpired during the academic year. Once the program is over, the research projects are likely to continue into the Fall 2024 semester and beyond.

Application process: Please fill out an on-line application form at the following link. The students at all the Rochester area colleges and universities are welcome to apply.


Projects:

 
Exact signal recovery

Project supervisors: Alex Iosevich and Azita Mayeli

1) Project description: Suppose that a signal of length N is transmitted via its discrete Fourier transform and some of the signal is lost in the transmission due to noise or interference. Under what conditions is it possible to recover the original signal exactly? This innocent looking question quickly leads to some interesting techniques and ideas from analysis, combinatorics and other areas of mathematics. We are going to investigate these types of questions from both the theoretical and computational points of view.

Project participants: Karam Aldahleh (kaldahle@u.rochester.edu), Gabe Hart (ghart3@u.rochester.edu), Alhussein Khalil (akhalil3@u.rochester.edu), Aiden Rohrbach (arohrbac@u.rochester.edu), Terrence Wong (twong15@u.rochester.edu), 

2) Buffon Needle Problem

Project supervisors: Alex Iosevich and Matthew Dannenberg

Project description: This project is continuing from last summer. The question is, which convex domain K in d-dimensional Euclidean space with a boundary with a fixed (d-1)-dimensional Hausorff content maximizes the Buffon probability? Here the Buffon probability p(K,r) is the probability that if one end of a needle of length r lands in K with uniform probability, then the other end also lands in K. Last summer, William Hagerstrom, Gabriel Hart, Tran Duy Anh Le, Isaac Li, and Nathan Skerett essentially resolved this question in two dimensions. They proved that given any convex set K in the plane where the length of the boundary is equal to 2 pi, there exists a threshold r_0 such that if r<r_0 and K is not the unit disk D, then p(K,r)<p(D,r). The purpose of this year's project is to extend this result to higher dimensions.

Project participants:

3) Automated theorem proving

Project supervisors: Alex Iosevich, Azita Mayeli and Stephanie Wang

Project description: The purpose of this project is to independently develop some automated proof methods. After an introduction to the subject matter based on materials provided by Professor Yifan Zou, we are going to write design and write some automated theorem proving code. More details will be added in the coming weeks.

Project participants:

4) Kolmogorov complexity and Hausdorff dimension

Project supervisors: Alex Iosevich, Azita Mayeli and Svetlana Pack

Project description: The purpose of this project is to understand the emerging connections between Kolomogorov complexity and Hausdorff dimension, with applications to configuration problems, complexity of graphs and machine learning.

Project participants:

5) Numerical solutions for partial differential equations

Project supervisors: Alex Iosevich

Project description: The group will use deep learning methods to investigate solutions of various partial differential equations. In particular, they will investigate how to solve the high latitude heat equation using a neural additive model. The group will also work on other SPDE related problems if time permits.

Project participants: Yuning Ren  (yren29@u.rochester.edu), Kunxu Song (ksong12@u.rochester.edu), Xiaoya Tan (xtan14@u.rochester.edu), Jingwen Xu (jxu85@u.rochester.edu)

6) Graph theory and cycle double-covers

Project supervisors: Gabe Hart and Alex Iosevich

Project description: A cycle double-cover of a graph G is a set of cycles in G such that every edge of G is included in exactly 2 of the cycles. The cycle double-cover conjecture states that every bridgeless graph has a cycle double-cover. We will investigate this conjecture and related problems using both theoretical and computational methods.

Project participants:

7) Math education modeling methods

Project supervisors: Anuurag Kumar and Stephanie Wang

Project description: At present, the United States severely underperforms in mathematics relative to its global standing, leading to countless attempts to aid STEM students. Lesser studied, however, is pedagogy at the undergraduate level, and the factors that go into attracting and retaining math major students. In this group, we will explore the variety of factors that contribute to the undergraduate mathematics experience through a first-generation lens, including, but not limited to: allocation of university resources, available mathematical support, community openness, perceived mathematical stigma and career trajectory, etc. Students in this group can expect to do mathematical modeling of demographics and statistical analysis, as well as learn how to apply quantitative measurements to subject data (interviews, surveys, etc.). Coding will be done in Python and/or R, and use of ChatGPT is welcome and even encouraged so far as the programmer understands the code produced.

Project participants:

8) Random walks and finite graphs

Project supervisors: Alex Iosevich and Anuraag Kumar

Project description: We are going to study random walks on graphs using Markov chain methods. The goal is to determine accurate distributions of hitting times.

Project participants:

9) Erdos distance problem on manifolds

Project supervisors: Alex Iosevich and Nathan Skerrett

Project description: The classical Erdos distance problem asks for the smallest possible number of distances determined by n points in Euclidean space in dimensions two and higher. In this project we are going to investigate this problem on Riemannian manifolds.

Project participants:

10) Physics project

Project supervisors: TBA

Project description: TBA

Project participants:

11) Sales modeling with economic indicators

Project supervisors: Alex Iosevich and Azita Mayeli

Project description: We are going to build and test neural network models with economic indicator regressors to effectively predict future sales in retail. A variety of neural network models will be built using tensorflow, keras, facebook prophet and others. Theoretical aspects of this problem will be considered as well.

Project participants:

12) Forecasting medical data using neural networks

Project supervisors: Alex Iosevich, Azita Mayeli and Svetlana Pack

Project description: We are going to work with large swaths of medical data, including EEG, seizures and others, and look for identifiable patterns using neural network analysis and more elementary statistical techniques.

Project participants: