**Jason Freitas and Joshua Huang**

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

August 3, 2023

**Jason Freitas and Joshua Huang**

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

May 18, 2023

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.

March 22, 2023

**Group Members:** Tyler Campos, Andrew Gannon, Benjamin Hanzsek-Brill, Connor Marrs, Alexander Neuschotz, Trent Rabe and Ethan Winters.

**Mentors:** Rachel Bailey, Fabrice Baudoin, Masha Gordina

**Overview:** We study and simulate on computers the fractional Gaussian fields and their discretizations on surfaces like the two-dimensional sphere or two-dimensional torus. The study of the maxima of those processes will be done and conjectures formulated concerning limit laws. Particular attention will be paid to log-correlated fields (the so-called Gaussian free field).

“Paths of the Fractional Gaussian Field on S1 and the Torus” by Andrew Gannon (University of Connecticut)

July 6, 2022

**Fair Pricing and Hedging Under Small Perturbations of the Numéraire on a Finite Probability Space**

Involve (2022), Vol. 15(4), pp. 649-668. [published version] [arXiv]

July 9, 2020

accepted in the Missouri Journal of Mathematical Sciences (2023)

August 3, 2019

Sarah Boese, Tracy Cui, Sam Johnston

Gianmarco Molino, Olekisii Mostovyi

In practice, financial models are not exact — as in any field, modeling based on real data introduces some degree of error. However, we must consider the effect error has on the calculations and assumptions we make on the model. In complete markets, optimal hedging strategies can be found for derivative securities; for example, the recursive hedging formula introduced in Steven Shreve’s “Stochastic Calculus for Finance I” gives an exact expression in the binomial asset model, and as a result the unique arbitrage-free price can be computed at any time for any derivative security.

In incomplete markets this cannot be accomplished; one possibility for computing optimal hedging strategies is the method of sequential regression. We considered this in discrete-time; in the (complete) binomial model we showed that the strategy of sequential regression introduced by Follmer and Schweizer is equivalent to Shreve’s recursive hedging formula, and in the (incomplete) trinomial model we both explicitly computed the optimal hedging strategy predicted by the Follmer-Schweizer decomposition and we showed that the strategy is stable under small perturbations.

December 20, 2017

Ayelet Amiran, Skylyn Brock, Ryan Craver, Ugonna Ezeaka, Mary Wishart

Fabrice Baudoin, Berend Coster, Phanuel Mariano

Financial markets have asymmetry of information when it comes to the prices of assets. Some investors have more information about the future prices of assets at some terminal time. However, what is the value of this extra information?

We studied this anticipation in various models of markets in discrete time and found (with proof) the value of this information in general complete and incomplete markets. For special utility functions, which represent a person’s satisfaction, we calculated this information for both binomial (complete) and trinomial (incomplete) models.

Journal reference: | Involve 12 (2019) 883-899 |

DOI: | 10.2140/involve.2019.12.883 |

tba

July 25, 2017

Lowen Peng, Anthony Sisti, Rajeshwari Majumdar

Phanuel Mariano, Masha Gordina, Sasha Teplyaev, Ambar Sengupta, Hugo Panzo

We study the Law of Large Numbers (LLN) and and Central Limit Theorems (CLT) for products of random matrices. The limit of the multiplicative LLN is called the Lyapunov exponent. We perturb the random matrices with a parameter and we look to find the dependence of the the Lyapunov exponent on this parameter. We also study the variance related to the multiplicative CLT. We prove and conjecture asymptotics of various parameter dependent plots.

Raji Majumdar and Anthony Sisti, will present 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 give talks on Saturday, January 13 at the AMS Contributed Paper Session on Research in Applied Mathematics by Undergraduate and Post-Baccalaureate Students.

May 12, 2016

Journal reference: |
Stochastics and Dynamics, Vol. 17, No. 6 (2017) 1750046 |

DOI: |
10.1142/S0219493717500460 |

Lance Ford, Derek Kielty, Rajeshwari Majumdar, Heather McCain, Dylan O’Connell

We investigated systems of complex-valued ordinary differential equations (ODEs) that blows up in finite time, which we refer to as *explosive* systems. The goal is to understand for what initial conditions does the system explode and will the addition of noise stabilize it; that is, if we were to perturb the system with an additive Brownian motion, will the system of stochastic differential equation (SDE) still be explosive? In fact, we were able to prove a toy model of the stochastic Burgers’ equation to be ergodic; that is, the SDE is nonexplosive and it has a unique limiting distribution.

April 21, 2016

David Wierschen and Becky Simonsen

This group considered the stability of matrix Lie group valued stochastic differential equations, dX_{t} = AX_{t}dt + BX_{t}dt. Random dynamical systems such as this arise in many applications (e.g., oceanic turbulence, helicopter blade motion, light in random channels, wireless networks) in which stability is of practical and theoretical concern. The stability of the zero solution, X_{t} = 0, is determined by the top Lyapunov exponent. But in practice, analytic calculations of the Lyapunov exponent are often impossible, so time discrete approximations and simulations are necessary. Oceledet’s famous multiplicative ergodic theorem ensures that the Lyapunov exponent of X_{t} is almost surely constant. But the Lyapunov exponent of a time discrete approximation is itself a random variable. The mean of this random variable has been studied. The group provided estimates on the variance, distribution and rate of convergence in certain numerical approximation methods. In addition, they expanded on and provided simulations for recent results regarding the top Lyapunov exponent of certain Lie group valued SDEs.