arXiv subjects
Publications and source records attributed to Thomas Willwacher.
We prove that the number of curves of a fixed genus g with n marked points over finite fields is a polynomial function of the cardinality of the field if and only if g = 0 or 3g + 2n is less than or equal to 24. This confirms a conjecture of Canning, Larson, and the authors.
We prove that any homogeneous family $σ_3,σ_5,\ldots$ in $\mathfrak{grt}_1(\mathbb K)$ with nonzero coefficients of $x^{2k}y$ in $σ_{2k+1}$ generates a free Lie subalgebra, building on the work of Brown.
We show that the category of lax involutive $n$-Frobenius algebras is Quillen equivalent to the category of right comodules of the $n$-Poisson cooperad. It follows in particular, that the cohomology of a parallelized $n$-manifold is naturally endowed with a homotopy involutive $n$-Frobenius structure extending the rational homotopy type of $M$, solving a long-standing question.
We prove that the number of curves of a fixed genus g over finite fields is a polynomial function of the size of the field if and only if g is at most 8. Furthermore, we determine for each positive genus g the smallest n such that the moduli space of curves of genus g with n marked points does not have polynomial point count. A key ingredient in the proofs, which is also a new result of independent interest, is the computation of the thirteenth cohomology group of the moduli spaces of stable curves of genus g with n marked points, for all g and n.
We explain how the Johnson homomorphism and the Enomoto-Satoh trace, as well as higher-loop-order generalizations, can be obtained from graph complexes originating in the Goodwillie-Weiss calculus. This paper can be seen as an addendum to our earlier work. It contains little new mathematical content, but is intended to give an overview of a different viewpoint on the Johnson homomorphism, for experts working mainly in the latter area.
We study the action of the orthogonal group on the little $n$-disks operads. As an application we provide small models (over the reals) for the framed little $n$-disks operads. It follows in particular that the framed little $n$-disks operads are formal (over the reals) for $n$ even and coformal for all $n$.
For fixed genus g and varying finite marking set A, the gluing and forgetful maps give the spaces of holomorphic forms on the moduli space of stable A-marked curves of genus g has the structure of an FA-module, i.e., a functor from the category of finite sets to vector spaces. We prove that the resulting FA-modules of holomorphic k-forms are simple, for k less than or equal to 18, whenever they are nonzero. Conditional upon the conjectured vanishing of holomorphic 19-forms and 20-forms in genus 3, for 15 and 16 marked points, respectively, this extends to k less than or equal to 20.
We compute the Kontsevich graph cohomology in loop order 11, and for some degrees in higher loop order. We apply our results to discuss a conjecture of F. Brown, providing a counterexample to a strong version of the conjecture.
We compute numerically the dimensions of the graded quotients of the linearized Kashiwara-Vergne Lie algebra lkv in low weight, confirming a conjecture of Raphael-Schneps in those weights. The Lie algebra lkv appears in a chain of inclusions of Lie algebras, including also the linearized double shuffle Lie algebra and the (depth associated graded of the) Grothendieck-Teichmüller Lie algebra. Hence our computations also allow us to check the validity of the Deligne-Drinfeld conjecture on the structure of the Grothendieck-Teichmüller group up to weight 29, and (a version of) the the Broadhurst-Kreimer conjecture on the number of multiple zeta values for a range of weight-depth pairs significantly exceeding the previous bounds. Our computations also verify a conjecture by Alekseev-Torossian on the Kashiwara-Vergne Lie algebra up to weight 29.
We discuss the cohomology of the bridgeless graph complex, that is, the subcomplex of the Kontsevich graph complex spanned by bridgeless graphs.
We show that a smaller version of the Kontsevich graph complex spanned by triconnected graphs is quasi-isomorphic to the full Kontsevich graph complex.
We study configuration spaces of framed points on oriented closed smooth manifolds. Such configuration spaces admit natural actions of the framed little discs operads, that play an important role in the study of embedding spaces of manifolds and in factorization homology. We construct real combinatorial models for these operadic modules, for orientable closed smooth manifolds.
We extend the rational homotopy theory of operads developed by B. Fresse to several types of modules over operads.
We prove that the Kontsevich graph complex $GC_d^{2}$ and its oriented version $OGC_{d+1}^2$ are quasi-isomorphic as dg Lie algebras.
We study the appearances of $\mathsf{LS}_{12}$ and $\mathsf{S}_{16}$ in the weight-graded compactly supported cohomology of moduli spaces of curves. As applications, we prove new nonvanishing results for the middle cohomology groups of $\mathcal{M}_9$ and $\mathcal{M}_{11}$ and give evidence to support the conjecture that the dimension fo $H^{2g + k}_c(\mathcal{M}_g)$ grows at least exponentially with $g$ for almost all $k$.
We study the weight 2 graded piece of the compactly supported rational cohomology of the moduli spaces of curves $M_{g,n}$ and show that this can be computed as the cohomology of a graph complex that is closely related to graph complexes arising in the study of embedding spaces. For $n = 0$, we express this cohomology in terms of the weight zero compactly supported cohomology of $M_{g',n'}$ for $g' \leq g$ and $n' \leq 2$, and thereby produce several new infinite families of nonvanishing unstable cohomology groups on $M_g$, including the first such families in odd degrees. In particular, we show that the dimension of $H^{4g-k}(M_g)$ grows at least exponentially with $g$, for $k \in \{ 8, 9, 11, 12, 14, 15, 16, 18, 19 \}$.
The real locus of the moduli space of stable genus-zero curves with marked points, $\overline{{\mathcal M}_{0,{n+1}}}({\mathbb R})$, is known to be a smooth manifold and is the Eilenberg-MacLane spaces for the so-called pure Cactus groups. We describe the operad formed by these spaces in terms of a homotopy quotient of an operad of associative algebras. Using this model, we identify various Hopf models for the algebraic operad of chains and homologies of $\overline{{\mathcal M}_{0,{n+1}}}({\mathbb R})$. In particular, we show that the operad $\overline{{\mathcal M}_{0,{n+1}}}({\mathbb R})$ is not formal. As an application of these operadic constructions, we prove that for each $n$, the cohomology ring $H^{\bullet}(\overline{{\mathcal M}_{0,{n+1}}}({\mathbb R}), {\mathbb{Q}})$ is a Koszul algebra, and that the manifold $\overline{{\mathcal M}_{0,{n+1}}}({\mathbb R})$ is not formal for $n\geq 6$ but is a rational $K(π,1)$-space. Additionally, we describe the Lie algebras associated with the lower central series filtration of the pure Cactus groups.
Effective and timely feedback in educational assessments is essential but labor-intensive, especially for complex tasks. Recent developments in automated feedback systems, ranging from deterministic response grading to the evaluation of semi-open and open-ended essays, have been facilitated by advances in machine learning. The emergence of pre-trained Large Language Models, such as GPT-4, offers promising new opportunities for efficiently processing diverse response types with minimal customization. This study evaluates the effectiveness of a pre-trained GPT-4 model in grading semi-open handwritten responses in a university-level mathematics exam. Our findings indicate that GPT-4 provides surprisingly reliable and cost-effective initial grading, subject to subsequent human verification. Future research should focus on refining grading rules and enhancing the extraction of handwritten responses to further leverage these technologies.