# VCU ANALYSIS, LOGIC AND PHYSICS SEMINAR (ALPS)

During the Fall 2022 -- Spring 2023 term, the ALPS seminar will be held both in-person and online (check schedule)

Online and streamed meetings will be conducted over Zoom. In-person meetings will be held in Harris 4145.

• Meeting ID: 964 9977 5895

• Either 3:00 - 4:00 PM or 9:00 - 10:00 AM on Fridays to accommodate speakers from different time zones.

• The password is "Euclid" (first letter capitalized) followed by the first 4 primes (total of 10 characters).

## Fall 2022 - Spring 2023 Schedule

Organizers: Marco Aldi, Brent Cody, Sean Cox, Alex Misiats, Allison Moore and Ihsan Topaloglu.

2021-2022 Schedule Coordinator: Allison Moore

The goal of the ALPS Seminar is to provide an informal venue where new ideas in all areas of pure and applied mathematics can be shared with VCU students and faculty. Particular emphasis is placed on topics connected to analysis, set theory and mathematical physics.

Everyone, especially graduate and advanced undergraduate students, is welcome to attend our talks. If you would like to contribute by giving a talk, please email your proposed title and abstract to one of the organizers.

## Abstracts

Sean Cox (VCU)

Title: How robustly can you predict the future?

Abstract: The Axiom of Choice has many bizarre consequences, the most famous of which are the Banach-Tarski Paradox and Vitali's discovery of sets of real numbers that are not Lebesgue measurable. Around 2008, Hardin and Taylor discovered another bizarre consequence: that any function on the reals can be predicted at almost every point in time, based solely on its behavior before that point. However, a few years later, Bajpai and Velleman proved that there are limitations on how time-invariant such predictors can possibly be, even with the Axiom of Choice at one's disposal. For example, while there are always such predictors that are invariant with respect to monotonic affine distortions of the time axis, there is no predictor that is invariant with respect to all monotonic $C^\infty$ distortions of the time axis. Naturally, they asked about the large gap between affine functions and $C^\infty$ functions.

I will give an overview of the Hardin-Taylor and Bajpai-Velleman work, and if time permits, briefly highlight some of our recent results on the Bajpai-Velleman question (joint with Cody, Elpers, and Lee).

Ludwig Striet (GMU and University of Freiburg)

Title: Approximation of fractional Operators and fractional PDEs using a sinc-basis

Abstract: We introduce a spectral method to approximate PDEs involving the fractional Laplacian with zero exterior condition. Our approach is based on interpolation by tensor products of sinc-functions, which combine a simple representation in Fourier-space with fast enough decay to suitably approximate the bounded support of solutions to the Dirichlet problem. This yields a numerical complexity of O(NlogN) for the application of the operator to a discretization with N degrees of freedom. Iterative methods can then be employed to solve the fractional partial differential equations with exterior Dirichlet condition. We show a number of example applications and establish rates of convergence that are in line with rates for finite element based approaches.

Brent Cody (VCU)

Title: Sparse Analytic Systems

Erdós proved that the Continuum Hypothesis (CH) is equivalent to the existence of an uncountable family F of (real or complex) analytic functions, such that {f(x) : f ∈ F} is countable for every x. We strengthen Erd ̋os’ result by proving that CH is equivalent to the existence of what we call sparse analytic systems of functions. Recall that Cantor proved that any two countably infinite dense subsets of R are order-isomorphic, and this order- isomorphism extends uniquely to a homeomorphism of R. A key part of constructing a sparse analytic system under CH, involves addressing the question of how nice this homeomorphism can be arranged to be. We use sparse analytic systems to construct, assuming CH, an equivalence relation ∼ on R such that any “analytic anonymous” attempt to predict the map x → [x]must fail almost everywhere. This is joint work with Sean Cox and Kayla Lee.

Peter McGrath (NCSU)

Title: Bending energy minimizers with prescribed genus and isoperimetric ratio.

The Bending (or Willmore) energy of a surface immersed in Euclidean three-space is the integral of the square of the surface's mean curvature and has been studied since the early 1800's---first by Germain and Poisson---in the study of elastic membranes. In the 1970's, Biologist Canham proposed to model red blood cells by surfaces minimizing bending energy with a constrained isoperimetric ratio. I will discuss the recent proof---by R. Kusner and myself---of the existence of a smooth surface with minimum bending energy amongst the class of surfaces of any prescribed genus and isoperimetric ratio.

Antonio De Rosa (University of Maryland)

Title: Regularity of anisotropic minimal surfaces

I will present a $C^{1,alpha}$-regularity theorem for m-dimensional Lipschitz graphs with anisotropic mean curvature bounded in $L^p$, $p > m$, in every dimension and codimension. This is based on a joint work with Riccardo Tione.