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.
Sperner’s lemma is a simple combinatorial result that is surprisingly powerful and useful—bringing together ideas in combinatorics, geometry, and topology while attracting interest from economists and game theorists. I’ll explain why, show some old and new proofs, and present some recent generalizations with diverse applications.
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.
We present a combinatorial analysis of fiber bundles of generalized configuration spaces on connected abelian Lie groups and discuss topological consequences. These bundles are akin to those of Fadell-Neuwirth for configuration spaces, and their existence is detected by a combinatorial property of an associated finite partially ordered set. Of particular focus is the case of a toric arrangement: a finite collection of codimension-one subtori in a complex torus. If the intersection pattern of the subtori satisfies the combinatorial condition of supersolvability, the complement of the toric arrangement sits atop a tower of fiber bundles. This structure provides insight into topological invariants of these toric arrangement complements, including the homotopy groups, cohomology, and topological complexity. Based on joint work with Daniel C. Cohen and Emanuele Delucchi.
Menger's theorem, arguably one of the most important theorems in graph theory, states that for any subsets $X$ and $Y$ of vertices of a graph, either there exist $k$ disjoint paths from $X$ to $Y$, or there exist a set of at most $k-1$ vertices hitting all such paths. We say that a graph or a graph class has the weak coarse Menger property if there exist functions $f$ and $g$ such that for any subsets $X$ and $Y$ of vertices and integers $k$ and $r$, either there exist $k$ paths from $X$ to $Y$ with pairwise at distance at least $r$, or there exists a union of $f(k,r)$ balls of radius $g(k,r)$ hitting all paths from $X$ to $Y$. Nguyen, Scott and Seymour proved that the class of all graphs does not have the weak coarse Menger property and asked whether minor-closed families have it. We answer this question affirmatively in a stronger form by showing that rooted fat $K_2$-minors have the coarse Erdos-Posa property in minor-closed families. Our result extends to every length space quasi-isometric to a locally finite infinite graph with an excluded finite minor, such as complete Riemannian surfaces of finite Euler genus, metric graphs with an excluded finite minor, string graphs, and Cayley graphs of finitely generated minor-excluded groups.
