Events of the Week

One of the oldest questions in algebraic geometry asks how many constraints a set of points imposes on homogeneous polynomials of a fixed degree. While "simple" points are still well-understood, the problem becomes much more complex when considering "fat points." I will discuss the difficulties in solving for "fat points" and share some expected developments in this direction.One of the oldest questions in algebraic geometry asks how many constraints a set of points imposes on homogeneous polynomials of a fixed degree. While "simple" points are still well-understood, the problem becomes much more complex when considering "fat points." I will discuss the difficulties in solving for "fat points" and share some expected developments in this direction.

Given an integer partition of $n$ into distinct parts, the sum of the reciprocal parts is an example of an Egyptian fraction. We study this statistic under the uniform measure on distinct parts partitions of $n$ and prove that, as $n \to \infty$, the sum of reciprocal parts is distributed away from its mean like a random harmonic sum. We will also spend some time introducing the Boltzmann model, which nicely captures known distributions for distinct parts partitions on which our proof relies.

We will complete the stability result relating the Gromov–Hausdorff distance and Rips complexes for finite metric spaces, and then turn to Chapter 4 of Polterovich et al., which discusses what information can be extracted from a barcode.

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This learning seminar introduces the foundations of persistent homology, a key tool in topological data analysis for capturing multiscale topological features of general metric spaces. We will study persistence modules, barcodes, and distances such as the bottleneck and interleaving metrics, with a focus on their geometric meaning.

A central goal of the seminar is to understand and prove stability theorems for persistent homology, including bounds relating bottleneck distance to the Gromov–Hausdorff distance. Core examples will come from Morse theory and Vietoris–Rips filtrations of metric spaces.

The seminar is structured as a guided, collaborative reading course and is aimed at graduate students and faculty generally interested in this aspect of topology or geometry.

Hyperbolic geometry is inseparably connected with the arithmetic of quadratic fields. For imaginary quadratic fields, ideal classes give rise to special points on the modular curve, whose values under the modular j-function are the classical singular moduli. For real quadratic fields, the corresponding objects are no longer points but closed geodesics; and in a newer geometric picture, they can also be enlarged to finite-area hyperbolic surfaces. The theme of this talk is that all three constructions carry arithmetic information of the same flavor, and I will describe how these invariants behave on average as the discriminant grows.

Developing a suitable notion of vector bundles in tropical geometry has recently attracted considerable interest. In a recent work, Khan and Maclagan introduced tropical vector bundles using matroids, inspired by Klyachko’s classification of toric vector bundles, and studied their stability properties. In this talk, we reinterpret their notion of stability through the framework of the categorical approach to stability proposed by André. This perspective clarifies the structure underlying their results and places them in a broader conceptual setting. This is joint work with Alex Sistko and Cameron Wright.

Understanding large systems, like those that describe ecosystems, trade networks, and software systems is a critical scientific need. Because these systems are difficult to instrument, tools that provide accurate, model-based fore- and retro-casting from noisy, sporadic data are valuable. The past few years have shown that sheaves are useful for building composite models of practical systems. These models can include both numerical and categorical data of substantial richness. Moreover, they allow for the ability to study both traditional ``forward'' (prediction) and ``inverse'' (inference) problems in the same framework, along with intermediate problems in which prediction and inference jointly play an important role.

What steady shapes are compatible with capillarity and fluid flow? I will discuss two recent results that give some answers in two and three dimensions. In two dimensions, stationary hollow vortices with irrotational exterior flow reduce to an overdetermined elliptic free-boundary problem in an exterior domain, with a jump condition coupling boundary curvature and the Neumann trace of the stream function. We prove that for Weber numbers We<=2, any Jordan-curve solution must be a circle; the linearized problem also isolates the discrete exceptional values We=3,4,5,… where the kernel becomes nontrivial. I will then turn to the axisymmetric two-phase Euler equations in three dimensions, where the corresponding steady problem near the explicit spherical solution built from Hill’s vortex admits smooth non-spherical bubble and drop solutions. Away from a discrete set of critical Weber numbers these solutions are obtained by the implicit function theorem, while at the critical values one gets bifurcating branches via the Crandall–Rabinowitz theorem.