Current & Upcoming

Eukaryotic cells produce thousands of different lipids—collectively known as the lipidome—whose composition is tailored to cellular needs. Lipids are distributed unevenly throughout biological systems, where they accumulate locally, forming membranes with specific compositions and thereby determining the identity and functional specialization of organelles. Due to their hydrophobicity, lipids cannot move freely out of cellular membranes through the cell’s aqueous environment and require transporters—lipid transfer proteins, or LTPs—to carry them. LTPs are soluble molecular machines responsible for transporting lipids, and they are found in all kingdoms of life. They have diverse structures, but many share a common mode of action: they extract specific lipids from membrane bilayers and load them into a hydrophobic pocket, forming water-soluble protein-lipid complexes that isolate cargoes from the aqueous phase. In addition to their cargo, some LTPs mobilize auxiliary lipids that function as exchange currencies or cofactors. They facilitate the uptake or release of the cargo, which would determine the direction of transport and its coupling to metabolism. However, for most LTPs, the identity of cargo and auxiliary lipids remains unknown. The fundamental biochemistry of LTPs remains poorly understood, limiting our ability to study how they function within cells. Our goal is to begin addressing this gap; I will present a recent systematic analysis of the lipid binding properties of human LTPs and discuss the general principles that we have derived from it.

When can we change the diagonal of a matrix without changing its spectrum? We completely answer this question over an algebraically closed field of characteristic zero or larger than the size of the matrix: A square matrix A admits a nonzero diagonal matrix D such that A and A+D have the same spectrum if and only if there are two principal minors of A of the same size that are not equal. This relates to the classical additive inverse eigenvalue problem in numerical analysis and has implications for existence and rigidity results in the theory of Floquet isospectrality of discrete periodic operators in solid state physics. The proof employs new techniques involving Hilbert schemes of points and the infinitesimal structure of the Hilbert--Chow morphism. This is joint work with John Cobb and Matt Faust.

We are now well into the era of AI in mathematics, and mathematical research data has never been more valuable. There has been a large increase in the data and software accompanying published research. Over the past five years, the Mathematical Research Data Initiative (MaRDI) has been building infrastructure to support mathematicians with their data in accordance with the FAIR principles (Findable, Accessible, Interoperable, Reusable). MaRDI spans a broad range of mathematical fields, from scientific computing and statistics to interdisciplinary research. In this talk, we give an overview of MaRDI, highlighting a few projects, and then take a deeper look at the MaRDI file format for serializing computer algebra data. Computer algebra data, from Gröbner bases to databases of phylogenetic models, is notoriously hard to store and share: objects depend on parent structures, refer to one another, and must remain readable as software evolves. We discuss why a dedicated format is necessary, how it can be used, and the design decisions we made to address these challenges.

TBA