Arxiv versions can be slightly out of date.

For preprints, the latest versions can be accessed via [pdf] links.

Eisenstein series via factorization homology of Hecke categories

with Penghui Li

More generally, for any pair of stacks \(\mathcal{Y}\to \mathcal{Z}\) satisfying some mild conditions and any
map between topological spaces \(N\to M\), we define \((\mathcal{Y}, \mathcal{Z})^{N, M} = \mathcal{Y}^N
\times_{\mathcal{Z}^N} \mathcal{Z}^M\) to be the space of maps from \(M\) to \(\mathcal{Z}\) along with a lift
to
\(\mathcal{Y}\) of its restriction to \(N\). Using the pair of pants construction, we define an
\(\mathrm{E}_n\)-category \(\mathrm{H}_n(\mathcal{Y}, \mathcal{Z}) =
\mathrm{IndCoh}_0\left(\left((\mathcal{Y},
\mathcal{Z})^{S^{n-1}, D^n}\right)^\wedge_{\mathcal{Y}}\right)\) and compute its factorization homology on
any \(d\)-dimensional manifold \(M\) with \(d\leq n\), \[
\int_M \mathrm{H}_n(\mathcal{Y}, \mathcal{Z}) \simeq \mathrm{IndCoh}_0\left(\left((\mathcal{Y},
\mathcal{Z})^{\partial (M\times D^{n-d}), M}\right)^\wedge_{\mathcal{Y}^M}\right), \] where
\(\mathrm{IndCoh}_0\) is the sheaf theory introduced by Arinkin--Gaitsgory and Beraldo. Our result naturally
extends previous known computations of Ben-Zvi--Francis--Nadler and Beraldo.

Higher representation stability for ordered configuration spaces and twisted commutative factorization algebras

The Atiyah–Bott formula and connectivity in chiral Koszul duality

Homological stability and densities of generalized configuration spaces

Free factorization algebras and homology of configuration spaces in algebraic geometry

Average Size of 2-Selmer Groups of Ellipitic Curves over Function Fields

with B.C. Ngo and B.V.H Le

These notes can be somewhat sketchy (i.e. use at your own risk). If you have any comments (mathematical or otherwise), please let me know.

Factorization algebras and categegories

Quot schemes

Smooth base change theorem

Trigonometric sums