Spectra are among the most fundamental objects in algebraic topology and appear naturally in the study of generalized cohomology theories, algebraic K-groups and cobordism invariants.  I will first explain that spectra define a homotopical enlargement of algebra known as “higher algebra,” where one has topological analogues of algebraic structures like rings, modules, and tensor products.

A striking feature of higher algebra is that there are additional “chromatic characteristics” interpolating between characteristic 0 and characteristic p.  These intermediate characteristics have shed light on mod p phenomena in geometry, number theory, and representation theory.  On the other hand, the extension of algebraic ideas to higher algebra has been fruitful within algebraic topology: I will discuss joint work with Robert Burklund and Tomer Schlank which proves a higher analogue of Hilbert’s Nullstellensatz, thus identifying the ‘’algebraically closed fields’’ of intermediate characteristic.  In addition to initiating the study of “chromatic algebraic geometry,” this work resolves a form of Rognes’ chromatic redshift conjecture in algebraic K-theory.

Property testing is a fundamental subject in theoretical computer science and combinatorics, which studies when and how global properties of large objects (such as a massive data set or a huge graph) can be robustly inferred when given only local views of the object. Famous examples of property testing include testing whether a given graph is triangle-free or whether a given boolean function is linear.

In this talk, I'll present a generalization of the property testing model where the "local views" of an object are not given by deterministic evaluations, but instead by the probabilistic outcomes of measurements on a quantum state. This gives rise to a noncommutative model of property testing, and raises many interesting questions at the interface of complexity theory, quantum information, operator algebras, and more. Finally, I'll describe how the recent quantum complexity result MIP* = RE can be viewed through the lens of noncommutative property testing.

For convex complete intersections, the Gromov-Witten invariants are often computed using the Quantum Lefshetz Hyperplane theorem, which relates the invariants to those of the ambient space. However,  the convexity condition often fails when the target is an orbifold, even for genus 0, hence Quantum Lefshetz is no longer guaranteed. In this talk, I will showcase a method to compute these invariants, despite the failure of Quantum Lefshetz, for orbifold complete intersections in stack quotients of the form [V // G]. This talk will be based on joint works with Felix Janda (Notre Dame) and Yang Zhou (Fudan), and with Rachel Webb (Berkeley).

The Bruhat order encodes algebraic and topological information of Schubert varieties in the flag manifold and possesses rich combinatorial properties. In this talk, we discuss three interrelated stories regarding the Bruhat order: self-dual Bruhat intervals, Billey-Postnikov decompositions, and automorphisms of the Bruhat graph. This is joint work with Christian Gaetz. 

We consider generic translation surfaces of genus g>0 with n>1 marked points and take covers branched over the marked points such that the monodromy of every element in the fundamental group lies in a cyclic group of order d. Given a translation surface, the number of cylinders with waist curve of length at most L grows like L^2. By work of Veech and Eskin-Masur, when normalizing the number of cylinders by L^2, the limit as L goes to infinity exists and the resulting number is called a Siegel-Veech constant. The same holds true if we weight the cylinders by their area. Remarkably, the Siegel-Veech constant resulting from counting cylinders weighted by area is independent of the number of branch points n. All necessary background will be given and a connection to combinatorics will be presented. This is joint work with Aaron Calderon, Carlos Matheus, Nick Salter, and Martin Schmoll.

I’ll talk about ongoing work in collaboration with the Ding lab of Biomedical Engineering at UCI. There are unresolved mysteries about the dynamics of RNA splicing, an important molecular process in the genetic machinery. These mysteries remain because the obtainable data for this process are not time series, but rather static spatial images of cells with stochastic particles.  From a modeling perspective, this creates a challenge of finding the right mathematical description that respects the stochasticity of individual particles but remains computationally tractable. I’ll share our approach of constructing a spatial Cox process with intensity governed by a reaction-diffusion PDE. We can do inference on this process with experimental images by employing variational Bayesian inference. Several outstanding issues remain about how to combine classical and modern statistical/data-science approaches with more exotic mechanistic models in biology.

The study of affine Deligne-Lusztig varieties (ADLVs) $X_w(b)$ and their certain union $X(\mu,b)$ has been crucial in understanding mod-$p$ reduction of Shimura varieties; for instance, precise information about the connected components of ADLVs (in the hyperspecial level) has proved to be useful in Kisin's proof of the Langlands-Rapoport conjecture. On the other hand, first introduced in the context of enumerative geometry to describe the quantum cohomology ring of complex flag varieties, quantum Bruhat graphs have found recent applications in solving certain problems on the ADLVs. I will survey such developments and report on my work surrounding a dimension formula for $X(\mu,b)$ in the quasi-split case, as well as some partial description of the dimension and top-dimensional irreducible components in the non quasi-split case.

[pre-talk at 1:20PM]

Complete Calabi-Yau metrics are fundamental objects in Kahler geometry arising as singularity models or "bubbles" ​in degenerations of compact Calabi-Yau manifolds.  The existence of these metrics and their relationship with algebraic geometry are the subjects of several long standing conjectures due to Yau and Tian-Yau.  I will describe some recent progress towards the question of existence, and explain some future directions, highlighting connections with notions of algebro-geometric stability.​