Fall 2026

This is an expository talk based on work of Déglise-Jin-Khan and Levine-Raksit. The classical Gauss-Bonnet formula states that the integral of the Euler class of the tangent bundle of a smooth, closed, oriented manifold agrees with its Euler characteristic. After introducing the categorical Euler characteristic and (twisted) Euler class, we discuss a motivic analogue of this formula, valid for any motivic commutative ring spectrum, and make some calculations along the way.

The solution of the Milnor conjectures on quadratic forms provides a complete set of invariants for quadratic forms. One would like to use this classification to attack a classical problem on the u-invariant of function fields of curves over totally imaginary number fields. The u-invariant is the maximum dimension of anisotropic quadratic forms over a field. Even finiteness is not known in this case. We explain how the finiteness of the u-invariant is reduced to bounding indices of the unramified Brauer group of the field. We run through the history and state some results in this direction.

Wonderful models of subspace arrangements, in the sense of De Concini and Procesi, provide smooth compactifications of arrangement complements and include, as notable examples, moduli spaces of curves. In this talk, we investigate the A1-homotopy-theoretic (motivic) properties of these models and develop tools for computing their cellular A1-homology. Our approach expresses the resulting invariants in terms of combinatorial data associated to the underlying arrangement.

Oriented cohomology theories provide a general framework to perform intersection-theory style calculations. Prototypical examples include Chow ring, $K$-theory, and algebraic cobordism, while Annala--Hoyois--Iwasa recently constructed the category $\mathsf{MS}_S$ for studying other non-$\mathbb A^1$-invariant theories. In this talk, we prove a blowup formula for oriented cohomology theories in $\mathsf{MS}_S$. We then give a ring presentation version of this formula and use it to compute oriented cohomology rings of blowups such as the moduli space of complete conics and $\overline M_{0,n}$. In particular, we demonstrate that one can recover the number $3264$ of conics simultaneously tangent to five given ones in $\mathbb P^2$ in algebraic cobordism. This is based on joint work with Arkamouli Debnath.

We'll provide a leisurely introduction to the basic ideas of enumerative geometry, following its history and development from Ancient Greece to the present day. We will explore Poncelet and Schubert's principle of "conservation of number," and see how it breaks over non-algebraically closed fields. The second half of the talk will explore how methods from motivic homotopy theory can be leveraged to repair conservation of number and provide quadratically enriched answers to classical enumerative questions. This talk will introduce some of the key players in enumerative geometry, including the Grothendieck-Witt ring of a field and quadratic Euler classes.