Searcharxiv⌕ Search

arXiv subjects

Thomas Willwacher

Publications and source records attributed to Thomas Willwacher.

At least 37 records · Page 2Linked to original sources
arXiv2024

The motivic structures $\mathsf{LS}_{12}$ and $\mathsf{S}_{16}$ in the cohomology of moduli spaces of curves

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 $k$.

math.AG↗
arXiv2024

Weight two compactly supported cohomology of moduli spaces of curves

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 $k \in \{ 8, 9, 11, 12, 14, 15, 16, 18, 19 \}$.

math.AG↗
arXiv2024

Real moduli space of stable rational curves revised

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.

math.AT↗
arXiv2024

AI-assisted Automated Short Answer Grading of Handwritten University Level Mathematics Exams

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.

math.HO↗
arXiv2024

Weight 2 cohomology of graph complexes of cyclic operads and the handlebody group

We compute the weight 2 cohomology of the Feynman transforms of the cyclic (co)operads $\mathsf{BV}$ and $\mathsf{HyCom}$, and the top$-2$ weight cohomology of the Feynman transforms of $D\mathsf{BV}$ and $\mathsf{Grav}$. Using a result of Giansiracusa, we compute, in particular, the top$-2$ weight cohomology of the handlebody group. We compare the result to the top$-2$ weight cohomology of the moduli space of curves $\mathcal{M}_{g,n}$, recently computed by Payne and the last-named author. We also provide another proof of a recent result of Hainaut-Petersen identifying the top weight cohomology of the handlebody group with the Kontsevich graph cohomology.

math.QA↗
arXiv2024

Cyclic operads through modules

We describe a way to compute mapping spaces of cyclic operads through modules. As an application we compute the homotopy automorphism space of the cyclic Batalin-Vilkovisky (Hopf co-)operad.

math.AT↗
arXiv2024

Automorphisms of framed operads

Let $\mathsf P$ be an operad acted upon by a group $G$, and let $\mathsf Q=\mathsf P\rtimes G$ be the corresponding framed operad. We relate the homotopy automorphism groups of $\mathsf P$ and $\mathsf Q$. We apply the result to compute the automorphisms of the framed little disks operad.

math.AT↗
arXiv2023

Graph homology computations

We compute numerically the homology of several graph complexes in low loop orders, extending previous results.

math.QA↗
arXiv2023

Weight 11 compactly supported cohomology of moduli spaces of curves

We study the weight 11 part of the compactly supported cohomology of the moduli space of curves $M_{g,n}$, using graph complex techniques, with particular attention to the case $n = 0$. As applications, we prove new nonvanishing results for the cohomology of $M_g$, and exponential growth with $g$, in a wide range of degrees.

math.AG↗
arXiv2023

Cyclic model for the dg dual of the BV operad

We describe a new small cyclic operad model QBV for the dg dual operad of the Batalin-Vilkovisky operad. As an application, we show that the Feynman transform of BV is quasi-isomorphic to the amputated Feynman transform of the dg dual, in complementary weights and degrees.

math.QA↗
arXiv2023

Configuration Spaces of Manifolds with Boundary

We study ordered configuration spaces of compact manifolds with boundary. We show that for a large class of such manifolds, the real homotopy type of the configuration spaces only depends on the real homotopy type of the pair consisting of the manifold and its boundary. We moreover describe explicit real models of these configuration spaces using three different approaches. We do this by adapting previous constructions for configuration spaces of closed manifolds which relied on Kontsevich's proof of the formality of the little disks operads. We also prove that our models are compatible with the richer structure of configuration spaces, respectively a module over the Swiss-Cheese operad, a module over the associative algebra of configurations in a collar around the boundary of the manifold, and a module over the little disks operad.

math.AT↗
arXiv2022

Stable cohomology of graph complexes in odd dimensions

We study graph complexes related to configuration spaces and diffeomorphism groups of highly connected manifolds of odd dimension. In particular we compute the cohomology in the "high genus" limit. This paper is a continuation of previous work by Felder, Naef and the second author in which the even dimensional case is studied.

math.QA↗
arXiv2022

A model for configuration spaces of points

We construct a real combinatorial model for the configuration spaces of points of compact smooth oriented manifolds without boundary. We use these models to show that the real homotopy type of configuration spaces of a simply connected such manifold only depends on the real homotopy type of the manifold. Moreover, we show that for framed $D$-dimensional manifolds these models capture a natural right homotopy action of the little $D$-disks operad.

math.QA↗
arXiv2021

Stable cohomology of graph complexes

We study three graph complexes related to the higher genus Grothendieck-Teichmüller Lie algebra and diffeomorphism groups of manifolds. We show how the cohomology of these graph complexes is related, and we compute the cohomology as the genus $g$ tends to $\infty$. As a byproduct, we find that the Malcev completion of the genus $g$ mapping class group relative to the symplectic group is Koszul in the stable limit (partially answering a question of Hain). Moreover, we obtain that any elliptic associator gives a solution to the elliptic Kashiwara-Vergne problem.

math.QA↗
arXiv2021

On the homotopy type of the spaces of spherical knots in $R^n$

We study the spaces of embeddings $S^m\hookrightarrow R^n$ and those of long embeddings $R^m\hookrightarrow R^n$, i.e. embeddings of a fixed behavior outside a compact set. More precisely we look at the homotopy fiber of the inclusion of these spaces to the spaces of immersions. We find a natural fiber sequence relating these spaces. We also compare the $L_\infty$-algebras of diagrams that encode their rational homotopy type, when the codimension $n-m\geq 3$.

math.AT↗
arXiv2021

On the delooping of (framed) embedding spaces

It is known that the bimodule derived mapping spaces between two operads have a delooping in terms of the operadic mapping space. We show a relative version of that statement. The result has applications to the spaces of disc embeddings fixed near the boundary and framed disc embeddings.

math.AT↗