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Jeremy Miller

Publications and source records attributed to Jeremy Miller.

At least 19 recordsLinked to original sources
arXiv2026

Codimensions one and two cohomology of Hecke congruence subgroups

For $n\geq 1$ and $p$ a prime, the Hecke congruence subgroup $\Gamma_{0,n}(p)\leq \mathrm{SL}_n(\mathbb{Z})$ is the subgroup of matrices whose first column is of the form $(*,0,\dots,0)^t\bmod p$. Borel--Serre showed that $\Gamma_{0,n}(p)$ has virtual cohomological dimension $\binom{n}{2}$. The first author proved that the rational cohomology in this top degree $\binom{n}{2}$ vanishes for $n$ sufficiently large compared to $p$. We prove analogous results in codimension $1$ and $2$.

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arXiv2026

Scissors automorphism groups II: Solomon-Tits theorems

The Solomon-Tits theorem says that the poset of proper non-trivial subspaces of a finite-dimensional vector space has realisation equivalent to a wedge of spheres. In this paper we prove a variant of this result for collections of geodesic subspaces of Euclidean, hyperbolic, or spherical geometry, assuming the collection is generated either by points or by hyperplanes. In the third paper of this series of papers, we will combine this with the homological stability theorems from the first paper to compute the homology of groups of scissors automorphisms in these geometries.

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arXiv2025

An analogue of Rognes' connectivity conjecture for free groups

We show that the common basis complex of a free group of rank $n$ has the homotopy type of a wedge of spheres of dimension $2n-3$. This establishes an $\mathrm{Aut}(F_n)$-analogue of the connectivity conjecture that Rognes originally stated for $\mathrm{GL}_n(R)$. To prove this, we provide several homotopy-equivalent models of the common basis complex, both in terms of free factors in free groups and in terms of sphere systems in 3-manifolds.

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arXiv2025

Representation stability for ordered Hurwitz spaces

In this paper, we study the topology of ordered Hurwitz space. These are moduli spaces of branched covers with a choice of ordering on the branched points. Answering a question of Ellenberg, we prove that the homology of ordered Hurwitz spaces exhibit representation stability.

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arXiv2025

Homological vanishing for the Steinberg representation II: reductive groups and integral conjectures

We prove that the homology groups of any connected reductive group over a field with coefficients in the Steinberg representation vanish in a range. The generalizes work of Ash-Putman-Sam on the classical split groups. We state a connectivity conjecture that would allow us to prove such a vanishing result for $SL_n(\mathbb{Z})$, as was conjectured by Church-Farb-Putman. We prove some special cases of this conjecture and use it to refine known results about the first and second of homology of $SL_n(\mathbb{Z})$ with Steinberg coefficients.

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arXiv2024

Scissors automorphism groups and their homology

In any category with a reasonable notion of cover, each object has a group of scissors automorphisms. We prove that under mild conditions, the homology of this group is independent of the object, and can be expressed in terms of the scissors congruence K-theory spectrum defined by Zakharevich. We therefore obtain both a group-theoretic interpretation of Zakharevich's higher scissors congruence K-theory, as well as a method to compute the homology of scissors automorphism groups. We apply this to various families of groups, such as interval exchange groups and Brin--Thompson groups, recovering results of Szymik--Wahl, Li, and Tanner, and obtaining new results as well.

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arXiv2024

A criterion for slope 1 homological stability

We show that for nice enough $\mathbb{N}$-graded $\mathbb{E}_2$-algebras, a diagonal vanishing line in $\mathbb{E}_1$-homology of gives rise to slope $1$ homological stability. This is an integral version of a result by Kupers-Miller-Patzt.

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arXiv2024

Comparison of 4.5PN and 2SF gravitational energy fluxes from quasicircular compact binaries

Recent years have seen significant advances in models of gravitational waveforms emitted by quasicircular compact binaries in two regimes: the weak-field, post-Newtonian regime, in which the gravitational wave energy flux has now been calculated to fourth-and-a-half post-Newtonian order (4.5PN) [Phys. Rev. Lett. 131}, 121402 (2023)]; and the small-mass-ratio, gravitational self-force regime, in which the flux has now been calculated to second perturbative order in the mass ratio (2SF) [Phys. Rev. Lett. 127, 151102 (2021)]. We compare these results and find agreement, showing consistency between the two (very distinct though both first-principle) perturbative calculations.

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arXiv2024

Partial bases and homological stability of $\operatorname{GL}_{n}(R)$ revisited

Let $R$ be a unital ring satisfying the invariant basis number property, that every stably free $R$-module is free, and that the complex of partial bases of every finite rank free module is Cohen--Macaulay. This class of rings includes every ring of stable rank $1$ (e.g. any local, semi-local or Artinian ring), every Euclidean domain, and every Dedekind domain $\mathcal{O}_S$ of arithmetic type where $|S| > 1$ and $S$ contains at least one non-complex place. Extending recent work of Galatius--Kupers--Randal-Williams and Kupers--Miller--Patzt, we prove that the sequence of general linear groups $\operatorname{GL}_n(R)$ satisfies slope-$1$ homological stability with $\mathbb{Z}[1/2]$-coefficients.

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arXiv2024

Uniform twisted homological stability

We prove a homological stability theorem for families of discrete groups (e.g. mapping class groups, automorphism groups of free groups, braid groups) with coefficients in a sequence of irreducible algebraic representations of arithmetic groups. The novelty is that the stable range is independent of the choice of representation. Combined with earlier work of Bergstr\"om--Diaconu--Petersen--Westerland this proves the Conrey--Farmer--Keating--Rubinstein--Snaith predictions for all moments of the family of quadratic $L$-functions over function fields, for sufficiently large odd prime powers.

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arXiv2023

Worldtube puncture scheme for first- and second-order self-force calculations in the Fourier domain

Second-order gravitational self-force theory has recently led to the breakthrough calculation of ``first post-adiabatic'' (1PA) compact-binary waveforms [Phys. Rev. Lett. 130, 241402 (2023)]. The computations underlying those waveforms depend on a method of solving the perturbative second-order Einstein equation on a Schwarzschild background in the Fourier domain. In this paper we present that method, which involves dividing the domain into several regions. Different regions utilize different time slicings and allow for the use of ``punctures'' to tame sources and enforce physical boundary conditions. We demonstrate the method for Lorenz-gauge and Teukolsky equations in the relatively simple case of calculating parametric derivatives (``slow time derivatives'') of first-order fields, which are an essential input at second order.

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arXiv2023

Accuracy Requirements: Assessing the Importance of First Post-Adiabatic Terms for Small-Mass-Ratio Binaries

We investigate the impact of post-adiabatic (1PA) terms on parameter estimation for extreme and intermediate mass-ratio inspirals using state-of-the-art waveform models. Our analysis is the first to employ Bayesian inference to assess systematic errors for 1PA waveforms. We find that neglecting 1PA terms introduces significant biases for the (small) mass ratio $\epsilon \gtrsim 10^{-5}$ for quasi circular orbits in Schwarzschild spacetime, which can be mitigated with resummed 3PN expressions at 1PA order. Moreover, we show that the secondary spin is strongly correlated with the other intrinsic parameters, and it can not be constrained for $\epsilon \lesssim 10^{-5}$. Finally, we highlight the need for addressing eccentric waveform systematics in the small-mass-ratio regime, as they yield stronger biases than the circular limit in both intrinsic and extrinsic parameters.

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arXiv2023

Enhancing the SEOBNRv5 effective-one-body waveform model with second-order gravitational self-force fluxes

We leverage recent breakthrough calculations using second-order gravitational self-force (2GSF) theory to improve both the gravitational-mode amplitudes and radiation-reaction force in effective-one-body~(EOB) waveform models. We achieve this by introducing new calibration parameters in the SEOBNRv5HM mode amplitudes, and matching them to the newly available 2GSF energy-flux multipolar data for quasicircular nonspinning binary black holes. We find that this significantly improves the SEOBNRv5HM energy flux, when compared to numerical-relativity (NR) simulations of binary black holes with mass ratios between 1:1 and 1:20. Moreover, we find that, once the conservative part of the SEOBNRv5 dynamics is calibrated, the SEOBNRv5HM waveform model with 2GSF information reproduces the binding energy of NR simulations more accurately, providing a powerful check of the consistency and naturalness of the EOB approach. While we only include nonspinning 2GSF information, the more accurate binding energy and energy flux carry over to the SEOBNRv5 waveform models for spinning binary black holes. Thus, our results improve the latest generation of SEOBNR waveform models (i.e., SEOBNRv5), which has been recently completed for use in the upcoming fourth observing (O4) run of the LIGO-Virgo-KAGRA Collaboration.

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arXiv2023

Polynomial stability of the homology of Hurwitz spaces

For a finite group $G$ and a conjugation-invariant subset $Q\subseteq G$, we consider the Hurwitz space $\mathrm{Hur}_n(Q)$ parametrising branched covers of the plane with $n$ branch points, monodromies in $G$ and local monodromies in $Q$. For $i\ge0$ we prove that $\bigoplus_n H_i(\mathrm{Hur}_n(Q))$ is a finitely generated module over the ring $\bigoplus_n H_0(\mathrm{Hur}_n(Q))$. As a consequence, we obtain polynomial stability of homology of Hurwitz spaces: taking homology coefficients in a field, the dimension of $H_i(\mathrm{Hur}_n(Q))$ agrees for $n$ large enough with a quasi-polynomial in $n$, whose degree is easily bounded in terms of $G$ and $Q$. Under suitable hypotheses on $G$ and $Q$, we prove classical homological stability for certain sequences of components of Hurwitz spaces. Our results generalise previous work of Ellenberg-Venkatesh-Westerland, and rely on techniques introduced by them and by Hatcher-Wahl.

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arXiv2023

On rank filtrations of algebraic K-theory and Steinberg modules

Motivated by his work on the stable rank filtration of algebraic K-theory spectra, Rognes defined a simplicial complex called the common basis complex and conjectured that this complex is highly connected for local rings and Euclidean domains. We prove this conjecture in the case of fields. Our methods give a novel description of this common basis complex of a PID as an iterated bar construction on an equivariant monoid built out of Tits buildings. We also identify the Koszul dual of a certain equivariant ring assembled out of Steinberg modules.

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