Upcoming Talks

Day showed that presheaf categories admit a symmetric monoidal structure, known as the convolution product. This structure satisfies a universal property in the ambient category of operads, where it arises as an exponential object. In this talk, we study the exponentiable objects in $\infty$-categories of operad-like structures, including $\infty$-operads, equivariant $\infty$-operads, and virtual double $\infty$-categories.

Seminar homepage: https://sites.google.com/view/question-seminar/home

In topology, cohomology is an invariant we can assign to spaces. In algebra, there are also cohomology theories for various algebraic structures, such as group cohomology, Lie algebra cohomology, and André-Quillen cohomology of commutative rings. Quillen cohomology provides a cohomology theory for any algebraic structure. It has been studied notably for divided power algebras and restricted Lie algebras, both of which are instances of divided power algebras over an operad: the commutative and Lie operad respectively.

Based on the recent result of Peter Koymans and Adam Morgan, who
proved that there are infinitely many nonisomorphic hyperelliptic curves of
any genus whose Jacobian has rank 1 over any number field, we prove the
same result for rank 2; however, our curves do not have simple Jacobians,
instead they factor into two rank 1 Jacobians. We also discuss results
about higher rank Jacobians of small genus curves. This is joint work with
Sun Woo Park.