Danica KosanovićPronounce c as zz in pizza, and ć as ci in ciabatta. Typeset the letter ć as \'{c} in LaTeX. LAGA Laboratoire analyse, géométrie et applications Université Sorbonne Paris Nord (Paris 13) 99 avenue Jeanbaptiste Clément 93430 Villetaneuse kosanovic[at]math[dot]univparis13[dot]fr 

I hold a postdoc position at LAGA (Paris 13) working with Geoffroy Horel, and funded by FSMP. In September 2021 I will move to ETH Zürich as a Hermann Weyl Instructor.
My interests include knot theory, 4manifolds, knotted surfaces in 4manifolds, homotopy types of embedding spaces, GoodwillieWeiss embedding calculus, operads, graph complexes. For more details, see this introduction or the slides from the public talk of my defense, or have a look at the tabs on the left.
I obtained my PhD degree from the University of Bonn, working at the MaxPlanck Insitut für Mathematik under the supervision of Peter Teichner. Previously, I studied in Belgrade (Serbia) and Cambridge (UK).
My partner Mihajlo Cekić is also a mathematician.
11.6.2021  A light bulb theorem for disks @ Georgia Topology Conference (Online) beamer slides 
  I'm organizing a "Building Bridges" learning seminar (currently on a long summer break). Here is its webpage. 
I like thinking about knots, 4manifolds, 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, TQFT’s, operads, as well as combinatorics of Feynman diagrams and configuration spaces, should merge together to give even more insight about lowdimensional manifolds.
In my thesis I studied finite type knot invariants and their relation to the GoodwillieWeiss embedding calculus. Here is a short introduction to these topics.
Finite type invariants (often called GusarovVassiliev, 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 nequivalence relations on knots. Then the nth term of the filtration is comprised of knots which are nequivalent to the unknot.
For example, two knots are 1equivalent if they can be related by a sequence of crossing changes. This means that the first term in the filtration is equal to the whole monoid of knots! To get an idea about 2equivalence, take a look at the operation on the left  grab some three strands of a knot and connectsum them with the Borromean rings.
Embedding calculus of Goodwillie and Weiss is another homotopytheoretic approach to spaces of embeddings. When applied to the embedding functor of long knots $\mathcal{K}$ in the 3space it yields a tower of spaces $T_n$ together with evaluation maps $ev_n\colon K\to T_n$. These spaces turn out to be very interesting. 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 $\pi_0$ which turns out to be a finite type invariant! It is conjectured to be universal such, in other words, the group of knots modulo relation of nequivalence is isomorphic to $\pi_0T_n$.
Therefore, the two stories should not be so separate after all. One unifying perspective is that of gropes. Namely, the trivalent vertices appearing in the diagrams for finite type theory (originating in quantum ChernSimons theory) correspond to the Borromean rings, and the isotopy depicted below hints at how this in turn relates to gropes. In the very last picture we clearly see a genus one surface with one boundary component emerging. This will represent the bottom stage of a grope.
Embedding calculus and grope cobordism of knots arxiv.org/abs/2010.05120 

A space level light bulb theorem for disks Joint with Peter Teichner arxiv.org/abs/2105.13032 
A geometric approach to the embedding calculus knot invariants. PhD Thesis. Download in Bonn Library.
21.4.2021 Online  Knotted families of arcs @ Münster Topology Seminar beamer slides 
15.3.2021 Online  Knotted families of arcs @ MIT Topology Seminar 
13.1.2021 Online  Knot invariants from homotopy theory @ Higher Structures & Field Theory Seminar 
4.12.2020 Online  Knot invariants from homotopy theory @ Colloquium LAGA Paris 13 
3.12.2020 Online  Knot invariants from homotopy theory @ Théorie des groupes, LAMFA Université d'Amiens 
26.11.2020 Online  Knot invariants from homotopy theory @ Séminaire AGATA, Université de Montpellier, beamer slides 
17.11.2020 Online  Knot invariants from homotopy theory @ Warwick algebraic topology seminar 
2.11.2020 Online  Knot invariants from homotopy theory @ G&T Seminar Glasgow 
16.10.2020  Knot invariants from homotopy theory @ Université de Lille 
31.7.2020 Online  Embedding calculus for knot spaces @ Oberwolfach Workshop Topologie 
29.5.2020 Online  Knot invariants from homotopy theory @ Topological Quantum Field Theory Seminar, Instituto Superior Técnico, Lisboa, video 
21.4.2020 Online  Knot invariants from homotopy theory @ jointly Séminaire de l'équipe Topologie Algébrique, LAGA, Paris 13 and Séminaire de Topologie, IMJPRG, Paris 7 
20.2.2020  A geometric approach to the embedding calculus @ Oberwolfach Workshop Lowdimensional Topology 
30.1.2020  Knot invariants from homotopy theory @ Topology Seminar Bochum 
20.1.2020  Knot theory meets the embedding calculus @ Copenhagen Algebra/Topology Seminar 
16.1.2020  Нове технике у теорији утапања (New techniques in the theory of embeddings) @ Mathematical Institute, Serbian Academy of Sciences and Arts, Belgrade 
2.12.2019  Knot theory meets the embedding calculus @ MPIM Topology Seminar, Bonn 
16.10.2019  Knots map onto components of the embedding calculus tower @ Spaces of Embeddings: Connections and Applications, Banff International Research Station, Canada 
16.9.2019  A gong show talk @ Workshop on 4manifolds, MPIM Bonn 
13.5.2019  A gong show talk @ Knots and Braids in Norway (KaBiN), Trondheim 
7.5.2019  A geometric approach to embedding calculus @ Utrecht Geometry Center Seminar 
25.12.2018  Инваријанте чворова и конфигурациони простори (Knot invariants and configuration spaces) @ Mathematical Institute, Serbian Academy of Sciences and Arts, Belgrade, slides (in Serbian) 
17.12.2018  Revisiting the Arf invariant @ Topology Seminar, MPIM Bonn 
6.12.2018  Extended evaluation maps from knots to the embedding tower @ Manifolds Workshop (part of Homotopy Harnessing Higher Structures Trimester) at Isaac Newton Institute, Cambridge 
28.11.2018  Knot theory meets homotopy theory @ IMPRS Seminar, MPIM Bonn, slides 
24.7.2018  Grope cobordism and the embedding tower for knots @ 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 @ Winter Braids, CIRM, Luminy, France 
6.5.2021  On a theorem of Kontsevich and ConantVogtmann @ Groupes de GrothendieckTeichmüller et applications 
11.3.2021  Chord diagram invariants of tangles @ Groupes de GrothendieckTeichmüller et applications, notes 
13.2.2020  On the punctured knots model for embedding spaces @ Configuration Categories Learning Seminar (Online) 
19.12.2019  On link maps @ Mojito’s Seminar (Online) 
13.2.2020  On the paper by BundeyGabai about knotted 3balls @ Online Student Seminar, notes 
19.12.2019  Watanabe's counting formula for classes in Diff(S^4) @ Hot Topic Seminar, MPIM 
5.11.2019  Milnor invariants and Whitney towers @ Milnor Invariants Learning Seminar, MPIM 
July 2019  Introduction to Milnor link invariants and relation to Massey products @ Milnor Invariants Learning Seminar, MPIM 
May 2019  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 ChernSimons theory for knots @ BV learning seminar, MPIM 
22.3.2018  Complex oriented cohomology theories @ Peter’s Seminar in Berkeley 
06.12.2017  Universal Knot Invariants @ The Chinese University of Hong Kong 
15.11.2017  How to draw a smooth 4−manifold? @ IMPRS seminar, MPIM 
25.09.2017  A categorical approach to quantum knot invariants @ Topology Seminar, MPIM 
04.08.2017  A survey of WittenReshetikhinTuraev invariants of 3manifolds @ Special Topology Seminar, MPIM 
02.06.2017  Topological reincarnations of the Arf invariant @ Cambridge Junior Geometry Tea Seminar, Cambridge, UK 
23.03.2017  Topological reincarnations of the Arf invariant @ Berkeley seminar 
See the tab seminar.
Geoffroy Horel and Bruno Vallette are organizing a learning seminar at Paris 13. See here.
Ben Ruppik and I were organising a series of talks on Milnor invariants. Ben made a cool website which contains our notes and references.
I was giving tutorials for this class. Here is the page with the class notes and homework assignments.
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:
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.