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► Continuous equivariant learning with linearly many frames

Nathaniel Kenschaft

Advisor: Florian Frick
Abstract: Many AI systems in science predict physical quantities, such as the forces between atoms in a molecule, and those predictions must respect symmetry: rotating the molecule should rotate the forces. A popular approach first turns every input into a standard orientation, runs an ordinary neural network, and turns the answer back. But a standard orientation cannot generally vary smoothly across all inputs, and abrupt switches can introduce artificial jumps in predicted forces. Like covering the Earth with overlapping maps, one can instead use several orientations and blend their predictions with gradually shifting weights. Each orientation, however, costs another run of the network.

Using topology, the mathematics of shape, this work shows that the number needed grows in proportion to the number of particles, and gives an explicit recipe that achieves this scaling, for example, 61 candidate orientations for 24 particles, compared with more than 1,000 in previous constructions.

► Gradient-flow methods for sampling and PDE approximation

Kausthub Satluri and Jerry Yin

Advisor:
Abstract: Gradient flows provide a common framework for studying evolution partial differential equations (PDEs) and developing sampling algorithms. This project investigated how regularizing the Wasserstein geometry on the space of probability measures changes the dynamics of Kullback–Leibler (KL) gradient flows, and whether these changes can lead to effective algorithms for sampling and PDE approximation. We implemented several gradient-flow algorithms and evaluated their performance, including comparisons with the “blob” method. In the examples tested, the numerical results suggest that algorithms based on the proposed flows may offer computational advantages over the ``blob" method.

► The $k$-Color Poset Ramsey Number of the Diamond is $2k$

Will Cong, Teresa Huang, John Li, and BangTam Ngo

Advisors: Tom Bohman and Quentin Dubroff
Abstract: Ramsey theory focuses on the appearance of order in a substructure given a structure of a specific size. The $k$-color Ramsey number of a poset $P$, a partially ordered set where some pairs of elements may not be comparable, is the smallest dimension $d$ such that any $k$-coloring of the $d$-dimensional Boolean lattice contains a monochromatic induced copy of $P$, our substructure. Our structure is the collection of all subsets of a set with $d$ elements, ordered by inclusion. Furthermore, an induced copy preserves exactly the ordering relations of $P$.

For $j\geq 2$, the $j$-diamond is the poset consisting of a maximum element, a minimum element, and an antichain of size $j$, a collection of $j$ elements that are pairwise incomparable, between them.

We prove that the $k$-color Ramsey number of the $j$-diamond is at most $jk$. Colorings of the Boolean lattice that assign sets of the same size the same color provide matching lower bounds for $j\in{2,3,4}$, showing that this bound is exact in these cases.

► New Bounds in Multicolor Ramsey Numbers of Hypergraphs

Will Schremmer and Lillian Xia

Advisors: Tom Bohman and Quentin Dubroff
Abstract: Traditional Ramsey Theory is the study of finding monochromatic subgraphs in an arbitrary coloring of complete graphs - graphs that have every possible edge given a set of vertices. We in particular looked at multicolor Ramsey numbers for uniform hypergraphs. An $r$-uniform hypergraph is a generalization of a graph where each edge connects $r$ vertices at once, rather than the standard two vertices per edge. Given an $r$-uniform hypergraph $H$, we define $R(k; H)$ as the Ramsey number of coloring the complete graph with $k$ colors and looking for a monochromatic $H$. Our main goal was to investigate the behavior of $R(k; H)$ for fixed $H$ and $k \to \infty$. We used Keevash's theorem on the existence of designs to find lower bounds of $R(k; H)$ for infinitely many values of $k$ for every hypergraph $H$, and developed a weighting lemma to find upper bounds. We ended up with matching upper and lower bounds for $R(k; H)$ for a large class of hypergraphs $H$.

► On the Banach–Mazur distance to the three-dimensional cube

Leo Zhuang and Clement Zhou

Advisor: Konstantin Tikhomirov
Abstract: We use computational methods that combine results from convex geometry with optimization algorithms to improve the best previously known uniform bound on the Banach–Mazur distance to the three-dimensional cube.

Specifically, we show that for every origin-symmetric 3-dimensional convex body K, its Banach–Mazur distance to the cube is at most 2.45, improving upon the Taschuk bound.