Fall 2023

Sep 29 - Forrest Miller (Northeastern)

Title: An Introduction to Mathematical Finance

Abstract: In 1952, Harry Markowitz developed mean variance portfolio management, forming the bedrock of modern and post-modern portfolio theory, as well as establishing finance as a field to be studied scientifically and mathematically. In this talk, we will highlight the main areas of mathematical finance, with a focus on the portfolio management side. We will build the Markowitz portfolio selection problem. Then, we will discuss the efficient frontier and what happens when a risk-free asset joins the risky assets of the market. Then, we will conclude with a discussion on risk measures and a brief introduction to the problem of option pricing.

Oct 13 - Arturo Ortiz San Miguel (Northeastern)

Title: Ballistic Annihilation and (Non)Universality

Abstract: Ballistic annihilation is a stochastic interacting particle system in which infinitely many particles with randomly assigned velocities move across the real line and mutually annihilate upon contact. We introduce a variant with superimposed clusters of multiple stationary particles. Our main finding is that the critical initial stationary particle density to ensure species survival depends on both the mean and variance of the cluster size. Our result contrasts with recent ballistic annihilation universality findings with respect to particle spacings.

Oct 27 - Elena Wang (Michigan State University)

Title: The Directed Merge Tree Distance and its Applications

Abstract: Geometric graphs appear in many real-world datasets, such as embedded neurons, sensor networks, and molecules. We investigate the notion of distance between graphs and present a semi-metric to measure the distance between two geometric graphs via the directional transform combined with the labeled merge tree distance. We introduce a way of rotating the sub-level set to obtain the merge trees and represent the merge trees using a surjective multi-labeling scheme. We then compute the distance between two representative matrices. Our distance is not only reflective of the information from the input graphs but also can be computed in polynomial time. We illustrate its utility by implementation on a Passiflora leaf dataset.

Nov 10 - Erika Beserra (Northeastern)

Title: A Surface Level Introduction to Khovanov Homology

Abstract: A knot is a 1-sphere embedded in 3 dimensional space. Put more simply, it is a loop of string twisted around itself with the ends connected. A central question of knot theory is how to tell knots apart up to ambient isotopy. This leads to the construction of knot invariants. A famous invariant is the Jones polynomial that associates each knot to a polynomial in Z[q, q^{-1}]. Although a fruitful invariant, it neither separates knots nor has it been proven to detect the unknot (the trivial embedding of the 1-sphere). To construct a better invariant, we turn to Khovanov homology, a method of categorifying the Jones Polynomial. The construction categorifies it by replacing the polynomial with a chain complex of graded vector spaces associated to the knot. The chain complex is constructed in such a way that its Euler Characteristic is the Jones Polynomial. The graded dimensions of the homology spaces produce a finer invariant and can detect the unknot.

Nov 27 - Fall break

Dec 1 - Alia Yusaini (Northeastern)

Title: Quantum Probabilities via the Method of Arbitrary Functions

Abstract: Quantum mechanics is inherently a probabilistic theory, wherein the particles it describes are modeled as waves of fluctuating probability densities. In reality, we observe these probabilistic outcomes in our experiments. While not deterministic, the measurement results are always definite. This suggests a physical process through which our probability distributions collapse to single, measurable outcomes. The pertinent question is: How does this physical process occur? This question underlies the measurement problem in quantum mechanics. Using the method of arbitrary functions, we seek to establish that Landsman and Reuver’s flea model might indeed provide a solution to the measurement problem in the double well potential system.