Author: Rogers, Luke

REU Seminars 2021

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July 23rd: Masha Gordina, University of Connecticut

Title: Emmy Noether, Symmetry and the Calculus of Variations

July 16th: Kasso Okoudjou, Tufts University

Title: TBA

Abstract:

July 9th: Sasha Teplyaev, University of Connecticut

Title: Spectral Dimension of the Universe

Abstract:
We know that our usual space-time is four dimensional, but is it the same on a quantum scale? There is a recent physics theory that there is a so called dynamical dimensional reduction in quantum gravity. This talk will describe the basic mechanism for this dimension reduction, and a surprising connection to the talk by Chris Hayes.

July 2nd: Patricia Alonso-Ruiz, Texas A&M

Title: Fractals are also in the market

Abstract: How does a financial investor decide which trading strategy to apply? Analyzing and classifying financial data is crucial to make a decision. One general classification involves two types of price behaviors: trending or mean-reverting. In the former, increasing (or decreasing) returns will likely lead to further increasing (or decreasing) returns, while in the latter price increases will likely lead to price decreases (or vice versa).
In this talk we will discuss a way to mathematically describe these regimes and discover that fractals play a fundamental role in that description.

June 25th: Chris Hayes, University of Connecticut

Title: Fractals and Dirichlet forms

Abstract: We will explore some simple examples of fractals, discuss dimensions of self-similar sets and then look at the basic ingredient in doing analysis (calculus) on some fractals: the construction of a self-similar Dirichlet form.

June 18th: Emily Gunawan, University of Oklahoma

Title: Box-ball systems and Tableaux

Abstract: Box-ball systems give a combinatorial model of a soliton, or solitary wave. We discuss some results obtained by past REU students in this area.

June 11th: Oleksii Mostovyi, University of Connecticut

Title: Introduction to financial mathematics

Abstract: We introduce some foundational ideas of financial mathematics, including the first and second fundamental theorems of asset pricing.

June 4th: Luke Rogers, University of Connecticut

Title: Measurement

Abstract: Since its earliest days, mathematics has been used to measure things. Usually so that someone could tax them! But measuring even simple things like the length of a curve in the plane can present many challenges. We will walk through some of the difficulties one might encounter and learn a little about the mathematics that arises from them.

Resistance scaling on affine carpets

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Group Members

Samantha Forshay, Leng Mawi, Matthew Peeks.

Supervisors

Chris Hayes, Luke Rogers, Sasha Teplyaev.

Overview

A celebrated result in analysis and probability on fractals is the construction of a diffusion on the standard Sierpinski Carpet by Barlow and Bass. One key part of their argument is a pair of upper and lower estimates for the resistances of precarpets: if \(K_n\) denotes the level \(n\) approximation of the carpet and \(E_n\) is the minimal Dirichlet energy of a function that is identically 1 on one side of the carpet and identically 0 on the other side, then there are constants \(0<c\leq C< \infty\) so that \(c\rho^n\leq E_n\leq C\rho^n\). Estimates for \(\rho\) are known but the exact value is not.

The Sierpinski-type carpets to which the preceding estimates of resistance have been extended are all self-similar. By contrast, in the setting of post-critically finite fractals, resistance scaling has been successfully studied also in the self-affine case, initially by Fitzsimmons, Hambly and Kumagai.

The goal of this project was to investigate what aspects of the Barlow-Bass approach to resistance estimation on carpets could be extended to the self-affine case, and to make numerical computations of the behavior of resistance in this setting and its dependence on the affine scalings.

Publication

Poster

REU Seminars 2020

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July 17th: Luke Rogers, UConn

Can I measure it with mathematics?

July 10th: Emily Meehan, Gallaudet University

Title: Baxter posets.

July 3rd: Daniel Labardini-Fragoso, UNAM

Title: Introduction to Hyperbolic Geometry

June 26th: Masha Gordina, University of Connecticut

Title: Isoperimteric inequalities

Abstract: The classical isoperimetric problem asks: among all figures with a given perimeter, which one encloses the greatest area? There are many different proofs of the isoperimetric inequality and interesting stories about the authors of these proofs. We will review different tools that can be used to prove such an inequality (plane geometry, calculus of variations and maybe even Fourier series), and if time permits we’ll describe how this type of problem is still an active area of research in analysis and geometry.

June 19th: Ivan Contreras, Amherst College

Dynamics and Topology on Graphs.

June 12th: Nadia Lafrenière, Dartmouth University

Derangements and their use in research (Algebraic Combinatorics)

June 5th: Greg Muller, University of Oklahoma

Title: Counting paths with linear algebra

Abstract: Counting paths in a graph is an elementary but important problem with many applications. Something remarkable happens if you assemble these counts into a matrix: the determinants of submatrices count certain collections of paths. This simple observation has a simple proof, but wide-ranging applications. Time permitting, I will review some applications to Pascal’s triangle, matrix factorizations, total positivity, and electrical networks.