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.

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.

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

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

please submit your application online at MathPrograms.org

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.