# Quoc P. Ho

## Papers/Preprints

### Mathematics


arXiv versions can be slightly out of date.
Click on titles to show/hide abstracts.

Revisiting mixed geometry

with Penghui Li
Preprint. Last updated: August 2022.

Abstract: We provide a uniform construction of “mixed versions” or “graded lifts” in the sense of Beilinson–Ginzburg–Soergel which works for arbitrary Artin stacks. In particular, we obtain a general construction of graded lifts of many categories arising in geometric representation theory and categorified knot invariants. Our new theory associates to each Artin stack of finite type $\mathcal{Y}$ over $\Fqbar$ a symmetric monoidal $\DG$-category $\Shv_{\gr, c}(\mathcal{Y})$ of constructible graded sheaves on $\mathcal{Y}$ along with the six-functor formalism, a perverse $t$-structure, and a weight (or co-$t$-)structure in the sense of Bondarko and Pauksztello, compatible with the six-functor formalism, perverse $t$-structures, and Frobenius weights on the category of (mixed) $\ell$-adic sheaves.

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 $G$.

[pdf] [arXiv]

Eisenstein series via factorization homology of Hecke categories

with Penghui Li
Advances in Mathematics, Vol. 404, Part A (Aug. 2022), 32 pages.

Abstract: Motivated by spectral gluing patterns in the Betti Langlands program, we show that for any reductive group $$\mathsfit{G}$$, a parabolic subgroup $$\mathsfit{P}$$, and a topological surface $$\mathsfit{M}$$, the (enhanced) spectral Eisenstein series category of $$\mathsfit{M}$$ is the factorization homology over $$\mathsfit{M}$$ of the $$\mathsf{E_2}$$-Hecke category $$\mathsf{H}_{\mathsfit{G}, \mathsfit{P}} = \mathsf{IndCoh}(\mathsf{LS}_{\mathsfit{G, P}}(\mathsf{D^2, S^1}))$$, where $$\mathsf{LS}_{\mathsfit{G, P}}(\mathsf{D^2, S^1})$$ denotes the moduli stack of $$\mathsfit{G}$$-local systems on a disk together with a $$\mathsfit{P}$$-reduction on the boundary circle.

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.

[pdf] [arXiv] [journal]

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

Preprint. Last updated: October 2020.

Abstract: Using factorization homology with coefficients in twisted commutative algebras (TCAs), we prove two flavors of higher representation stability for the cohomology of (generalized) configuration spaces of a scheme/topological space $$\mathsfit{X}$$. First, we provide an iterative procedure to study higher representation stability using actions coming from the cohomology of $$\mathsfit{X}$$ and prove that all the modules involved are finitely generated over the corresponding TCAs. More quantitatively, we compute explicit bounds for the derived indecomposables in the sense of Galatius–Kupers–Randal-Williams. Secondly, we prove that when certain $$\mathsf{C}_\infty$$-operations on the cohomology of $$\mathsfit{X}$$ vanish, the cohomology of its configuration spaces form a free module over a twisted commutative algebra built out of the configuration spaces of the affine space. This generalizes a result of Church–Ellenberg–Farb on the freeness of $$\mathsf{FI}$$-modules arising from the cohomology of configuration spaces of open manifolds and, moreover, resolves the various conjectures of Miller–Wilson in this case.

[pdf] [arXiv]

The Atiyah–Bott formula and connectivity in chiral Koszul duality

Advances in Mathematics, Vol. 392 (Dec. 2021), 71 pages.

Abstract: The $$\otimes^\star$$- structure on the category of sheaves on the Ran space is not pro-nilpotent in the sense of Francis–Gaitsgory. However, under some connectivity assumptions, we prove that Koszul duality induces an equivalence of categories and that this equivalence behaves nicely with respect to Verdier duality on the Ran space and integrating along the Ran space, i.e. taking factorization homology. Based on ideas sketched by Gaitsgory, we show that these results also offer a simpler alternative to one of the two main steps in the proof of the Atiyah–Bott formula.

[pdf] [arXiv] [journal]

Homological stability and densities of generalized configuration spaces

Geometry & Topology, Vol. 25 (2021), No. 2, pp. 813–912 (100 pages).

Abstract: We prove that the factorization homologies of a scheme with coefficients in truncated polynomial algebras compute the cohomologies of its generalized configuration spaces. Using Koszul duality between commutative algebras and Lie algebras, we obtain new expressions for the cohomologies of the latter. As a consequence, we obtain a uniform and conceptual approach for treating homological stability, homological densities, and arithmetic densities of generalized configuration spaces. Our results categorify, generalize, and in fact provide a conceptual understanding of, the coincidences appearing in the work of Farb–Wolfson–Wood. Our computation of the stable homological densities also yields rational homotopy types which answer a question posed by Vakil–Wood. Our approach hinges on the study of homological stability of cohomological Chevalley complexes, which is of independent interest.

[pdf] [arXiv] [journal]

Free factorization algebras and homology of configuration spaces in algebraic geometry

Selecta Mathematica (N.S.), Vol. 23 (2017), No. 6, pp. 2437–2489 (53 pages).

Abstract: We provide a construction of free factorization algebras in algebraic geometry and link factorization homology of a scheme with coefficients in a free factorization algebra to the homology of its (unordered) configuration spaces. As an application, this construction allows for a purely algebro-geometric proof of homological stability of configuration spaces.

[pdf] [arXiv] [journal]

Average size of 2-Selmer groups of elliptic curves over function fields

with B.C. Ngo and B.V.H Le
Mathematical Research Letters, Vol. 21 (2014), No. 6, pp. 1305–1339 (35 pages).

Abstract: Employing a geometric setting inspired by the proof of the Fundamental Lemma, we study some counting problems related to the average size of 2-Selmer groups and hence obtain an estimate for it.

[pdf] [arXiv] [journal]

### Physics

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
Preprint. Last updated: March 2022.

Abstract: We present a low-scaling diagrammatic Monte Carlo approach to molecular correlation energies. Using combinatorial graph theory to encode many-body Hugenholtz diagrams, we sample the Møller-Plesset (MPn) perturbation series, obtaining accurate correlation energies up to $\mathsf{n=5}$, with quadratic scaling in the number of basis functions. Our technique reduces the computational complexity of the molecular many-fermion correlation problem, opening up the possibility of low-scaling, accurate stochastic computations for a wide class of many-body systems described by Hugenholtz diagrams.

[arXiv]

## Expository Notes

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

for the conference in local geometric Langlands (Paris, Jan. 2018)
[pdf]

Quot schemes

for the Topic Examination at UChicago
[pdf]

Smooth base change theorem

for the etale cohomology student seminar at UChicago
[pdf]

Trigonometric sums

for the etale cohomology student seminar at UChicago
[pdf]