Random geometric graphs are random graph models defined on metric spaces. Such a model is defined by first sampling points from a metric space and then connecting each pair of sampled points with probability that depends on their distance, independently among pairs. In this work we show how to efficiently reconstruct the geometry of the underlying space from the sampled graph under the {\em manifold} assumption, i.e., assuming that the underlying space is a low dimensional manifold and that the connection probability is a strictly decreasing function of the Euclidean distance between the points in a given embedding of the manifold in $\mathbb{R}^N$. Our work complements a large body of work on manifold learning, where the goal is to recover a manifold from sampled points sampled in the manifold along with their (approximate) distances.

(Joint work with P. Jiradilok and E. Mossel).

Given a sequence of relatively prime integers \(a_0 = 0 < a_1 < \dots < a_n = d\) and a field \(k\), consider the projective monomial curve \(\mathcal{C}\subset\mathbb{P}_k^{\,n}\) of degree \(d\) parametrically defined by \(x_i = u^{a_i}v^{d-a_i}\) for all \(i \in \{0,\ldots,n\}\) and its  coordinate ring \(k[\mathcal{C}]\). The curve \(\mathcal{C}_1 \subset \mathbb A_k^n\) with parametric equations \(x_i = t^{a_i}\) for \(i \in \{1,\ldots,n\}\) is an affine chart of \(\mathcal{C}\) and we denote its coordinate ring by \(k[\mathcal{C}_1]\).

In this talk, we will discuss the equality of the Betti numbers of \(k[\mathcal{C}]\) and \(k[\mathcal{C}_1]\) and present a combinatorial criterion that ensures equality. This is joint work with I. García-Marco and P. Gimenez.

We highlight a perhaps important but hitherto unobserved insight: The attention module in a ReLU-transformer is a cubic spline. Viewed in this manner, this mysterious but critical component of a transformer becomes a natural development of an old notion deeply entrenched in classical approximation theory. Conversely, if we assume the Pierce--Birkhoff conjecture, then every spline is also an encoder.

Let \(A = \{a_0=0<a_1<\dots<a_{n-1}=d\}\) be a finite set of relatively prime integers. For all \(s\in \mathbb{N}\), the \(s\)-fold sumset of \(A\) is the set \(sA\) of integers obtained by collecting all sums of \(s\) elements in \(A\). On the other hand, given a field \(k\), one can associate with \(A\) the projective monomial curve \(\mathcal{C}_A\) parametrized by (A\), i.e., the Zariski closure of \( \{(v^d:u^{a_1}v^{d-a_1}:\cdots:u^{a_{n-2}}v^{d-a_{n-2}}:u^d) \mid (u:v) \in \mathbb{P}_k^1\} \subset \mathbb{P}_k^{\, n-1} \, .\)
In this talk, I will present some results relating properties of \(\mathcal{C}_A\) to the behavior of the sumsets of \(A\), revealing a new interplay between commutative algebra and additive number theory. This is joint work with P. Gimenez.

Given a large network one is often interested in efficiently estimating various local statistics. In this talk, we'll discuss the distribution of one possible estimator arising from counting monochromatic subgraphs in a random vertex colorings. We focus on the    asymptotic normality of these counts, particularly for monochromatic triangles, and provide new, local influence-based necessary and sufficient conditions.  The conditions we obtain combine ideas from Boolean analysis as well as classical fourth-moment theorems originating from normal approximation results in the Wiener space.

For parameterized systems, one standard problem is to determine the set of parameters which "best" fits given data.  Two examples of this will be summarized in this talk, both of which can be solved using homotopies.  The first is variational inference in which one searches in a parameterized family of probability distributions for a probability distribution that best fits the given data.  The second is synthesizing a linkage whose coupler curve best approximates the given data.  This talk is joint work with Emma Cobian, Fang Liu, and Daniele Schiavazzi (variational inference) and Aravind Baskar and Mark Plecnik (approximate synthesis).

 

What is the most singular possible (reduced) hypersurface in positive characteristic? One answer to this question comes from finding a lower bound on an invariant called the F-pure threshold of a polynomial in terms of its degree. In this talk, I'll introduce the F-pure threshold and then discuss a class of hypersurfaces which obtain a minimal F-pure threshold.  We’ll further investigate some of their surprising algebraic and geometric properties including their connection to surfaces with many lines.  This talk is based on joint work with Zhibek Kadyrsizova, Jennifer Kenkel, Jyoti Singh, Karen E. Smith, Adela Vraciu, and Emily E. Witt.

In this talk, we’ll present some recent work on traveling waves in water that carry vortices in their bulk. We show that for any supercritical Froude number (non-dimensionalized wave speed), there exists a continuous one-parameter family of solitary waves with a submerged point vortex in equilibrium. This family bifurcates from an irrotational laminar flow, and, at least for large Froude numbers, it extends up to the development of a surface singularity. These are the first rigorously constructed gravity wave-borne point vortices without surface tension, and notably our formulation allows the free surface to be overhanging. Through a separate numerical study, we find strong evidence that many of the waves do indeed have an overturned air—water interfaces. Finally, we prove that generically one can perform a desingularization procedure to obtain a solitary wave with a submerged hollow vortex. Physically, these can be thought of as traveling waves carrying spinning bubbles of air in their bulk.

We will also discuss some work in progress on the existence of imploding vortex configurations that experience finite-time self-similar collapse.

This is joint work with Ming Chen, Kristoffer Varholm, and Miles Wheeler.
 

Density estimation for Gaussian mixture models is a classical problem in statistics that has applications in a variety of disciplines. Two solution techniques are commonly used for this problem: the method of moments and maximum likelihood estimation. This talk will discuss both methods by focusing on the underlying geometry of each problem.

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