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, 4-manifolds, knotted surfaces in 4-manifolds, homotopy types of embedding spaces, Goodwillie-Weiss 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 Max-Planck 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.

### News and Upcoming talks

I'm organizing a "Building Bridges" learning seminar, here is its webpage.

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, TQFT’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 studied finite type knot invariants and their relation to the Goodwillie-Weiss embedding calculus. Here is a short introduction to these topics.

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.

*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 is equal to the whole monoid of knots! To get an idea about 2-equivalence, take a look at the operation on the left - grab some three strands of a knot and connect-sum them with the Borromean rings.

Embedding calculus of Goodwillie and Weiss is another homotopy-theoretic approach to spaces of embeddings. When applied to the embedding functor of long knots $\mathcal{K}$ in the 3-space 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 n-equivalence 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 Chern-Simons 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.

### Preprints

*Embedding calculus and grope cobordism of knots.* See arxiv.org/abs/2010.05120.

We show that embedding calculus invariants $ev_n$ are surjective for long knots in an arbitrary $3$-manifold. This solves some remaining open cases of Goodwillie--Klein--Weiss connectivity estimates, and at the same time confirms one half of the conjecture that for classical knots $ev_n$ are universal additive Vassiliev invariants over the integers. In addition, we give a sufficient condition for this conjecture to hold over a coefficient group, which is by recent results of Boavida de Brito and Horel fulfilled for the rationals and for the $p$-adic integers in a range. Therefore, embedding calculus invariants are strictly more powerful than the Kontsevich integral.

Furthermore, our work shows they are more computable as well. Namely, the main theorem computes the first possibly non-vanishing invariant $ev_n$ of a knot which is grope cobordant to the unknot to be precisely equal to the equivalence class of the underlying decorated tree of the grope in the associated graph complex. Actually, our techniques apply beyond dimension $3$, offering a description of the layers in embedding calculus for long knots in a manifold of any dimension, and suggesting that certain generalised gropes realise the corresponding graph complex classes.

### In preparation

*On certain homotopy groups of spaces of embedded arcs and circles. * To appear soon.

For the space of embedded arcs (long knots) in manifolds of dimension $d\geq4$ we describe the lowest homotopy group which potentially distinguishes it from the space of immersions. To this end we both collect some existing results and also describe explicit generators of these groups. For $d=3$ we obtain invariants of isotopy classes of knots, related to Vassiliev type $\leq1$. We also discuss embedded circles, answering a question posed by Arone and Szymik: *in a simply connected $4$-manifold the fundamental group of the space of embedded circles is the same as that of the space of immersed circles*.

*A space level light bulb theorem for disks.* To appear soon, joint with Peter Teichner.

We describe the homotopy type of the space of embedded $k$-disks in a $d$-manifold $M$ with fixed boundary in $\partial M$, for which there exists a geometric dual sphere, in terms of the space of embedded $(k-1)$-disks in the manifold obtained from $M$ by attaching a handle to the dual sphere. As a consequence, for $k=2$ and $d=4$ we can use the Dax invariant of knotted arcs to *completely classify such $2$-disks in a $4$-manifold up to isotopy*, answering questions posed in \cite{Gabai-disks}. Moreover, we recover the classification of homotopic $2$-spheres up to isotopy, with the Dax invariant reducing to the Freedman-Quinn invariant of \cite{Schneiderman-Teichner}. This lifts the light bulb theorem of \cite{Gabai-spheres, Schneiderman-Teichner} to the level of spaces.

*Spaces of gropes and the embedding calculus.* Work in progress, joint with Yuqing Shi and Peter Teichner.

*On Borromean link families in all dimensions. * Work in progress, joint with Peter Teichner.

### Thesis

*A geometric approach to the embedding calculus knot invariants.* PhD Thesis. Download.

### Building Bridges Seminar: Invariants of embedding spaces

(Winter 2020)
See the tab seminar.

### Working Group on Grothendieck-Teichmüller Group (and Applications)

(Winter 2020)
Geoffroy Horel and Bruno Vallette are organizing a learning seminar at Paris 13. See here.

### Milnor Invariants Learning Seminar

(July & November 2019)
Ben Ruppik and I were organising a series of talks on Milnor invariants. Ben made a cool website which contains our notes and references.

### The class on 4-manifolds by Peter Teichner and Aru Ray

(Winter 2018)
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.