346 citations · 1.2k across the 23 of their papers we have counts for
25 papers
Which groups are amenable to proving exponent two for matrix multiplication?
Jonah Blasiak, Thomas Church, Henry Cohn +2
The Cohn-Umans group-theoretic approach to matrix multiplication suggests embedding matrix multiplication into group algebra multiplication, and bounding in terms of the repres…
On finite generation of the Johnson filtrations
Thomas Church, Mikhail Ershov, Andrew Putman
We prove that every term of the lower central series and Johnson filtrations of the Torelli subgroups of the mapping class group and the automorphism group of a free group is finit…
Linear and quadratic ranges in representation stability
Thomas Church, Jeremy Miller, Rohit Nagpal +1
We prove two general results concerning spectral sequences of -modules. These results can be used to significantly improve stable ranges in a large portion of the stab…
Generating the Johnson filtration II: finite generation
Thomas Church, Andrew Putman
We prove that every term of the lower central series and Johnson filtrations of the Torelli subgroups of the mapping class group and the automorphism group of a free group are fini…
Bounding the homology of FI-modules
Thomas Church
The main theorem of Church-Ellenberg [arXiv:1506.01022] is a sharp bound on the homology of FI-modules, showing that the Castelnuovo-Mumford regularity of FI-modules over Z can be…
On cap sets and the group-theoretic approach to matrix multiplication
Jonah Blasiak, Thomas Church, Henry Cohn +4
In 2003, Cohn and Umans described a framework for proving upper bounds on the exponent of matrix multiplication by reducing matrix multiplication to group algebra multiplicatio…
The codimension-one cohomology of SL_n Z
Thomas Church, Andrew Putman
We prove that H^{d-1}(SL_n Z; Q) = 0, where d = n-choose-2 is the cohomological dimension of SL_n Z, and similarly for GL_n Z. We also prove analogous vanishing theorems for cohomo…
Homology of FI-modules
Thomas Church, Jordan S. Ellenberg
We prove an explicit and sharp upper bound for the Castelnuovo-Mumford regularity of an FI-module V in terms of the degrees of its generators and relations. We use this to refine a…
Integrality in the Steinberg module and the top-dimensional cohomology of SL_n(O_K)
Thomas Church, Benson Farb, Andrew Putman
We prove a new structural result for the spherical Tits building attached to SL_n(K) for many number fields K, and more generally for the fraction fields of many Dedekind domains O…
Generating the Johnson filtration
Thomas Church, Andrew Putman
For k >= 1, let Torelli_g^1(k) be the k-th term in the Johnson filtration of the mapping class group of a genus g surface with one boundary component. We prove that for all k, ther…
Representation stability in cohomology and asymptotics for families of varieties over finite fields
Thomas Church, Jordan S. Ellenberg, Benson Farb
We consider two families X_n of varieties on which the symmetric group S_n acts: the configuration space of n points in C and the space of n linearly independent lines in C^n. Give…
Rotor-routing and spanning trees on planar graphs
Melody Chan, Thomas Church, Joshua A. Grochow
The sandpile group Pic^0(G) of a finite graph G is a discrete analogue of the Jacobian of a Riemann surface which was rediscovered several times in the contexts of arithmetic geome…
FI-modules over Noetherian rings
Thomas Church, Jordan S. Ellenberg, Benson Farb +1
FI-modules were introduced by the first three authors in [CEF] to encode sequences of representations of symmetric groups. Over a field of characteristic 0, finite generation of an…
A stability conjecture for the unstable cohomology of SL_n Z, mapping class groups, and Aut(F_n)
Thomas Church, Benson Farb, Andrew Putman
In this paper we conjecture the stability and vanishing of a large piece of the unstable rational cohomology of SL_n Z, of mapping class groups, and of Aut(F_n).
FI-modules and stability for representations of symmetric groups
Thomas Church, Jordan S. Ellenberg, Benson Farb
In this paper we introduce and develop the theory of FI-modules. We apply this theory to obtain new theorems about: - the cohomology of the configuration space of n distinct ordere…
Invariance properties of Miller-Morita-Mumford characteristic numbers of fibre bundles
Thomas Church, Martin Crossley, Jeffrey Giansiracusa
Characteristic classes of fibre bundles in the category of closed oriented manifolds give rise to characteristic numbers by integrating the classes over the base.…
Orbits of curves under the Johnson kernel
Thomas Church
This paper has two main goals. First, we give a complete, explicit, and computable solution to the problem of when two simple closed curves on a surface are equivalent under the Jo…
The rational cohomology of the mapping class group vanishes in its virtual cohomological dimension
Thomas Church, Benson Farb, Andrew Putman
Let Mod_g be the mapping class group of a genus g >= 2 surface. The group Mod_g has virtual cohomological dimension 4g-5. In this note we use a theorem of Broaddus and the combinat…
Homological stability for configuration spaces of manifolds
Thomas Church
Let C_n(M) be the configuration space of n distinct ordered points in M. We prove that if M is any connected orientable manifold (closed or open), the homology groups H_i(C_n(M); Q…
On the geometric nature of characteristic classes of surface bundles
Thomas Church, Benson Farb, Matthew Thibault
Each Morita--Mumford--Miller (MMM) class e_n assigns to each genus g >= 2 surface bundle S_g -> E^{2n+2} -> M^{2n} an integer e_n^#(E -> M) := <e_n,[M]> in Z. We prove that when n…
Representation theory and homological stability
Thomas Church, Benson Farb
We introduce the idea of *representation stability* (and several variations) for a sequence of representations V_n of groups G_n. A central application of the new viewpoint we intr…
Parametrized Abel-Jacobi maps and abelian cycles in the Torelli group
Thomas Church, Benson Farb
Let I_g,* denote the (pointed) Torelli group. This is the group of homotopy classes of homeomorphisms of the genus g >= 2 surface S_g with a marked point, acting trivially on H :=…
Infinite generation of the kernels of the Magnus and Burau representations
Thomas Church, Benson Farb
Consider the kernel Mag_g of the Magnus representation of the Torelli group and the kernel Bur_n of the Burau representation of the braid group. We prove that for g >= 2 and for n…
Some groups of mapping classes not realized by diffeomorphisms
Mladen Bestvina, Thomas Church, Juan Souto
Let S be a closed surface of genus g >= 2 and z in S a marked point. We prove that the subgroup of the mapping class group Map(S,z) corresponding to the fundamental group pi_1(S,z)…