Current Projects: 2023-2024

Project 1: Mathematical Modeling of Social Dynamics

Description: The intent of this project is to find a species that contains some sort of social interaction and try and predict movement throughout the system. The student is encouraged to think of possible areas with social dynamics. Possible areas of interest (but not limited to) are: Economics, Epidemiology (disease modeling), 1-2 species biological systems, etc.

Mentees: Arron Bui and Charlie Diaz

Mentor: Jonathan Engle

Project 2: The Theory of Optical Thermodynamics

Description: In this project, we will establish the fundamental principles of the recently developed Theory of Optical Thermodynamics within the realm of discrete nonlinear optics. Our goal is to examine how this theory aligns with its predecessors, including the Wave Turbulence Theory.

Mentees: Ryan Scott and Wonmin Song

Mentor: Savvas Sardelis

Project 3: Fast graph-based algorithms for analyzing networks

Description: We will work on some spectral methods and/or some other methods (like modularity based) for graph problems. We will show some applications like Image Segmentation.

Mentee: Jenny Petrova

Mentor: Yue Shen

Project 4: Exploration of Nonnegative Matrix Factorization and Regularization in Music

Description: The goals of this project would be to understand NMF and regularization, apply the NMF/regularized NMF to some music audio files, and potentially research/develop a graph regularized NMF in Python.

Mentees: Jingwen Zhou and Max Varela Torres

Mentor: Jonathan Valyou

Project 5: A Stroll Through the Algebro-Geometric Landscape

Description: Depending on student background it would be a look at the basics of algebraic geometry in the classical or scheme-theoretic paradigm. If the student already has a background in AG we could look at Intersection Theory. The texts would be either Shafarevich's Basic Algebraic Geometry, Vakil's The Rising Sea or Fulton's Intersection Theory.

Mentee: Ayotuntosimi Loye

Mentor: Franquiz Caraballo Alba

Project 6: Conformal Mapping Visualization

Description: A classical problem of map-making is transforming a spherical shape onto a two dimensional surface since often a faithful representation will be somewhat distorted. One way to enforce some resemblance is to require the angles in the original shape to preserved; these are called conformal mappings. This project will be visualizing a conformal map of the circle to a square.

Mentee: Tyler Goldman

Mentor: Jonathan Schillinger

Project 7: Solving ODE/PDE using artificial Neural Networks

Description: The primary goal of this project is to apply machine learning for solving initial/boundary value problems, and subsequently, compare the performance of these machine learning-based approaches with established techniques like finite difference methods.

Mentee: Connor Brown

Mentor: Shirin Provat

Project 8: Combinatorial game theory

Description: We will explore some modern results in game theory, beginning by reading recent papers concerning the Zeckendorf Game, Infinite Hex, and Othello.

Mentee: Jeremiah Hockaday

Mentor: Nicholas Ossi

Project 9: Applications of Machine Learning in Finance

Description: In recent times, Machine learning has become more prominent in finance industry due to the availability of vast amounts of data and more affordable computing power. Prominent financial institutions, including major banks and financial services companies, have started to incorporate artificial intelligence technologies, notably machine learning (ML), into their operations. Some of the current applications of AI in finance are Algorithmic trading, Risk Management, Fraud detection, Derivative Pricing etc. We will be focusing in one of those topics.

Mentees: Pietro Candiani and Alexander Khan

Mentor: Md Arafatur Rahman

Project 10:

Description:

Mentees: Samuel Vecchio and Sophie Allen

Mentor: Emmanuel Hartman