Session A1
Speaker: Georgia Harbor-Collins (UConn)
(9-9:30 am MONT 111)
Commuting Polynomials
Abstract: For polynomials f(x) and g(x) we do not usually expect f(g(x)) = g(f(x)), but it does happen sometimes! In this talk we will see there are only two interesting ways polynomials can commute, and one way has an unexpected relationship to trigonometry.
Session A2:
Speaker: Heidi Benham (UConn)
(9:40-10:10 am MONT 111)
The Problem with Problems
Abstract: Every day, we are surrounded by problems. So, what is a problem? We can think of a problem as anything that occurs which requires a solution. Mathematically, we can think of a problem as a collection of instances, each of which have a collection of corresponding solutions. The instances are some type of mathematical object that may have certain properties and the solutions are some (possibly different) type of object that relates to a given instance in a prescribed way. Many theorems can be restated as problems. For example, the Intermediate Value Theorem can be viewed as a problem where an instance is a continuous function f on an interval [a, b] along with a y between f(a) and f(b) and a solution is an x value between a and b such that f(x) = y. What if we know how to solve one problem? Could this ever help us solve any other problems? The answer is, sometimes, yes. Computable reducibility and Weihrauch reducibility are two ways of formalizing this notion that incorporate ideas from computability theory.
I’m Having a Hamburger Moment
Abstract: The “moments” of a function f(x) are a sequence of numbers defined as the integral of powers of x times the function. More than just a calculus exercise, moments are fundamental in a variety of applications from physics to statistics. In mathematics we study what we call “moment problems” in which we ask: To what extent, if at all, does the sequence of moments determine the function? In other words, if you are given only a sequence of moments, can you figure out what function they came from? In this talk we will try to answer this question (*spoiler alert* the answer is: it depends) and discuss some related problems as well as one application in image processing.
Session B1:
Speaker: Jasper Toman-Yih (UConn)
(9-9:30 am MONT 112)
The Life of a Ph.D. Candidate in Statistics: Looking Back and Looking Forward
Abstract: The goal of this talk is to inspire the undergraduate students to pursue the advanced degree by sharing with them my past academic and working experiences and current academic and research life as a Ph.D. candidate in statistics at UConn. This talk consists of three sections. The first section is about my past academic and working experience -- the career of the machine learning track. The second section talks about the Ph.D. life from Jan. 2022 until present. The third section discusses my current research topic, that is leveraging the historical information via power prior in the Bayesian design of clinical trials. The concept of power prior, clinical trial and the Bayesian design will be demonstrated. Finally, we show the advantages by doing so with an example of a clinical trial.
Session B2:
Speaker: Zhe Guan (UConn)
(9:40 - 10:10 am MONT 112)
Session A3:
Speaker: Herbert Siewert (Fairfield)
(1:10-1:40 pm MONT 111)
Combinatorial game theory within The Game of the Amazons
Abstract: Combinatorial Game Theory is a branch of mathematics that has only recently garnered popularity as a branch of inquiry after the seminal publications "On Numbers and Games" by John Conway and "Winning Ways for Your Mathematical Plays" by Elwyn Berlekamp, John Conway, and Richard Guy. These publications demonstrated that games between two players with no chance elements and no hidden information, including games similar to Go, Chess, and Hex, contain a rich mathematical structure which has been further expanded upon and explored in recent decades. This talk primarily focuses on exploring the foundation of this theory and then applying the theory to The Game of the Amazons by demonstrating optimal strategies within given positions.
Session A4:
Speakers: Sanjana Paul and Amy Pinargote (Smith)
(1:50-2:20 pm MONT 111)
Number Theoretic and Combinatorial Properties of Increasing Sequences of Positive Integers
Abstract: This talk focuses on increasing sequences of positive integers, in particular Complementary Sequences, which are pairs of sequences whose union gives the positive integers and which have no elements in common. Complementary sequences are related to combinatorial games, which are two-player, alternating, deterministic games with perfect information. The objective of the talk is to present some results about complementary sequences, their generation methods and their relationship with combinatorial games, as well as to present some of the research our group has been doing in this area.
Knots and Cluster Algebras: a Link Between Worlds
Abstract: Knots and links are fascinating objects with many widely studied algebraic, combinatorial, geometric, and topological properties. One important question concerning knots is that of finding invariants, which are statistics associated with knots that don't depend on how they are embedded in three-dimensional space. These are useful because they can help us tell certain knots apart; if an invariant takes one value on some knot and a different value on another, then those two knots must be fundamentally different. This talk will focus on two famous invariants that come in the form of polynomials: the Alexander polynomial and the Jones polynomial. We'll first look at a few traditional ways to describe these polynomial invariants and see some of what they're capable of. Following this, we'll take a detour into the mysterious world of cluster algebras, a relatively young but highly active area of current research with deep connections to Lie theory, Poisson geometry, representation theory, combinatorics, and much more. It turns out that, in addition to providing us with this already-lengthy list of relationships to other branches of math, cluster algebras are also capable of reproducing the Alexander polynomials of knots and links. We'll sketch this "roadmap through the clusterverse" from a knot diagram to its Alexander polynomial, and time permitting, we'll take a peek at a conjectural roadmap toward the Jones polynomial as well. This talk is based on a key result of V. Bazier-Matte and R. Schiffler, as well as the speaker's ongoing joint work with R. Schiffler.
Session B3:
Speaker: Benjamin Grant (UConn)
(1:10-1:40 pm MONT 112)
Stochastic Calculus and its applications in Finance
Abstract: Consider a scenario in the financial market where an investor has some extra ”in- sider” information about the future price of a stock, or another one where the stock market can face sudden “jumps” in price due to various events across the world. As an investor, can we mathematically model these scenarios and find an optimal investment strategy?
Following the work of American economists/mathematicians Fischer Black, Myron Scholes and Robert Merton on option pricing theory, which used random phenomena known as stochastic processes to model stock prices, Mathematical Finance emerged as a prominent discipline. In this talk, we shall discuss the foundations of Stochastic Calculus, which extends the methods of ordinary calculus to stochastic processes. It provides a rich set of tools to model various financial scenarios discussed above and formulate them as mathematical research problems.
Session B4:
Speaker: Rahul Krishnakumar (UConn)
(1:50-2:20 pm MONT 112)
Undergraduate Talk Abstracts:
| Triangular Mosaic Knots by Niko Ramirez (Wesleyan University) | Knot mosaics are knot diagrams constructed from a set of tiles with simple arcs or crossings. This talk will walk through an ongoing research project to develop a triangular variant of mosaic knots that could be used to model knots embedded in thickened surfaces via surface triangulation. Triangular mosaic knots offer a less rigid system for modeling knots without the constraint of tiling regular polygons in the plane. |