Mathematical Sciences

► Improving the Gaussian Correlation Inequality
Isaac Browne

Advisor:  Tomasz Tkocz
Abstract:  The Gaussian Correlation Inequality is the statement that the standard Gaussian measure of the intersection of two convex symmetric sets is at least the product of their measures. This project is devoted to extending and improving this inequality. In particular, we show several cases of a conjectured strengthening involving a Minkowski sum, with the unconditional case extending to product measures with arbitrary nonincreasing densities. In addition, we provide a counterexample to another correlation inequality for complements of symmetric strips.

► Modeling Thermodynamic Properties in Mixtures
Nevan Giuliani

Advisor:  Noel Walkington
Abstract:  In this project, I will create a library that allows for the computation of physical properties of greenhouse gases in various states of matter. The physical properties that I will be focused on such as free energy and entropy usually do not have a closed form function that computes their values.

The computation of these quantities is required by climate change models to determine the distribution of greenhouse gases into the various states of matter at particular conditions. The functions we will devise must be consistent with certain properties guaranteed by physics. Thus, the functions must be increasing in certain inputs, convex/concave in certain inputs, and homogeneous. Using these restrictions, along with experimental data, I will develop methods for computing these quantities at a given condition. I will then program the functions so that climate change software can call them. Lastly, I will test the functions against measured data to evaluate performance and make adjustments as needed.

► Energy-Stable Numerical Schemes for the Q-Tensor Flow of Liquid Crystals
Max Hirsch

Advisor:  Franziska Weber
Abstract:  Liquid crystals are a state of matter with properties intermediate to those of liquids and those of crystals. Like a crystal, liquid crystals have an orientation, and we can model how this orientation changes over time by a system of PDEs. Gudibanda, Weber, and Yue describe a convergent, fully discrete energy-stable finite difference scheme for solving this system of PDEs. We extend this work by adding a term to the PDE system which accounts for inertia, and we present two different finite difference schemes for this system and prove that they are energy-stable.

We then present a different scheme for solving this system of PDEs using the finite element method and show that this scheme obeys an energy law analogous to those obeyed by the finite difference schemes. We implement the schemes in Python.

► Learning Governing Equations with Randomized Basis
Youngjoo Lee

Advisor:  Hayden Schaeffer
Abstract:  Our goal is to approximate the governing equations of some unknown dynamical system from data. We consider first order in time parabolic and hyperbolic PDE, whose semi-discrete approximations are (localized) ODEs. We approximate the unknown differential system using a data-driven dictionary with randomized features and train the system using the LASSO problem. As an example, notice that in the image, the simulation based on our learned governing equation (orange) is close to the true dynamics (blue) and thus we have found a good approximation to the velocity field.

► Random feature maps for learning agent-based models
Yuxuan Liu

Advisor:  Hayden Schaeffer
Abstract:  In agent-based systems, the evolution of multiple agents is determined by their relative distance and position, with magnitude given by an interacting kernel . Given observed (noisy) trajectory samples of all agents, we derived a method to recover the kernel using a new random feature method. This allows us to model smooth trajectories without knowing the exact parameterize form.

Using the learned interaction kernel, we can re-simulate the system and make predictions of the future states. The figures compare the actual trajectories (left) and our learned trajectories (right) for the Predator-Prey system.

► Trace of homogeneous fractional Sobolev spaces in infinite strip domains of constant width
Khunpob Sereesuchart

Advisor:  Giovanni Leoni
Abstract:  In this project, we analyze the trace operator for homogeneous fractional Sobolev spaces for infinite strip domains of constant width. We look at semi-norms over the boundary where the trace is bounded, and from this we derive an inverse. The semi-norms imposed on the trace resemble those present in homogeneous Sobolev spaces, which was established by G. Leoni and I. Tice in 2019.

However, due to more global properties of fractional Sobolev spaces, we are required to impose another seminorm to regulate the trace outside of what the screened seminorm regulates. The proof techniques mainly stem from the analysis of the trace of fractional Sobolev spaces on half-spaces, relying on slicing the domain along different axes to establish bounds that are easier to work with. This result suggests that fractional Sobolev spaces on infinite strip domains of Lipschitz width will be similar to its Sobolev space counterpart, with an additional seminorm bounding distant differences.

► The subword complexity of finite binary words
Andres Burch, Joseph Kalarickal, Jonathan Shoung

Advisor:  Irina Gheorghiciuc
Abstract:  The subword complexity of a binary word $w$ of length $n$ is the sequence $(f(1),f(2),\ldots,f(n))$, where $f(i)$ is the number of distinct subwords of $w$ of length $i$. The problem of finding the necessary and sufficient conditions for a sequence of $n$ natural numbers to be the subword complexity sequence of a finite binary word is still open. In this paper we use several necessary conditions to find an upper bound for the number of distinct subword complexity sequences of binary words of length $n$. According to computer results the number of actual distinct subword complexity sequences of binary words of length $n$ is asymptotic to $(1.4142)^n$.

Our upper bound is asymptotic to $0.22597486(1.495)^n$, and thus very close to the suspected number of distinct subword complexity sequences of binary words of length $n$.

► Isoperimetric Inequalities
Jonathan Jenkins, David Rotunno

Advisor:  Raghavendra Venkatraman
Abstract:  We review relationships of the isoperimetric inequality to other classical inequalities in functional analysis, such as the Sobolev inequality and the Brunn-Minkowski inequality. We then consider the relative isoperimetric problem, which involves minimizing perimeter subject to a volume constraint among sets contained in a given domain. We study this problem in the simplified setting of polygonal sets.

► Chirality in convex geometry
Laura Stemmler, Mirabel Hu, Nick Scheel, Samuel Murray

Advisor:  Florian Frick and Steven Simon (Bard College)
Abstract:  An object is chiral if it cannot be transformed into its mirror image. For example, a Möbius strip is chiral: It has a left-handed and a right-handed version. In this project we prove chiral versions of some of the main theorems of convex geometry.

These results take the following general form: Even if a collection of convex sets or of points in d-space does not meet the prerequisites of some theorem in convex geometry, the conclusion might still hold under weaker requirements at some point in time, if we transform the collection into its mirror image. For example, Radon's theorem asserts that any d+2 points in d-space may be split into two parts whose convex hulls intersect. The chiral version says that if we transform just d+1 points into their mirror image, at some time we can split them into two sets whose convex hulls intersect.

► The prophet problem with corrupted distributions
Braden Yates, Zhian Zhang, Sheena Zhu

Advisors:  David Offner and C. J. Argue
Abstract:  In the classical prophet problem, a player is given a sequence of $n$ distributions $\mathcal{D}_1, \mathcal{D}_2,\dots, \mathcal{D}_n$. After seeing all the distributions, the player sees independent random draws $X_1,X_2,\dots, X_n$ in sequence. Upon seeing $X_i$, the player must accept $X_i$ and end the game or reject $X_i$ in order to see $X_{i+1}$. The player's objective is to select an $X_i$ of maximum value. The classical performance metric is the competitive ratio, which is the ratio of the player's chosen value to the retrospective optimum value, namely the maximum $\max_i X_i$. We investigate a model of the problem with corrupted distributions. Namely, there is a fixed number $r$ (known to the player) of indices $i$ in $[n]$ for which the value $X_i$ is drawn from a distribution $\tilde{\mathcal{D}}_i$, which may be arbitrarily different from $\mathcal{D}_i$.

We give an algorithm with competitive ratio $\frac{1}{8r+4}$, and prove that no algorithm has competitive ratio greater than $\frac{1}{2r+1}$.

For a generalized version of the problem, where the player selects $k$ items with the objective of maximizing the sum of their values, we obtain an algorithm with competitive ratio $\frac{k}{8r+4k}$.

► Dispersion in dimension two
Zaixing Zhang, Eric Zheng

Advisor:  Boris Bukh
Abstract:  Dispersion of a set $P$ is the area of the largest axis-parallel rectangle containing no point of $P$. In this project, we investigated dispersion of sets in the two-dimensional torus. We found lattices of largest dispersion and searched for lattices of small dispersion with a computer.

28-point lattice of smallest dispersion

Non-lattice 4-point set of smallest dispersion, and a largest empty box therein