# REU Sigma Seminar 2019

Seminars convene every Thursday during the REU program at 12:30 pm in MONT 214.

August 1st at 10:30 am: Keith Conrad, The Biggest Known Prime Number.

Abstract: Since there are infinitely many prime numbers, there is no biggest prime. But there is always a biggest known prime, currently over 24 million digits. It belongs to a family of primes called Mersenne primes. The talk will discuss the history of Mersenne primes and how their primality is checked.

July 25: Gianmarco Molino, Heat Kernels and Index Theory.

Abstract: TBA.

July 18: Matthew Badge, Curves, and Parameterizations.

Abstract: I will introduce some ideas in metric geometry. A curve is the image of a continuous map from [0,1] into a metric space. How do we measure the length of a curve? How do we tell whether or not a set of points is a curve of finite length, and if it is, how do we build a map whose image is the curve? If time permits, I will also describe analogues of these questions for higher-dimensional curves, which are open for research.

July 11: Behrang Forghani, Harmonic Functions, and Trees.

Abstract: In this talk, I will introduce the notion of bounded harmonic functions and discuss some of their properties for Markov chains.
D. Blackwell classified the space of bounded harmonic functions for random walks on Z. I will present his elegant proof and other results related to regular trees.

July 4: No Seminar.

June 27: Arthur Parzygnat, Computing the square root of a positive matrix.

Abstract: Given any function f on some domain and a diagonalizable nxn matrix A whose eigenvalues are in the domain of f, we will prove there exists a polynomial p such that p(A)=Pf(D)P^{-1}, where P is a matrix of eigenvectors of A and D is the corresponding matrix of eigenvalues. This may sound strange when you look at examples. For instance, given a (positive) matrix A, we will find a polynomial p such that p(A)=sqrt(A), the square-root of A. Along the way of proving this theorem, we will learn several useful techniques and results. For example, we will prove that if B is a matrix that commutes with A, i.e. AB=BA, then f(A)B=Bf(A) for any function f (satisfying the above assumptions).

June 20: Reed Solomon, Graduate School Panel.

Abstract: Want to learn about graduate school in mathematics and the application process. Ask Professor Reed Solomon (Director of Graduate Studies) and Ph.D. students Waseet Kazmi, Gianmarco Molino, Sean Eustace, and Lisa Naples!

June 13: Christopher Hayes, Introduction to self-similar structures.

Abstract: One category of a fractal is the self-similar set. Famous examples include the Sierpinski Carpet and Koch Curve and analysis of these types of fractals is currently an active field of research. In this talk we will go over a variety of examples, discuss some construction methods, definitions, commonly used properties such as the open set condition and having a finite post-critical set, and end with comments on how analysis can be done on these sets, in particular, the construction of a Dirichlet form on a post-critically finite set.

# UCONN Math REU at REU Conference at UMass Amherst 2019

Our REU participants will be giving talks on July 23rd, 2019 at the REU Conference at UMass Amherst 2019.

Fractals projects:

• Resistance scaling on the Octacarpet  (Claire Canner, William Huang, Michael Orwin)
• Decimation structure of the spectra of self-similar groups (Brett Hungar, Madison Phelps, Johnathan Wheeler)
• Can we hear the shape of a fractal? Spectral analysis of self-similar sets (Elizabeth Melville, Nikhil Nagabandi)

Stochastic and Financial Mathematics project:

• Hedging by Sequential Regression in Generalized Discrete Models and the Follmer-Schweizer
decomposition (Sarah Boese, Tracy Cui, Sam Johnston)

Markov Chains projects:

• A Version of the Elephant Random Walk and Additive Functionals of Finite State Markov Chains (Jonah Green, Taylor Meredith, Rachel Tan)
• The Voter Model on Complete Bipartite Graphs ( Philip Speegle, R Oliver Vandenberg)
• On a Nonlinear Random Walk on Graphs (Jonah Botvinick-Greenhouse, Connor Fitch, Mark Kong)

Additional information and abstract list: REU Conference at UMass Amherst 2019

# Math UConn REU at JMM 2018

Two of our REU (2017 Stochastics) participants, Raji Majumdar and Anthony Sisti, will be presenting posters Applications of Multiplicative LLN and CLT for Random Matrices and Black Scholes using the Central Limit Theorem on Friday, January 12 at the MAA Student Poster Session, and both of them will be giving talks on Saturday, January 13 at the AMS Contributed Paper Session on Research in Applied Mathematics by Undergraduate and Post-Baccalaureate Students.

Their travel to the 2018 JMM has been made possible with the support of the MAA and UConn’s OUR travel grants.

# 2017 Announcements

## Phanuel Mariano – The volume of the unit ball in n dimensions

July 28, 2017

Phanuel Mariano from the University of Connecticut will be giving a talk computing the volume of the unit ball in arbitrary dimension.

## Michelle Rabideau – Continued Fractions and the Fibonacci Sequence

July 21, 2017

Michelle Rabideau from the University of Connecticut will be giving a talk related to Continued Fractions.

## Hugo Panzo – Laplace’s method and applications to probability

July 14, 2017

Hugo Panzo from the University of Connecticut will be giving a talk related to Laplace’s method.

## Patricia Alonso Ruiz – Resistance metric – an electric interpretation of measuring distances

July 7, 2017

Any weighted graph can be seen as an electric linear network where the current flows between nodes (vertices) connected by resistors (weighted edges). This electric interpretation provides a special way to measure distances in a graph via the so-called effective resistance metric. What does this metric actually do, how it is related to energy minimizers and why it is so helpful when graphs become infinite are some of the questions we will address in this talk.

## Keith Conrad – An algebraic characterization of differentiation

July 30, 2017

The derivative is defined using limits while the basic rules of differentiation (sum rule, product rule, chain rule) have an algebraic flavor. We will see how differentiation, and more generally differential operators, can be characterized purely algebraically by putting all the analytic conditions into the functions that we want to differentiate.

## Ambar Sengupta – Random Matrices: Pictures From Traces and Products

July 23, 2017

Professor Ambar Sengupta will give a talk based on Random Matrices.

## Zihui Zhao – Harmonic Measure

June 16, 2017

Zihui Zhao form the University of Washington will be an introductory talk in harmonic measure.

## Luke Rogers – Length and volume

June 9, 2017

I will talk a little about Euclidean length and volume, the lengths of curves, Peano curves, the positive area curves of Osgood, and a delightful theorem of Hajlasz and Strzelecki that shows one can measure volume with a string.

## Daniel Kelleher – The metric space of metric spaces

June 2, 2017

Metric spaces are sets which have a notion of distance. We will compare two different metric spaces, and see that this comparison makes the set of metric spaces into a metric space (Don’t worry, after that it’s turtles the rest of the way down). The focus will be put on length spaces — metric spaces where distance is given as the length of the shortest curve connecting two points. For these spaces we discover a sense of curvature

# Archived 2015 Announcements

## Alexander Teplyaev – The Spectral Dimension of the Universe

May 29, 2015

Professor Alexander Teplyaev will explain some ideas behind the notion of spectral dimension and how they are related to research being done in our department.

## Masha Gordina – Random thoughts on Brownian motion

June 5, 2015

Professor Masha Gordina will talk about the fascinating history of the Brownian motion and its applications in the real world.

## Keith Conrad – Continued Fractions

June 12, 2015

Professor Keith Conrad will talk about continued fractions, how to compute them, some of their properties, and how to answer seemingly unanswerable questions like this: if an unknown fraction is roughly 2.32558, what is it? (The answer is not 232558/100000.)

## Thomas Laetsch – From Brownian motion cometh

June 19, 2015

Following Dr. Gordina’s talk developing Brownian motion, Thomas Laetsch will take us on a short drunkard’s walk through several theories stemming from or related to Brownian motion. R(E)U ready?

## Joe Chen – Drunkard, Octopus, and Electrical Networks

June 26, 2015

Joe Chen  will summarize the main ideas behind electrical networks and describe two unexpected applications to probability.

# Archived 2013 Announcements

## Brian Hartmann

July 25, 2013

Professor Brian Hartmann of the University of Connecticut will speak.

## Graduate School Panel

July 19, 2013

There will be a panel discussion on the process of applying to grad school, life as a graduate student and beyond.

## Johanna Franklin

July 12, 2013

Professor Johanna Franklin of the University of Connecticut will speak.

## Aaron Bailey – π, e and Other Irrational Numbers

July 9, 2013

Professor Aaron Bailey will give a talk based on the epinonimous 1986 paper of Alan Parks.

## Chen-Yun Lin – Calculus of Variations

June 27 & 28, 2013

Professor Chen-Yun Lin will give a two part discussion of variational methods in optimization problems.

## Daniel Kelleher – Infinite Dimensional Linear Algebra

June 21 & 25, 2013

Daniel Kelleher of the University of Connecticut will give a two part discussion of the complications that arise in concepts of linear algebra in infinite dimensional vector spaces.

## Reed Solomon – The Continuum Hypothesis

June 18, 2013

Professor Reed Solomon of the University of Connecticut will give a talk.

## Rebecca Tramel – Projective Space and Bezout’s Theorem

June 14, 2013

Rebecca Tramel of the University of Edinburgh will give a presentation about curves in the projective plane, and Bezout’s theorem concerning their intersection.

## Keith Conrad – The ABC Conjecture

June 11, 2013

Professor Keith Conrad of the University of Connecticut will give a presentation on the current state of the ABC conjecture.

## Introduction to Mathematica

June 7, 2013

Alex Baldenko will be conducting an introduction to the computer algebra system Mathematica.

## Introduction to Latex

June 5, 2013

Gabe Feinberg will be conducting an introduction to the typesetting program LaTeX

# Archived 2012 Announcements

## Find a graduate program

The American Mathematical Society has a new website “Find a graduate program” which is a good central source for basic statistics on graduate programs in mathematics and related fields. This is a useful source when applying to graduate schools.

## 2012 Project page

The List of projects from the summer 2012 is live.

## Summer Research Symposium at Mount Holyoke

August 2, 2012

At 2:30pm Jed Chou and Ben Whitney of our REU will present their research at Mount Holyoke College in South Hadley, MA.

## UConn Summer Research Symposium

August 2, 2012

At 1pm REU students will participate in a poster session at UConn with the Chemistry REU where they will share their results from the summer.

## Young Mathematicians Conference at Ohio State

July 27 – July 29, 2012

Three of the four groups will travel to Ohio State to give talks or present posters at this year’s Young Mathematicians Conference!

## Sigma: “Arrow’s Theorem” by Lucas David-Roesler

July 20, 2012

Lucas David-Roesler will talk about Arrow’s Theorem.

## Sigma: “Brownian Motion and Levy Processes” by John Haga

July 13, 2012

John Haga will talk about Brownian motion and Levy processes.

## Sigma: “The Modular Group and the Upper Half Plane” by Harris Daniels

July 6, 2012

Harris Daniels will talk about the Modular Group and the Upper Half Plane.

## Sigma: “Marvels in Mathematica” by Alex Baldenko

June 29, 2012

Alex will talk about the Mathematica.

## Education Group Survey

July 6, 2012

At 12:15pm the Math Education group will be administering a survey as part of their project before the regularly scheduled SIGMA seminar.

## 4th of July Picnic!

July 3, 2012

At around 3:30pm we will meet in MSB118 and then drive to Mansfield Hollow for a holiday barbecue!

## Sigma: “Fractal Dimension of the Universe” by Alexander Teplyaev

June 22, 2012

Alexander Teplyaev, of the University of Connecticut, will talk about the Fractal dimension of the Universe.

## Sigma: “Grassmannians and Projective Space” by Rebecca Tramel

June 14, 2012

Rebecca will talk about Grassmannians and Projective space.

## Sigma: “The Projective Plane” by Keith Conrad

June 8, 2012

Keith Conrad, of the University of Connecticut, will talk about the Projective Plane.

## Sigma: Spectral Graph Theory

June 1, 2012

Dan Kelleher will be giving a seminar on Spectral Graph Theory.

## Introduction to LaTeX

The typesetting language LaTeX. On Thursday, Gabe will be giving a short talk about the basics on Thursday at 10. The files to load LaTeX can be found here.

## Introduction to Mathematica

Most of the projects this summer will make use of the CAS Mathematica. Wednesday morning at 10, Alex will be holding an introduction to mathamtica. Here is the notebook from the presentation.

## Welcome to Storrs!!!

The program has begun!

# New Site Launched

We are proud to announce the launch of our new website. Built on the the university Aurora service, this new version of our site sports a modern look, faster loading times, and works on all mobile and tablet devices.