Schedule, Notes/Handouts & Videos

Given a square system of real polynomials, we should never expect all solutions to the system to be real. Remarkably, a family of such solutions emerges in geometry from simple Schubert problems osculating rational normal curves (this is a deep and recent theorem previously known as the Shapiro-Shapiro conjecture). This has been a central topic in real algebraic geometry the past few decades, and reduces in many cases to the study of a map of real algebraic manifolds called the Wronski map. This map is not oriented in many cases, which resists our attempts to approach it through the lens of quadratically enriched enumerative geometry. We introduce the idea of twisted A1-degrees and A1-chambers, which allow us to repair conservation of number in many cases when orientability hypotheses may fail. We apply these ideas to the study of the Wronski map, exploring its twisted degrees, which unifies work from many authors. This is based on joint work with Jake Levinson.

I will discuss recent joint work with Satriano and Sundbo, in which we use non-A¹-invariant motivic homotopy theory to construct new realization homomorphisms from the Grothendieck ring of varieties in positive characteristic, conditional on resolution of singularities. These include realizations associated with Hodge polynomials, among others. These new realization homomorphisms reveal previously unknown properties of the Grothendieck ring of varieties. For example, we show that universally homeomorphic varieties need not have the same motivic class, and that the classes of curves of different genus are pairwise algebraically independent. Under the same assumption, we also show that K-equivalent varieties have the same Hodge numbers in positive characteristic.

The category of strictly $\mathbb{A}^1$-invariant sheaves is central to the development of $\mathbb{A}^1$-algebraic topology over a field. It is a nontrivial fact that this is an abelian, symmetric monoidal category. I will outline a fundamental computation in this category, namely that of the tensor product of the Milnor-Witt K-theory sheaf with the so-called free strictly $\mathbb{A}^1$-invariant sheaf on a point, defined over a finite separable field extension of the base field. The computation is made possible by the use of cellular $\mathbb{A}^1$-homology. If time permits, I will discuss a few applications of our main results.

This talk is based on joint work with Fabien Morel.

Beilinson's conjectures predict that special values of L-functions are periods. In the case of number fields, Zagier verified numerically that these periods should arise more precisely as special values of polylogarithms. Following Beilinson's philosophy of regulators, this suggested that polylogarithms should come from motivic cohomology, and some instances of these "motivic polylogarithms" were indeed first constructed by Beilinson, Deligne, and Soulé. In this talk, I want to explain a new construction of the polylogarithm functions in algebraic K-theory and in motivic cohomology, which naturally recovers the motivic classes of Beilinson--Deligne--Soulé as a special case, and which also suggests a surprising relation to chromatic homotopy theory. This is a joint work in progress with Arpon Raksit, partly based on a joint work with Benjamin Antieau, Federico Binda, and Brian Shin.

In this talk, I will introduce the category of real isotropic motivic spectra, obtained from SH(R) by annihilating anisotropic quadrics. Then, I will show that its cellular subcategory can be identified with F2-synthetic spectra. Finally, I will use this identification to study the real isotropic localization of algebraic symplectic cobordism.

Modularity of theta functions is an important cornerstone in the theory of automorphic forms and L-functions. A few years ago, joint work of myself with Zhiwei Yun and Wei Zhang constructed objects called higher theta functions, and conjectured their modularity. We recently completed the proof of this conjecture. So far this sounds like an abstract for a number theory talk, but the proof involves a crucial use of motivic homotopy theory, and I will highlight that. Specifically, motivic homotopy theory enters through a formalism that we call the “sheaf-cycle correspondence”, an enhancement of Grothendieck’s sheaf-function correspondence.