In this talk, I will present a detailed study of the curvature and symplectic asphericity properties of symmetric products of curves. I show that these spaces can be used to answer nuanced questions arising in the study of closed Riemannian manifolds with positive scalar curvature. For example, symmetric products of curves sharply distinguish between two distinct notions of macroscopic dimension introduced by Gromov and Dranishnikov. As a natural generalization of this circle of ideas, I will also address the Gromov–Lawson and Gromov conjectures in the Kaehler projective setting and draw new connections between the theories of the minimal model, positivity in algebraic geometry, and macroscopic dimensions. This is joint work with Alexander Dranishnikov and Ekansh Jauhari.
The permutahedral and Peterson varieties are subvarieties of the flag variety defined by how a fixed matrix acts on the components of a flag. As subvarieties of the flag variety, their cohomology classes (which are the same) have a positive expansion in the basis of Schubert classes. In a remarkable series of papers, Nadeau and Tewari describe the coefficients of this expansion in terms of a parking (as in parking function) procedure on reduced words for the indexing permutation. The subregular Peterson is a reducible scheme with the same cohomology class, and we describe the cohomology classes of the irreducible components (which must add up to the class described by Nadeau and Tewari) in terms of this parking procedure. This is joint work with Lucas Gagnon (USC) and Carole Zhuang (WUSTL).
Given a matroid $M$ and a building set $B$ on its lattice of flats, we prove that
the associated nested set complex $N$ is shellable. This generalizes a classical
result of Bj¨orner that the order complex of the lattice of flats of a matroid
is shellable (the case when $B$ is the maximum builiding set), and strengthens
a result of Feichtner--Müller that $N$ is Cohen-Macaulay for arbitrary $B$. Our
approach is geometric in nature utilizing the Bergman fan $\Sigma_{M,B}$, and is inspired
by Bruggesser and Mani’s line shellings of polytopes. We prove that, given a
normal complex $P$ for $\Sigma_{M,B}$, as introduced by Nathanson--Ross, and a
particularly well-behaved vector ω, the order of the vertices of P induced by ω
is a shelling order for $N$. This is joint work with Spencer Backman, Anastasia
Nathanson, Ethan Partida, and Noah Prime.
Each day your actions generate data, and that data is being used at some cost to your privacy. "Differentially private" algorithms seek to protect the privacy of individual data, often by injecting some randomness. Such mechanisms have been used by Apple, Google, Uber, and the US Census Bureau. I'll describe how such algorithms work and discuss recent efforts to quantify how much randomness is needed to guarantee privacy but still give accurate answers. Surprisingly, this analysis involves the geometry of sets positioned in space in clever ways.
