Projects Summer 2023

Cool picture!




Buffon Needle Problem

Supervisors: Matthew Dannenberg and Alex Iosevich

Prerequisites: Honors multi-variable calculus and strong python programming skills

Participants: William Hagerstrom, Gabriel Hart, Tran Duy Anh Le, Isaac Li, Nathan Skerett

Description: This problem is related to the material that will be touched upon in probability and geometry mini-courses. Consider a bounded convex set K in the plane of fixed perimeter. Drop a needle of length r such that one end is inside K with equal probability and the orientation of the needle is random. Let p(K,r) denote the probability that the other end of the needle also lands inside K. Our goal is to prove that if r is small enough, then among all the convex sets of a given perimeter, p(K,r) is maximized if K is a disk. Furthermore, we will conduct computer experiments to estimate the value of r for which the disk maximizes the Buffon probability. Chat GPT will be used as an auxiliary tool.

Suggested steps to understand this problem:

i) Compute p(K,r) when K is a disk of radius 1.

ii) Compute p(K,r) when K is a square of the same perimeter as the disk of radius 1.

iii) Compute p(K,r) when K is an ellipse of the same perimeter as the disk of radius 1. This calculation may need to be done numerically.

iv) Write python code that estimates p(K,r) with the input of K accomplished in some reasonable way.

v) Try to conceptualize the optimization of p(K,r) as a reinforcement learning problem.

vi) Look up and understand the Fourier series proof (due to Hurewicz) of the iso-perimetric inequality in the plane.


Systems of nonlinear equations: reinforcement learning versus classical methods

Supervisors: Alex Iosevich and Azita Mayeli

Prerequisites: Multi-variable calculus (not necessarily honors) and strong python programming skills

Participants: Xiaolu Chen, Joshua Iosevich, Stephanie Wang, Tae Ho Yoo

Description: This problem is related to the material from the geometry mini-course. There is a variety of classical methods for estimating solutions of systems of non-linear equations. For example, in this case of a single equation, the Newton method taught in most first calculus sequences is an example of such an approach. For systems of equations, the Gauss-Newton method is extremely effective. When the functions involved in a nonlinear system are very rough, or discrete, which could mean a variety of different things, classical methods tend to break down and other approached are needed. One possible approach is reinforcement learning.

An example of a situation where reinforcement learning and classical methods co-exist is the study of shortest paths in graphs. The Dykstra algorithms yields the shortest path, but in many situations, reinforcement learning based algorithms do better, and it would be interesting to understand this phenomenon systematically.

In this project, we are going to create a reinforcement learning code for a variety of situation, including approximating solutions to systems of nonlinear equations, grid search, graph algorithms, and others, and compare the results to the classical techniques in cases of overlap. Chat GPT will be used as an auxiliary tool.

There is plenty of room for theoretical work on this problem as well.

Suggested steps to understand this problem:

i) Remind yourself exactly how the Newton method works.

ii) Look up and understand related methods for estimating solutions of systems of equations.

iii) Write your own python code pertaining to i) and ii).

iv) Read up on reinforcement learning. An excellent source is a book, entitled "Reinforcement Learning" by Sutton and Barto.

v) Write reinforcement learning code to estimate solutions of systems of nonlinear equations.

vi) Find a reasonable way to compare the performance of your code in parts v) and iii).


Geometric combinatorcs and complexity:

Supervisors: Alex Iosevich and Azita Mayeli

Prerequisites: Strong Python programming skills

Participants: Joshua Iosevich, Alhussein Khalil, Anuurag Kumar, Nathan Skerrett

Description: The main thrust of this research group is to investigate two concepts that keep arising in geometric combinatorics. The first is understanding zeroes of the Fourier transform of an indicator function of a set in vector spaces over integers modulo a prime. The second is to understand the complexity of graphs that arise in the study of finite point configurations in vector spaces over finite fields.



Suggested steps to understand this problem:

i) Read the paper where the triangle result is proved. It can be found on arxiv.org under the designation arXiv: 1311.4788. If you run into questions, which is almost inevitable, please contact me immediately and I will point you in the right direction.

ii) Write a code that encodes triangles in a two-dimensional vector spaces over integers modulo p, where p is a prime, as a python class. Within that class, write a function that determines whether two given triangles (with vertices as input) are or not are congruent. Note that two triangles are congruent if one can map the vertices of one onto the vertices of the other using a translation and a rotation. What a translation is in this context is clear. For the right notion of a rotation, read the paper mentioned above.

iii) Consider the following theoretical exploration. One can find an infinite sequences of primes p such that the two dimensional vector space over the integer modulo p does not contain an equilateral triangle. By the way, what an equilateral triangle means in this context is in itself an interesting question. By following the same idea, you can construct a prime p that excludes other congruence classes of triangles. An interesting question is just how many congruence classes (relative to p) can we exclude by imposing conditions on the prime p?


Modeling seizures using machine learning:

Supervisors: Svetlana Pack and Yujia Zhai

Prerequisites: Good knowledge of statistics and strong Python programming skills.

Participants: Andrew Isaacson, Ajax Benander, Xiangyi Chen, Wentao Jiang, Jenifer Kim, Svetlana Pack, Yining Qian, Scott Sun, Gus Vietze, Matthew Xie, Jonathan Zhang

Description: It is widely believed in the medical community that epileptic seizures do not follow a particular daily or weekly time patterns. We believe that the techniques of modern data science have not yet been deployed in the study of this problem in a systematic way. We are going to experiment with a variety of techniques, including reinforcement learning, to look for patterns in the commercially available data sets. Chat GPT will be used as an auxiliary tool.

Suggested steps to understand this problem:

i) Learn how to use TensorFlow or similar to make time series forecasts.

ii) Read up on what it means for a time series to be forecastable.


Time series and retail analytics:

Supervisors: Alex Iosevich, Azita Mayeli and Evan Witz

Prerequisites: Good knowledge of statistics and strong Python programming skills.

Participants: Moez Boussarsar, Colin Hascup, Jennifer Kim, Jiaming Lyu, Peter MacNeil, Cooper Orio, Jiamu Tang, Yining Qian

 Description: We are going to experiment with a variety of methods of determining whether a time series can be accurately predicted. We are also going set up retail forecasting models using a variety of regressors.

Suggested steps to understand the problem: You should study the initial notebook for the Python sessions this week carefully. It contains a variety of forecasting mechanisms that will serve as the the basis for our analysis.


Fractal structures in large data sets:

Supervisors: Alex Iosevich and Azita Mayelia

Prerequisites: Good knowledge of analysis at the level of Rudin's book and strong python programming skills

Participants: Yuesong Huang, Xinyi Liu, Peter MacNeil, Svetlana Pack, Zhelin Sheng, Nathan Skerrett,
 
Description: This is a followup on one of the projects from Tripods 2021 which resulted in a research paper which can be found at https://arxiv.org/pdf/2209.12079.pdf. The paper describes an approach to detecting fractal structures in large data sets using discrete energy. In this project we are going to develop this theory further and create synthetic data sets to test the results. One of the applied avenues we may pursue is the idea of using the Chaitin-Komogorov-Solomonoff complexity of a time series to determined the degrrr yo which it can be accurately forecasted. Chat GPT will be used as an auxiliary tool.

Suggested steps to understand this problem:

i) Read arxiv2209.12079 (the predecessor paper from Tripods 2021) and start asking questions about anything that is not clear.

ii) At the same time, read up on basic properties of the Minkowski and Hausdorff dimension.

iii) Try to duplicate the codes used to produce the diagrams in the paper above.


VC dimension and applications:

Supervisors: Alex Iosevich, Donovan Snyder and Emmett Wyman

Prerequisites: Linear algebra, real analysis and elementary probability

Participants: Xiaolu Chen, Lily Testa,Tran Duy Anh Le, Scott Sun, Tae Ho Yoo

 Description: Given a collection of subsets in the plane, what is the smallest set which contains an element of each? We use the notion of the VC-dimension to find such witness sets in both the finite and infinite settings. One of the main focus for this project will be the setting of fractal sets, though other settings of interested are welcome! Chat GPT will be used as an auxiliary tool.

Suggested steps to understand this problem:

i) Become familiar with the definition of VC Dimension and the theory of epsilon nets (see the enclosed pdf).

ii) Be able to calculate the VC dimension of common hypothesis classes.

iii) Learn about fractal sets and the Minkowski dimension (example notes are available).


Discrete Neural Nets

Supervisor: Charlotte Aten

Prerequisites: Good programming skills and ability to write rigorous arguments

Participants: Karam Aldahleh, Ajax Benander, Alhussein Khalil, Jennifer Kim, Isaac Li, Lilian Stolberg, TJ Weaver, Kevin Xue

Description: This project is a follow-up of the discrete neural nets project from the 2021 Tripods REU. That project, which is introduced in a video linked below, was finally coded and tested (with promising results) as part of Rachel Dennis's 2023 senior thesis, also linked below. Discrete neural nets are neural nets whose activation functions can only take on a finite set of values. This prevents us from using established techniques like gradient descent but opens up new avenues not available for continuous functions. In this year's project, we will explore how well this technique can be extended from classifying/transforming black-and-white images to arbitrary data types (such as images with color).

Suggested steps to understand this problem:

i) Watch the initial video pitch (https://youtu.be/nr0KbcloYW4) for the discrete neural nets project from the 2021 Tripods REU.


ii) Read Rachel Dennis's 2023 senior thesis (https://www.sas.rochester.edu/mth/undergraduate/honorspaperspdfs/r_dennis23.pdf). Start asking questions about any parts you don't understand.
   
iii) A preprint of a paper will be posted soon, and a cleaned-up version of the code will be available to examine, as well. Take a look at these and start playing around when they appear.
   
Don't worry if some of the category theory or universal/abstract algebra language is a bit intimidating. All the necessary algebra will be introduced during the abstract algebra mini-course.