Quoc P. Ho



Arxiv versions can be slightly out of date.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
Preprint. Last updated: March 2021.

Abstract: Motivated by spectral gluing patterns in the Betti Langlands program, we show that for any reductive group \(G\), a parabolic subgroup \(P\), and a topological surface \(M\), the (enhanced) spectral Eisenstein series category of \(M\) is the factorization homology over \(M\) of the \(\mathrm{E}_2\)-Hecke category \(\mathrm{H}_{G, P} = \mathrm{IndCoh}(\mathrm{LS}_{G, P}(D^2, S^1))\), where \(\mathrm{LS}_{G, P}(D^2, S^1)\) denotes the moduli stack of \(G\)-local systems on a disk together with a \(P\)-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 \(N\to M\), 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.

[pdf] [arXiv]

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 \(X\). First, we provide an iterative procedure to study higher representation stability using actions coming from the cohomology of \(X\) 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 \(C_\infty\)-operations on the cohomology of \(X\) 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 \(\mathrm{FI}\)-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 Ellipitic 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]

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)

Quot schemes

for the Topic Examination at UChicago

Smooth base change theorem

for the etale cohomology student seminar at UChicago

Trigonometric sums

for the etale cohomology student seminar at UChicago