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Revisiting mixed geometry

with Penghui Li

Classically, mixed versions were only constructed in very special cases due to non-semisimplicity of Frobenius. Our construction sidesteps this issue by semi-simplifying the Frobenius action itself. However, the category $\Shv_{\gr, c}(\mathcal{Y})$ agrees with those previously constructed when they are available. For example, for any reductive group $G$ with a fixed pair $T\subset B$ of a maximal torus and a Borel subgroup, we have an equivalence of monoidal $\DG$ weight categories $\Shv_{\gr, c}(B\backslash G/B) \simeq \Ch^b(\SBim_W)$, where $\Ch^b(\SBim_W)$ is the monoidal $\DG$-category of bounded chain complexes of Soergel bimodules and $W$ is the Weyl group of $G$.

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 \(\mathsfit{N}\to \mathsfit{M}\), we define \((\mathcal{Y}\mathsf{,} \mathcal{Z})^{\mathsfit{N, M}} = \mathcal{Y}^\mathsfit{N} \times_{\mathcal{Z}^\mathsfit{N}} \mathcal{Z}^\mathsfit{M}\) to be the space of maps from \(\mathsfit{M}\) to \(\mathcal{Z}\) along with a lift to \(\mathcal{Y}\) of its restriction to \(\mathsfit{N}\). Using the pair of pants construction, we define an \(\mathsf{E}_\mathsf{n}\)-category \(\mathsf{H}_\mathsf{n}(\mathcal{Y}\mathsf{,} \mathcal{Z}) = \mathsf{IndCoh}_\mathsf{0}(((\mathcal{Y}\mathsf{,} \mathcal{Z})^{\mathsf{S}^{\mathsfit{n-1}}\mathsf{,} \mathsf{D}^\mathsf{n}})^\wedge_{\mathcal{Y}})\) and compute its factorization homology on any \(\mathsfit{d}\)-dimensional manifold \(\mathsfit{M}\) with \(\mathsfit{d\leq n}\), \[\int_\mathsfit{M} \mathsf{H}_\mathsf{n}(\mathcal{Y}\mathsf{,} \mathcal{Z}) \simeq \mathsf{IndCoh}_\mathsf{0}\left(\left((\mathcal{Y}\mathsf{,} \mathcal{Z})^{\partial (\mathsfit{M}\times \mathsf{D}^{\mathsfit{n-d}})\mathsf{,} \mathsfit{M}}\right)^\wedge_{\mathcal{Y}^\mathsfit{M}}\right), \] where \(\mathsf{IndCoh_0}\) 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 elliptic curves over function fields

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

Diagrammatic Monte Carlo for electronic correlation in molecules: high-order many-body perturbation theory with low scaling

with G. Bighin, M. Lemeshko, and T. V. Tscherbul

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