Georgia Institute of Technology School of Mathematics | Georgia Institute of Technology | Atlanta, GA

Tropicalization is usually applied to algebraic or semi-algebraic sets, but I would like to introduce a different category of sets with well-behaved tropicalization: sets with the Hadamard property, i.e. subsets of the positive orthant closed under coordinate-wise (Hadamard) multiplication. Tropicalization (in the sense of logarithmic limit sets) of a set S with the Hadamard property is a convex cone, whose defining inequalities correspond to pure binomial inequalities valid on S.

I will do several examples of sets S with the Hadamard property coming from combinatorics, such as counts of independent sets in matroids, counts of faces in simplical complexes, and counts of graph homomorphisms. In all of our examples we observe a fascinating polyherdrality phenomenon: even though the sets S we are dealing with are not semilagebraic (they are infinite subsets of the integer lattice) the tropicalization is a rational polyhedral cone. Also, the pure binomial inequalities valid on S are often combinatorially interesting.

Joint work with Annie Raymond, Rekha Thomas and Mohit Singh.

Given a knot $K$ in the 3-sphere, one can ask: what kinds of surfaces in the 3-sphere are bounded by $K$? One can also ask: what kinds of surfaces in the 4-ball (which is bounded by the 3-sphere) are bounded by $K$? In this talk we will discuss how to construct surfaces in both the 3-sphere and in the 4-ball bounded by a given knot $K$, how to obstruct the existence of such surfaces, and explore what is known and unknown about surfaces bounded by so-called torus knots.

For a planar graph $H$, let ${\bf N}_{\mathcal P}(n,H)$ denote the maximum number of copies of $H$ in an $n$-vertex planar graph. The case where $H$ is the path on $3$ vertices, $H=P_3$, was established by Alon and Caro. The case of $H=P_4$ was determined, also exactly, by Gy\H{o}ri, Paulos, Salia, Tompkins, and Zamora. In this talk, we will give the asymptotic values for $H$ equal to $P_5$ and $P_7$ as well as the cycles $C_6$, $C_8$, $C_{10}$ and $C_{12}$ and discuss the general approach which can be used to compute the asymptotic value for many other graphs $H$. This is joint work with Debarun Ghosh, Ervin Győri, Addisu Paulos, Nika Salia, Chuanqi Xiao, and Oscar Zamora and also joint work with Chris Cox.

Random feature methods have been successful in various machine learning tasks, are easy to compute, and come with theoretical accuracy bounds. They serve as an alternative approach to standard neural networks since they can represent similar function spaces without a costly training phase. However, for accuracy, random feature methods require more measurements than trainable parameters, limiting their use for data-scarce applications or problems in scientific machine learning. This paper introduces the sparse random feature expansion to obtain parsimonious random feature models. Specifically, we leverage ideas from compressive sensing to generate random feature expansions with theoretical guarantees even in the data-scarce setting. We provide generalization bounds for functions in a certain class (that is dense in a reproducing kernel Hilbert space) depending on the number of samples and the distribution of features. The generalization bounds improve with additional structural conditions, such as coordinate sparsity, compact clusters of the spectrum, or rapid spectral decay. We show that the sparse random feature expansions outperform shallow networks in several scientific machine learning tasks. Applications to signal decompositions for music data, astronomical data, and various complicated signals are also provided.