Title: Multiplicative comparison of K-Theories and homology

• Abstract: K-Theory is a machine taking certain types of categories as input and producing spectra (or spaces) as its output. There are two notable versions, Segal and Waldhausen’s K-theory, depending on the specific kind of input, with a precise comparison between the two, due to A.-M. Bohmann and A. Osorno. We fully extend Bohmann and Osorno’s comparison to include a fourth object, spectra in chain complexes. This is joint work with Brandon Doherty and Arash Karimi.

Title: Optimal Transport and Its Metrics: From the Classical Wasserstein Distance to Sliced Approaches

• Abstract: The goal of this talk is to introduce the Optimal Transport (OT) problem and touch on different metrics that we can create from it. Our objects will be measures or, more specifically, probability distributions. We will begin with the classical Wasserstein distance, discuss its linearized version (including known results and open questions), and then move toward the so-called Sliced-Wasserstein (SW) metric. The latter is a computationally efficient alternative to the Wasserstein distance, comparing probability measures via their one-dimensional projections ("slices"). In particular, for probability measures supported on a compact set, SW induces a topology equivalent to that of the Wasserstein distance, while being substantially cheaper to compute. Despite this topological equivalence, the geometry can differ significantly: unlike the Wasserstein space, the SW space is not geodesic in general. Indeed, classical SW-type approaches quantify dissimilarity but do not provide an explicit "assignment" between the original measures, making it nontrivial to define analogues of Wasserstein "displacement interpolations". This motivates projection-based constructions guided by two questions: can one recover a meaningful transport plan within the sliced framework, and can such constructions yield metrizations with good topological behavior relative to the Wasserstein distance?

Title: Stability patterns in the rational homology of (ordered) Hurwitz spaces

• Abstract: I will give an introduction to Hurwitz spaces, which are certain covers of unordered configuration spaces of the plane and have applications in algebraic geometry and number theory. I will talk about a pattern called homological stability that these spaces have, due to work of Ellenberg--Venkatesh--Westerland. I will also talk about joint work with Jeremy Miller and Jennifer Wilson concerning ordered Hurwitz spaces and a pattern that these spaces have called representation stability.

Title: Rognes’ connectivity conjecture

• Abstract: Rognes’ connectivity conjecture concerns the connectivity of a simplicial complex called the common basis complex. Rognes proved that the equivariant homology of this complex is the E^1 page of a spectral sequence converging to the homology of the algebraic K-theory spectra. I will describe joint work with Patzt and Wilson where we prove the connectivity conjecture for fields. I will explain a connection between the homology the common basis complex and the André–Quillen homology of a certain equivariant ring built out of Steinberg modules. Time permitting, I will mention variants of these results for symplectic groups (joint with Scalamadre and Sroka) and automorphism groups of free groups (joint with Bruck and Piterman).

Title: Algebraic vs. holomorphic vector bundles

• Abstract: A basic question in complex algebraic geometry is to characterize the ``algebraic'' objects amongst ``holomorphic'' objects, when the two notions make sense, e.g., (cohomology classes, K-theory classes, vector bundles). We will discuss recent progress on this question for vector bundles on smooth affine complex varieties. This talk is based on joint work with Tom Bachmann, Jean Fasel and Mike Hopkins.

Title: Symmetric products of curves, scalar curvature, and positivity in algebraic geometry

• Abstract: In this talk, I will present a detailed study of the curvature and symplectic asphericity properties of symmetric products of curves. I demonstrate that these spaces can be utilized to address 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.

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Title: Principal minors, stable polynomials and tropical geometry

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April 14

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April 21

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