This is a kind of a learning seminar, whose goal is to bridge geometric and algebraic approaches to spaces of embeddings. This includes invariants of classical knots, 2knots in 4manifolds, homotopy groups of diffeomorphism groups. One of the goals is to learn embedding calculus through its applications.
The seminar is held online on Wednesdays at 15:00 CET. Contact me if you wish to be added to the mailing list and obtain the meeting details.
[Weiss] Embeddings from the point of view of immersion theory: Part I
[GoodwillieWeiss] Embeddings from the point of view of immersion theory: Part II
[Lambrechts] Primer on homotopy limits (.pdf)
[Weiss] Immersion theory for homotopy theorists (fancy .pdf)
The first block: 


Nov 11 
Introduction and motivation I will talk about why a topologist might care about spaces of embeddings, and sketch why embedding calculus might help understand them. 
handwritten slides  
Nov 18  Pedro Boavida de Brito 
Configuration categories and embedding calculus An introduction to embedding calculus, and its relation to the little disks operad \(\mathrm{E}_n\). Partly based on doi/abs/10.1112/topo.12048. 
tex notes 
Nov 25  Geoffroy Horel 
Action of GT on the tower for knots I will explain how the action of the GrothendieckTeichmüller group \(GT\) on braid groups, originally studied by arithmetic geometers, induces an action on the embedding calculus tower for long knots. This can be used to deduce some integral results about the universal finite type invariant for knots. This is joint work with Pedro Boavida de Brito. Based on arxiv.org/abs/2002.01470. 
beamer slides 
Dec 02  Gregory Arone 
Operad formality and rational homology of embedding spaces The GoodwilileWeiss tower can be described as a space of maps between modules of \(\mathrm{E}_n\). The formality of \(\mathrm{E}_n\) has rather far reaching consequences for the rational homotopy type of embedding spaces. I will focus on rational homology. Based on projecteuclid.org/euclid.gt/1513732795. 
beamer slides 
Dec 09  Victor Turchin 
Rational homotopy type of embedding spaces I will talk about my joint work with Benoit Fresse and Thomas Willwacher. Using embedding calculus and methods of the rational homotopy theory we construct \(L_\infty\)algebras of diagrams that encode the rational type of connected components of embedding spaces in \(\mathbb{R}^n\). This type depends on the component. Different known invariants of embeddings seem to be responsible for the rational homotopy type. Some examples will be discussed. Based on arxiv.org/abs/2008.08146. 
handwritten slides (.svg) 
Dec 16  Pascal Lambrechts 
Bonus Talk: Back to basics Homotopy limits for a working lowdimensional/differential topologist 
beamer slides 
~ Winter Break ~


Jan 6  Dev Sinha 
Algebraic topology of embedding spaces and its application to knot theory, from a geometric perspective
Geometric algebraic topology makes use of representations of homology and cohomology by manifolds along with explicit maps from spheres for homotopy. My interest in this aspect of algebraic topology has grown in part from my study of spaces of embeddings. 
handwritten slides 1 handwritten slides 2 
The second block: 


Jan 13  Slava Krushkal 
Embedding obstructions in 4space from the GoodwillieWeiss calculus and Whitney disks Given a 2complex K, I will explain how to use a version of the GoodwillieWeiss tower to formulate an obstruction theory for embeddings into \(\mathbb{R}^4\). I will also introduce a geometric analogue, based on intersections of Whitney disks. Focusing on the first obstruction beyond the classical embedding obstruction of van Kampen, I will show that the two a priori very different approaches in fact give the same result, and also relate it to the Arnold class in the cohomology of configuration spaces. The obstructions are realized in a family of examples. Joint work with Greg Arone. 

Jan 20  Ben Ruppik 
Unknotting 2knots with Finger and Whitney moves
This is joint work arxiv.org/abs/2007.13244 with Jason Joseph, Michael Klug, Hannah Schwartz. Any smoothly knotted 2sphere in the 4sphere is regularly homotopic to the unknot. This means that every 2knot K in \(\mathbb{S}^4\) can be obtained by first performing a number of trivial finger moves on the unknot, and then removing the resulting intersection points in pairs via Whitney moves along possibly complicated Whitney discs. We define the CassonWhitney unknotting number of the 2knot K as the minimal number of finger moves needed in such a process to arrive at K. 

Jan 27  Rob Schneiderman 
Whitney towers, capped gropes and the higherorder Arf invariant conjecture This selfcontained talk will introduce a theory of Whitney towers which `measures’ the failure of the Whitney move in dimension four and is closely related to certain 2complexes called capped gropes which are geometric embodiments of commutators. The main goal of the talk is to describe a naturally arising family of link concordance invariants which are conjectured to be nontrivial finite type invariants generalizing the Arf invariant of a knot. Accompanying material for this talk can be found in the first two sections of the expository paper arxiv.org/abs/2012.01475. 

Feb 3  Maggie Miller 
TBA (Smooth surfaces in 4manifolds) ... 

Feb 10  Anthony Conway 
TBA (On topologically isotopic surfaces in 4manifolds) ... 

Feb 17  Aru Ray 
TBA (On concordance of links) ... 

Feb 24  Mark Powell 
TBA (A sruvey of open problems in 4manifold theory) ... 