**Jason Freitas and Joshua Huang**

**Mentor: ****Oleksii Mostovyi**

**Jason Freitas and Joshua Huang**

**Mentor: ****Oleksii Mostovyi**

REU participants:

Bobita Atkins, Massachusetts College of Liberal Arts

Ashka Dalal, Rose-Hulman Institute of Technology

Natalie Dinin, California State University, Chico

Jonathan Kerby-White, Indiana University Bloomington

Tess McGuinness, University of Connecticut

Tonya Patricks, University of Central Florida

Genevieve Romanelli, Tufts University

Yiheng Su, Colby College

Mentors: Bernard Akwei, Rachel Bailey, Luke Rogers, Alexander Teplyaev

**Convergence, optimization**** and stabilization**** of singular eigenmaps**

B.Akwei, B.Atkins, R.Bailey, A.Dalal, N.Dinin, J.Kerby-White, T.McGuinness, T.Patricks, L.Rogers, G.Romanelli, Y.Su, A.Teplyaev

Eigenmaps are important in analysis, geometry and machine learning, especially in nonlinear dimension reduction.

Versions of the Laplacian eigenmaps of Belkin and Niyogi are a widely used nonlinear dimension reduction technique in data analysis. Data points in a high dimensional space \(\mathbb{R}^N\) are treated as vertices of a graph, for example by taking edges between points separated by distance at most a threshold \(\epsilon\) or by joining each vertex to its \(k\) nearest neighbors. A small number \(D\) of eigenfunctions of the graph Laplacian are then taken as coordinates for the data, defining an eigenmap to \(\mathbb{R}^D\). This method was motivated by an intuitive argument suggesting that if the original data consisted of \(n\) sufficiently well-distributed points on a nice manifold \(M\) then the eigenmap would preserve geometric features of \(M\).

Several authors have developed rigorous results on the geometric properties of eigenmaps, using a number of different assumptions on the manner in which the points are distributed, as well as hypotheses involving, for example, the smoothness of the manifold and bounds on its curvature. Typically, they use the idea that under smoothness and curvature assumptions one can approximate the Laplace-Beltrami operator of \(M\) by an operator giving the difference of the function value and its average over balls of a sufficiently small size \(\epsilon\), and that this difference operator can be approximated by graph Laplacian operators provided that the \(n\) points are sufficiently well distributed.

In the present work we consider several model situations where eigen-coordinates can be computed analytically as well as numerically, including the intervals with uniform and weighted measures, square, torus, sphere, and the Sierpinski gasket. On these examples we investigate the connections between eigenmaps and orthogonal polynomials, how to determine the optimal value of \(\epsilon\) for a given \(n\) and prescribed point distribution, and the dependence and stability of the method when the choice of Laplacian is varied. These examples are intended to serve as model cases for later research on the corresponding problems for eigenmaps on weighted Riemannian manifolds, possibly with boundary, and on some metric measure spaces, including fractals.

Approximation of the eigenmaps of a Laplace operator depends crucially on the scaling parameter \(\epsilon\). If \(\epsilon\) is too small or too large, then the approximation is inaccurate or completely breaks down. However, an analytic expression for the optimal \(\epsilon\) is out of reach. In our work, we use some explicitly solvable models and Monte Carlo simulations to find the approximately optimal value of \(\epsilon\) that gives, on average, the most accurate approximation of the eigenmaps.

Our study is primarily inspired by the work of Belkin and Niyogi “*Towards a theoretical foundation for Laplacian-based manifold methods*.”

Talk: Laplacian Eigenmaps and Chebyshev Polynomials

Talk: A Numerical Investigation of Laplacian Eigenmaps

Talk: Analysis of Averaging Operators

Intro Text: Graph Laplacains, eigen-coordinates, Chebyshev polynomials, and Robin problems

Intro Text: A Numerical Investigation of Laplacian Eigenmaps

Intro Text: Comparing Laplacian with the Averaging Operator

Poster: Laplacian Eigenmaps and Orthogonal Polynomials

Results are presented at the 2023 Young Mathematicians Conference (YMC) at the Ohio State University, a premier annual conference for undergraduate research in mathematics, and at the 2024 Joint Mathematics Meetings (JMM) in San Francisco, the largest mathematics gathering in the world.

The University of Connecticut’s summer program brings together a small group of undergraduates to explore what it is like to do research in pure and applied mathematics. Over the course of 10 weeks, we follow research projects from beginning to end — starting with reading about the project, writing proofs/programs, performing calculations, and ending with writing up results. In the past, many of our projects have culminated in published articles and conference talks.

**Applications will be accepted until the end of March, but e****arly applications are strongly encouraged.
**

The program runs for 10 weeks, from late May to early August. Summer 2023 dates: 05/29/2023 — 08/05/2023

The REU program is primarily aimed at math and science majors, but your experience and interests are more important than your major. We primarily enroll students who have completed their sophomore or junior year, but sometimes we admit unusually well-qualified freshmen. We have NSF funding to support students who are U.S. citizens or permanent residents. This covers lodging and provides a stipend; the latter has previously been in the range of $4500-5000 for the summer. As this grant is intended to increase the participation of undergraduate students in mathematics research, we are particularly interested in applications from members of underrepresented groups, including women and minorities.

All applications are processed through the NSF REU application website, https://etap.nsf.gov. We will begin reviewing applications starting in early March; applications received after the deadline will be considered on a rolling basis.

Math REU groups from Amherst College, Tufts University, UConn, UMass and Yale University are getting together online on Thursday July 29th for a conference to share their summer research work. Check out the 2021 Math REU conference website

Title: Emmy Noether, Symmetry and the Calculus of Variations

Title: TBA

Abstract:

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.

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.

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.

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.

Title: Introduction to financial mathematics

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

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.

Math REU groups from Amherst College, UConn, UMass, Williams College and Yale University are getting together online on Thursday July 30th for a conference to share their summer research work. The conference website is at

Can I measure it with mathematics?

Title: Baxter posets.

Title: Introduction to Hyperbolic Geometry

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.

Dynamics and Topology on Graphs.

Derangements and their use in research (Algebraic Combinatorics)

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.

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.