Events of the Week

Graduate Student Chapters General meeting

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.

My dissertation studies the "Multiplicity = Volume" formula in commutative algebra, extending its reach to Newton non-degenerate ideals in regular local rings and to multi-p-families of ideals in prime characteristic. My goal is to make the multiplicities of these families more accessible from a combinatorial perspective. This research was completed under the guidance of Professor Tài Huy Hà, whose support was invaluable throughout this journey.

Everyone is welcome—I would be truly honored by your presence.

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.