Thomas Willwacher
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View article: Real models for the framed little n$n$‐disks operads
Real models for the framed little n$n$‐disks operads Open
We study the action of the orthogonal group on the little ‐disks operads. As an application we provide small models (over the reals) for the framed little ‐disks operads. It follows in particular that the framed little ‐disks operads are f…
View article: The 11-loop graph cohomology
The 11-loop graph cohomology Open
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.
View article: Numerical computation of linearized KV and the Deligne-Drinfeld and Broadhurst-Kreimer conjectures
Numerical computation of linearized KV and the Deligne-Drinfeld and Broadhurst-Kreimer conjectures Open
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 inclusio…
View article: Rational homotopy theory of operad modules through colored operads
Rational homotopy theory of operad modules through colored operads Open
We extend the rational homotopy theory of operads developed by B. Fresse to several types of modules over operads.
View article: The oriented graph complex revisited
The oriented graph complex revisited Open
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.
View article: The motivic structures $\mathsf{LS}_{12}$ and $\mathsf{S}_{16}$ in the cohomology of moduli spaces of curves
The motivic structures $\mathsf{LS}_{12}$ and $\mathsf{S}_{16}$ in the cohomology of moduli spaces of curves Open
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 $\m…
View article: Moduli spaces of curves with polynomial point counts
Moduli spaces of curves with polynomial point counts Open
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 modu…
View article: AI-assisted Automated Short Answer Grading of Handwritten University Level Mathematics Exams
AI-assisted Automated Short Answer Grading of Handwritten University Level Mathematics Exams Open
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 se…
View article: Cyclic operads through modules
Cyclic operads through modules Open
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.
View article: Automorphisms of framed operads
Automorphisms of framed operads Open
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 th…
View article: Cyclic model for the dg dual of the BV operad
Cyclic model for the dg dual of the BV operad Open
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 …
View article: Weight 2 cohomology of graph complexes of cyclic operads and the handlebody group
Weight 2 cohomology of graph complexes of cyclic operads and the handlebody group Open
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…
View article: A model for configuration spaces of points
A model for configuration spaces of points Open
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 suc…
View article: Graph homology computations
Graph homology computations Open
We compute numerically the homology of several graph complexes in low loop orders, extending previous results.
View article: Models for configuration spaces of points via obstruction theory I
Models for configuration spaces of points via obstruction theory I Open
We derive rational (Sullivan) models for configuration spaces of points on manifolds purely from algebraic considerations via obstruction theory, essentially without the use of analytic or geometric techniques.
View article: Stable cohomology of graph complexes
Stable cohomology of graph complexes Open
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 ge…
View article: Weight 11 compactly supported cohomology of moduli spaces of curves
Weight 11 compactly supported cohomology of moduli spaces of curves Open
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 fo…
View article: Stable cohomology of graph complexes in odd dimensions
Stable cohomology of graph complexes in odd dimensions Open
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 …
View article: The weight two compactly supported Euler characteristic of moduli spaces of curves
The weight two compactly supported Euler characteristic of moduli spaces of curves Open
We derive a formula for the generating function for the weight two compactly supported $\mathbb S_n$-equivariant Euler characteristics of the moduli spaces of curves $\mathcal M_{g,n}$, using graph complexes and calculations inspired by op…
View article: Models for the n-Swiss Cheese operads
Models for the n-Swiss Cheese operads Open
We describe combinatorial Hopf (co-)operadic models for the Swiss Cheese operads built from Feynman diagrams. This extends previous work of Kontsevich and Lambrechts-Voli ' c for the little disks operads.
View article: Weight two compactly supported cohomology of moduli spaces of curves
Weight two compactly supported cohomology of moduli spaces of curves Open
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 a…
View article: Stable cohomology of graph complexes
Stable cohomology of graph complexes Open
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 ge…
View article: Pre-Lie Pairs and Triviality of the Lie Bracket on the Twisted Hairy Graph Complexes
Pre-Lie Pairs and Triviality of the Lie Bracket on the Twisted Hairy Graph Complexes Open
We study pre-Lie pairs, by which we mean a pair of a homotopy Lie algebra and a pre-Lie algebra with a compatible pre-Lie action. Such pairs provide a wealth of algebraic structure, which in particular can be used to analyze the homotopy L…
View article: On the delooping of (framed) embedding spaces
On the delooping of (framed) embedding spaces Open
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 fi…
View article: On the homotopy type of the spaces of spherical knots in $R^n$
On the homotopy type of the spaces of spherical knots in $R^n$ Open
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 t…
View article: Classification of universal formality maps for quantizations of Lie bialgebras
Classification of universal formality maps for quantizations of Lie bialgebras Open
We settle several fundamental questions about the theory of universal deformation quantization of Lie bialgebras by giving their complete classification up to homotopy equivalence. Moreover, we settle these questions in a greater generalit…
View article: On the rational homotopy type of embedding spaces of manifolds in $R^n$
On the rational homotopy type of embedding spaces of manifolds in $R^n$ Open
We study the spaces of embeddings of manifolds in a Euclidean space. More precisely we look at the homotopy fiber of the inclusion of these spaces to the spaces of immersions. As a main result we express the rational homotopy type of conne…
View article: Oriented hairy graphs and moduli spaces of curves
Oriented hairy graphs and moduli spaces of curves Open
We discuss a graph complex formed by directed acyclic graphs with external legs. This complex comes in particular with a map to the ribbon graph complex computing the (compactly supported) cohomology of the moduli space of points $\mathcal…
View article: The intrinsic formality of $E_n$-operads
The intrinsic formality of $E_n$-operads Open
We establish that E_n -operads satisfy a rational intrinsic formality theorem for n\geq 3 . We gain our results in the category of Hopf cooperads in cochain graded dg-modules which defines a model for the rational homotopy of operads in sp…