Laplacian eigenmaps 2023
May 18, 2023
May 18, 2023
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).
July 9, 2022
University of Wisconsin-Madison, USA
University of Notre Dame, USA
Harvard University, Cambridge, USA
University of Connecticut, Storrs, USA
July 6, 2022
Involve (2022), Vol. 15(4), pp. 649-668. [published version] [arXiv]
July 9, 2020
accepted in the Missouri Journal of Mathematical Sciences (2023)
July 8, 2020
Séminaire Lotharingien de Combinatoire (2021)
Sém. Lothar. Combin. 85B (2021), Art. 14, 12 pp.
Proceedings of the 33rd Conference on Formal Power
Series and Algebraic Combinatorics
Ben Drucker, Eli Garcia, Emily Gunawan, and Rose Silver
A box-ball system is a collection of discrete time states representing a permutation,
on which there is an action called a BBS move. After a finite number of BBS moves
the system decomposes into a collection of soliton states; these are weakly
increasing and invariant under BBS moves. The students proved that when this
collection of soliton states is a Young tableau or coincides with a partition of a type
described by Robinson-Schensted (RS), then it is an RS insertion tableau. They also
studied the number of steps required to reach this state.
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.
August 13, 2018
project page:
July 9, 2018
Cory McCartan, Laura LeGare, Caitlin Davis.
Geodesics (shortest paths) on manifolds such as planes and spheres are well understood. Geodesics on fractal sets such as the Sierpinski Triangle are much more complicated. We begin by constructing algorithms for building shortest paths and provide explicit formulas for computing their lengths. We then turn to the question of interpolation along geodesics—given two subsets of the Sierpinski Triangle, we “slide” points in one set along geodesics to the other set. We construct a measure along the interpolated sets which formalizes a notion of the interpolation of a distribution of mass, and we prove interesting self-similarity relations about this measure.