Minimum-norm interpolators have recently gained attention primarily as an analyzable model to shed light on the double descent phenomenon observed for neural networks. The majority of the work has focused on analyzing interpolators in Hilbert spaces, where typically an effectively low-rank structure of the feature covariance prevents a large bias. More recently, tight vanishing bounds have also been shown for isotropic high-dimensional data for $\ell_p$-spaces with $p\in[1,2)$, leveraging the sparse structure of the ground truth. This work takes a first step towards establishing a general framework that connects generalization properties of the interpolators to well-known concepts from high-dimensional geometry, specifically, from the local theory of Banach spaces.

We will discuss the importance of Chow rings in algebraic geometry, as well as one of the central issues in their computation: can one find the Chow ring of a space given the Chow ring of an open and its complement? This is the so called patching problem, and we will discuss multiple ways of solving it. Our main examples will be the moduli stack of curves, as well as root gerbes and root stacks.

The boundary regularization effect for hypoelliptic kinetic equations is limited. The solution with the simplest zero inflow boundary conditions exhibits at most Hölder continuity near the singular set of the boundary. We will discuss recent results on hypoelliptic regularity and explain the smoothness properties of solutions in the presence of boundary conditions in certain cases.

An inhomogeneous system of linear equations over a ring $R$ is partition regular if for any finite coloring of $R$, the system has a monochromatic solution. In 1933, Rado showed that an inhomogeneous system is partition regular over $\mathbb{Z}$ if and only if it has a constant solution. Following a similar approach, Byszewski and Krawczyk showed that the result holds over any integral domain. In 2020, Leader and Russell generalized this over any commutative ring $R$, with a more direct proof than what was previously used. We analyze some of these combinatorial results from a computability-theoretic point of view, starting with a theorem by Straus used directly or as a motivation to many of the previous results on the subject.

How much information first-order logic can detect is a natural question in logic. In this talk, I will present Thomas Koberda's proof of first-order rigidity of homeomorphism groups of compact manifolds.

Given a closed surface S of genus at least 2, Bers' simultaneous uniformization provides a way to holomorphically associate pairs of oppositely oriented complex structures on S to quasi-Fuchsian hyperbolic 3-manifolds. In this talk, we'll discuss joint work with Christian El Emam in which we give a procedure for holomorphically associating (many, but not all) pairs of Hitchin representations into SL(3,R) to representations to SL(3,C). Along the way, we'll introduce conformal harmonic maps into holomorphic Riemannian symmetric spaces and other related objects.

Definitive Screening Designs (DSDs) were introduced in 2011. Since then they have become popular for industrial applications. This talk describes what a DSD is. It then explains why engineers prefer them to standard two-level fractional factorial designs. Finally, it shows how to construct them and block them.

A theorem due to Erd\H{o}s and Gallai fully answers the question of when one can realize a provided integer sequence as the degree sequence of a graph with an easy-to-check, necessary and sufficient condition. If one considers a random integer partition as the provided integer sequence, a theorem of Pittel's shows that with high probability a random partition is not the degree sequence of a graph. We consider the analogous question for 3-uniform hypergraphs and show that with high probability a random partition can indeed be realized as the degree sequence of a 3-uniform hypergraph. We will further consider the realizability question of whether one can realize three random integer partitions as the marginals of an associated three-dimensional contingency table. We will briefly discuss how this latter result resolves two conjectures of Pak and Panova regarding Kronecker coefficients

We show that the combinatorial matter of graph coloring is, in fact, quantum in the sense of satisfying the sum over all the possible intermediate state properties of a path integral. In our case, the topological field theory (TFT) with defects gives meaning to it. This TFT has the property that when evaluated on a planar trivalent graph, it provides the number of Tait-Coloring of it. This can be considered a generalization of groups. With the Klein-four group as a 1-defect condition, we reinterpret graph coloring as sections of a certain bundle, distinguishing a coloring (global-sections) from a coloring process (local-sections.) These constructions also lead to an interpretation of the word problem, for a finitely presented group, as a cobordism problem and a generalization of (trivial) bundles at the level of higher categories. See https://arxiv.org/abs/2410.00378.

Given a Fano manifold M with extremal contractions to Fano manifolds A and B, it is expected that the derived category of M contains two semi-orthogonal decompositions, related by the action of the braid group, which refine the semi-orthogonal decompositions of the derived categories of A and B. I will discuss strategies for proving this expectation when the Fano manifolds have moduli interpretations. As one application, we construct a semi-orthogonal decomposition of the derived category of the moduli space of stable rank 2 vector bundles on a smooth projective curve, as conjectured by Narasimhan and by Belmans, Galkin, and Mukhopadhyay.

It was first observed by Alon, Fox and Zhao that VC-dimension can be used to characterize a dichotomy in the growth of regular partitions of graphs. Specifically, if a hereditary graph property H has finite VC-dimension, then results of Alon-Fischer-Newman and Lov'{a}asz-Szegedy imply all graphs is H have $\epsilon$-regular partitions of size polynomial in $1/\epsilon$. On the other hand, if H has infinite VC-dimension, then results of Gowers and Fox-Lov\'{a}sz show there are graphs in H whose smallest $\epsilon$-regular partition has size at least an exponential tower of height polynomial in $1/\epsilon$. In this talk, I present several analogous dichotomies in the setting of hereditary properties of 3-uniform hypergraphs. These dichotomies are all characterized by various analogues of VC-dimension for 3-uniform hypergraphs.

Dumas, Lannes, and Szeftel have introduced a family of equations designed to model waves in nonlinear optics which have less symmetry than solutions of the nonlinear Schrödinger equation, in that they are not axisymmetric about the axis of propagation. Their equations can include regularization terms which are present in some but not all spatial directions. We study the stability of standing-wave solutions of such regularized nonlinear Schrödinger equations, and find that the regularization can increase the range of nonlinearities for which standing waves are stable. This effect is similar to that seen in the Benjamin-Bona-Mahony regularization of the Korteweg-de Vries equation, but notably, the effect is present even when the regularization terms are not present in all directions.

In this talk, I will introduce a novel weighted l2 projection method for estimating covariance functions, with an emphasis on estimation of autocovariance sequences from reversible Markov chains. Shape-constrained estimation of a function with discrete support has been investigated and successfully applied to various application problems. Notably, Berg and Song (2023) connected this idea with uncertainty quantification in Markov chain Monte Carlo (MCMC) samples and proposed a shape-constrained estimator for autocovariance sequences. While the least-squares objective is commonly used in shape-constrained regression, it can be suboptimal due to correlation and unequal variances in the input function. To address this, we introduce a weighted least-squares method that defines a weighted norm on transformed data. Our approach involves transforming input data into the frequency domain and weighting the input sequence based on their asymptotic variances, exploiting the asymptotic independence of periodogram ordinates. I will discuss the computational aspects, theoretical properties, and the improved performance of this method compared to its non-weighted counterpart.

We present a new method to determine asymptotics in Markov processes, using orthogonal polynomial duality. This method allows for certain expected values to be decomposed over an orthogonal basis of duality functions in a "simpler" dual Markov process. Estimates for these expectations are then calculated in terms of estimates on the duality functions, which result in "universal" asymptotics. We demonstrate this method for asymmetric dynamic interacting particle systems, where there had not even been conjectures for asymptotics. These asymptotics are Tracy--Widom fluctuations for certain values of the asymmetry. We will also touch upon the algebraic and combinatorial structures underlying the probabilistic models.

We introduce a powerful scan statistic and the corresponding test for detecting the presence and pinpointing the location of a change point within the distribution of a data sequence with the data elements residing in a separable metric space (Ω, d). These change points mark abrupt shifts in the distribution of the data sequence as characterized using distance profiles, where the distance profile of an element ω ∈ Ω is the distribution of distances from ω as dictated by the data. This approach is tuning parameter free, fully non-parametric and universally applicable to diverse data types, including distributional and network data, as long as distances between the data objects are available. We obtain an explicit characterization of the asymptotic distribution of the test statistic under the null hypothesis of no change points, rigorous guarantees on the consistency of the test in the presence of change points under fixed and local alternatives and near-optimal convergence of the estimated change point location, all under practicable settings. To compare with state-of-the-art methods we conduct simulations covering multivariate data, bivariate distributional data and sequences of graph Laplacians, and illustrate our method on real data sequences of the U.S. electricity generation compositions and Bluetooth proximity networks.

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We consider the electron magnetohydrodynamics (MHD) with static background ion flow. A special situation was studied numerically by physicists. In this paper we show the existence of determining wavenumber for the electron MHD, and establish a regularity condition only on the low modes of the solution.

In education, health, and human services, an intervention program is usually implemented by many local organizations. Determining which organizations are more effective is essential for theoretically characterizing effective practices and for intervening to enhance the capacity of ineffective organizations. In multisite randomized trials, site-specific intention-to-treat (ITT) effects are likely invalid indicators for organizational effectiveness and may lead to inequitable decisions. This is because sites differ in their local ecological conditions including client composition, alternative programs, and community context. Applying the potential outcomes framework, this study proposes a mathematical definition for the relative effectiveness of an organization. The estimand contrasts the performance of a focal organization with those that share the features of its local ecological conditions. The identification relies on relatively weak assumptions by leveraging observed control group outcomes that capture the confounding impacts of alternative programs and community context. We propose a two-step mixed-effects modeling (2SME) procedure. Simulations demonstrate significant improvements when compared with site-specific ITT analyses or analyses that only adjust for between-site differences in the observed baseline participant composition. We illustrate its use through an evaluation of the relative effectiveness of individual Job Corps centers by reanalyzing data from the National Job Corps Study, a multisite randomized trial that included 100 Job Corps centers nationwide serving disadvantaged youths. The new strategy promises to alleviate consequential misclassifications of some of the most effective Job Corps centers as least effective and vice versa.

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In oncology clinical trials, subjects prematurely discontinuing from the assigned treatment prior to experiencing an event of interest are often handled by noninformative censoring under censor-at random assumption. Such methods can be challenged with respect to the robustness of the ignorable or noninformative censoring and sensitivity analyses using informative censoring are often required. In a recently published article (Jin and Fang, 2024), reference-based methods (including Jump to Reference and Copy Reference) and tipping point analysis for time-to-event data with possibly informative censoring were proposed. These are novel methods to fit the gap in literature for time-to-event analysis with applications in oncology clinical trials. We will describe and facilitate the implementation of these methods in this presentation. Illustrative examples are provided to demonstrate the reference-based methods and tipping point analysis.

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I'll discuss joint work with Alessio Caminata, Luca Schaffler, and Han-Bom Moon in which we study equations defining (the closure of) the locus of n points in projective space that lie on a rational normal curve and apply these equations to resolve a question of Lior Pachter and David Speyer from 2004 on the tropical geometry of the space of phylogenetic trees.

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