Danica Kosanović

Max-Planck Institute für Mathematik
Office B27
Vivatsgasse 7, Bonn, DE 53111

I am a second year PhD student at the Max-Planck Insitut für Mathematik, under supervision of Peter Teichner. My studies are funded by the Max Planck Society, and I am a member of the International Max Planck Research School on Moduli Spaces (IMPRS). I have previously studied in Belgrade (Serbia) and Cambridge (UK).

My interests include knot theory, 4-manifolds, embedding calculus, operads, graph complexes.

For more details, have a look at the tabs on the left.

I like thinking about knots, 4-manifolds, surfaces inside, and in general about topology in low dimensions! However, I also believe that formalism and tools of higher topology, i.e. homotopy theory, higher categories, TFT’s, operads, as well as combinatorics of Feynman diagrams and configuration spaces, should merge together to give even more insight about low-dimensional manifolds.

In my thesis I am looking at finite type invariants of knots and their relation to the embedding calculus of Goodwillie and Weiss. Below I give a short introduction into this topic; more about my research can be found in the tab ''Talks''.

Finite type invariants (often called Gusarov-Vassiliev, or just Vassiliev, invariants) give a certain filtration on the set of all invariants by their type. A dual point of view is, however, more geometric: there is a filtration on the monoid of knots itself, which arises by looking at a certain sequence of n-equivalence relations on knots. Then the n-th term of the filtration is comprised of knots which are n-equivalent to the unknot.

Connect-sum with Borromean rings

For example, two knots are 1-equivalent if they can be related by a sequence of crossing changes. This means that the first term in the filtration will be equal to the whole monoid of knots! The 2-equivalence is a bit more interesting. Just to get an idea about it, take a look at the operation on the left - grab some three strands of a knot and connect-sum them with the Borromean rings.

A completely separate story can be told about the Goodwillie-Weiss embedding calculus applied to the embedding functor of long knots in the 3-space. This yields a tower of spaces together with evaluation maps from the space of knots to the levels of the tower. These spaces turn out to be very interesting and mysterious. For example, they can be shown to be double loop spaces of the mapping spaces between some (truncated) operads. Hence, their components form an abelian group and the evaluation map from knots gives a map on π0 which turns out to be -- a finite type invariant.

Hence, two stories are not so separate after all. Moroever, the trivalent vertices appearing in the diagrams for finite type theory (originating in quantum Chern-Simons theory) correspond to the Borromean rings, and the isotopy below hints how this in turn relates the n-equivalence with gropes. Namely, in the very last picture we clearly see a genus one punctured surface, which will represent the bottom stage of a grope.

Borromean rings isotopy

In the winter semester 2018 I was giving tutorials for Peter's and Aru's class on 4-manifolds.

1. Here are the level sets of the Boy's surface.

2. Here is the proof that Bing double of any knot is a boundary link: Bing

3. Here are solutions to some of the homework exercises we didn't have time to cover in the tutorials.

4. See also interior and boundary twists.

Research talks and posters:

A gong show talk – at Knots and Braids in Norway (KaBiN)
A geometric approach to embedding calculus – at Utrecht Geometry Center Seminar
Инваријанте чворова и конфигурациони простори – at Mathematical Institute SANU, Belgrade
Revisiting the Arf invariant – at Topology Seminar, MPIM
Extended evaluation maps from knots to the embedding tower – at Manifolds Workshop (part of Homotopy Harnessing Higher Structures Trimester) at Isaac Newton Institute, Cambridge
Knot theory meets homotopy theory, IMPRS Seminar, MPIM
Grope cobordism and the embedding tower for knots – at ICM 2018 Satellite Conference: Braid Groups, Configuration Spaces and Homotopy Theory, in Salvador, Brazil
Feb 2018
Poster A homotopy theoretic approach to finite type knot invariants – presented at Winter Braids at CIRM, Luminy, France

Expository talks:

May 2018
Formality of little disks operads – IMPRS seminar, MPIM
Sep/Oct 2018
Two talks about the paper of Ihara on automorphisms of pure sphere braid group, GT learning seminar, MPIM
Apr/May 2018
Two talks on perturbative quantization and Chern-Simons theory for knots, BV learning seminar, MPIM
Complex oriented cohomology theories – at Peter’s Seminar in Berkeley
Universal Knot Invariants – at The Chinese University of Hong Kong
How to draw a smooth 4−manifold? – at IMPRS seminar, MPIM
A categorical approach to quantum knot invariants – at Topology Seminar, MPIM
A survey of Witten-Reshetikhin-Turaev invariants of 3-manifolds – at Special Topology Seminar, MPIM
Topological reincarnations of the Arf invariant – at Cambridge Junior Geometry Tea Seminar, Cambridge, UK
Topological reincarnations of the Arf invariant – Berkeley seminar