## About me

I am a PhD student at the mathematical institute in
Göttingen and a member of the research training
group 1493 mathematical structures in modern quantum physics.

For my PhD project I work with Chenchang Zhu here in Göttingen and Bruno Vallette from Nice.

## Research interests

Generally, I am interested in higher structures in geometry and topology. More specifically, I work in homotopy-algebraic structures arising in geometry and topology.

### What are homotopy-algebraic structures?

Let's look at an example. Assume $(A,m,d_A)$ is a dg associative algebra and $(V,d_V)$ is a deformation
retract, i.e.
$$ h \circlearrowright (A,d_A) \underset{i}{\overset{p}{\rightleftarrows}} (V,d_V), \qquad
pi = \text{Id}_V , \qquad \text{Id}_A - ip = [d_A,h] . $$
One may transfer the multiplication $m\colon A\otimes A \to A$ to a multiplication on $V$,
$$ \mu_2\colon V\otimes V \overset{i\otimes i}{\longrightarrow} A\otimes A
\overset{m}{\longrightarrow} A \overset{p}{\longrightarrow} V .$$
However, this multiplication $\mu_2$ is in general not associative when $ip \neq \text{Id}_A$. In fact, if
we define $\mu_3 = \mu_2(h\mu_2\otimes 1) - \mu_2(1\otimes h\mu_2)$, then $\partial\mu_3$ measures the
defect of associativity,
$$ \mu_2(\mu_2\otimes 1) - \mu_2(1\otimes \mu_2) = \partial\mu_3 .$$
Similarly one obtains higher multiplications $\mu_n\colon V^{\otimes n} \to V$, which are subject to a bunch
of coherence conditions. The structure $(V,\{\mu_n\},d_V)$ defined in this way is an
*associative algebra up to homotopy* or *$A_\infty$-algebra.*

This notion was introduced by Jim Stasheff in
J. Stasheff, *Homotopy associativity of H-spaces I*,
Trans. Am. Math. Soc. 108 (1963), 275-292.

Similar notions of algebra up to homotopy exist for other types of algebras, e.g. for Lie algebras, see
T. Lada, J. Stasheff, *Introduction to sh Lie algebras for physicists*,
Int. J. Theor. Phys. 32 (1993), 1087-1103.
For a modern textbook introduction to the subject, see
J.-L. Loday, B. Vallette, *Algebraic operads*,
Grundlehren der mathematischen Wissenschaften 346, Springer-Verlag, 2012.

### Where do these structures appear?

Since the setting is a very general one, these structures appear in many places. One very prominent example is the theory of deformation quantization of Poisson manifolds.

**Theorem (Kontsevich):**Every Poisson manifold has a formal deformation quantization.

The proof of this theorem makes heavy use of algebraic operads and homotopy algebras. In particular Deligne's Hochschild cohomology conjecture plays a role, i.e. the fact that the usual Gerstenhaber structure on Hochschild cohomology $\text{HH}^\bullet(A,A)$ of an associative algebra $A$ actually comes from a homotopy Gerstenhaber structure on the Hochschild cochain complex $\text{HC}^\bullet(A,A)$.

## Publications

### Weak Lie 3-algebras

(In preparation. pdf)

The usual definition of Lie algebra up to homotopy involves the cobar-bar resolution of the operad
$\text{Lie}$. The resulting definition is that of $\text{L}_\infty$ algebras. These feature strictly
antisymmetric structure maps satisfying a generalized Jacobi identity. The goal of this work is to relax
both the Jacobi identity and the antisymmetry simultaneously in a coherent way, and obtain a new definition
of *weak Lie $k$-algebras* as approximation to the full notion of *$\text{EL}_\infty$
algebras* on a $k$-term complex.

Based on joint work with Bruno Vallette on *symmetric homotopy theory for operads* and the new
cobar-bar resolution developed there, I define a dg subcooperad $\text{Lie}^\vee_k$ of
$\tilde{\text B}\text{Lie}$ for each $k$, governing (via the Maurer-Cartan equation) weak Lie $k$-algebras.
Considering this structure in the case $k = 2$, one recovers the notion of weak Lie 2-algebra as defined in
D. Roytenberg, *On weak Lie 2-algebras*,
Geometrical methods in physics 956 (2007), 180-198.
[arxiv]
For the case $k = 3$ we make the definition of a *weak Lie 3-algebra* explicit and continue to
develop a fully weak notion of morphism.

It was shown in
J. Baez, A. Hoffnung, C. Rogers,
*Categorified Symplectic Geometry and the Classical String*,
Comm. Math. Phys. 293 (2010), 701-725.
Geometrical methods in physics 956 (2007), 180-198.
[arxiv]
that given a 2-plectic manifold $(M,\omega)$, the associated Lie 2-algebra $\text{L}_\infty(M,\omega)$ is
isomorphic to the associated Leibniz 2-algebra $\text{Leib}(M,\omega)$ in the sense of weak Lie
2-algebras. Using the developed theory of weak Lie 3-algebras, I show that this result easily extends to
the case of 3-plectic manifolds.

### Symmetric homotopy theory for operads

(Joint work with Bruno Vallette. arxiv)

We consider a colored operad $\mathcal{O}$ governing non-unital symmetric operads. We apply the curved
Koszul duality theory as developed in
J. Hirsh, J. Millès, *Curved Koszul duality theory*,
Math. Ann. 354:4 (2012), 1465-1520.
[arxiv]
This gives rise to a new notion of symmetric homotopy operad as
$\Omega\mathcal{O}^{\bf\text{¡}}$-algebra. It is different from the notion of homotopy operad defined
in
P. van der Laan, *Coloured Koszul duality and strongly homotopy operads*,
Preprint, December 2003.
[arxiv]
since in our construction both composition and symmetry are relaxed up to homotopy. On the other hand, van
der Laan relaxes operadic composition but keeps strict $\mathbb{S}$-modules underneath.

In the next part we apply Koszul duality for operadic algebras as described in
J. Millès, *The Koszul complex is the cotangent complex*,
Internat. Math. Res. Notices 2012:3 (2012), 607-650.
[arxiv]
to obtain a new type of cobar-bar adjunction
$\tilde\Omega : \mathcal{O}^{\bf\text{¡}}\text{-coalgebras}
\rightleftarrows \mathcal{O}\text{-algebras} : \tilde{\text B}$
between operad and a certain notion of *higher cooperads*.
We show that the counit of this adjunction gives a cofibrant resolution over any ring, and prove that
$\text{I} \oplus \tilde\Omega\tilde{\text B}\bar{\mathcal{P}}
\cong \Omega\text{B}(\mathcal{P} \otimes \text{BE})$,
where $\text{BE}$ denotes the dg Barratt-Eccles operad. In particular, for $\mathcal{P} = \text{Com}$, this
gives a new construction for a cofibrant $\text{E}_\infty$ operad.

This preprint contains an appendix on group representation up to homotopy.

### Shifted $\text{L}_\infty$ bialgebras

(Master thesis, unpublished, 2011. pdf, poster)

A Lie bialgebra is a pair of Lie algebra structures on dual vector spaces $(\mathfrak{g},\mathfrak{g}^*)$. The two structures are required to satisfy a certain compatibility condition which can be expressed in different ways. Lie bialgebras appear naturally as the infinitesimal versions of Poisson Lie groups.

Categorification of Lie algebras was done in
J. Baez, A. Crans, *HDA6: Lie 2-algebras*, Theory Appl. Categ. 12 (2004), 492-528.
[journal,
arxiv]
In fact, the semi-strict Lie $k$-algebras are precisely $k$-term $L_\infty$ algebras.

In this article I give several equivalent definitions of Lie $k$-bialgebras corresponding to the ways in which the compatibility can be expressed in the classical case.

Two closely related articles on the 2-case are

C. Bai, Y. Sheng, C. Zhu,*Lie 2-bialgebras*, Comm. Math. Phys. 320:1 (2013), 149-172. [arxiv] Z. Chen, M. Stienon, P. Xu,

*Weak Lie 2-bialgebras*, J. Geom. Phys. 68 (2013), 59-68. [arxiv]

## Teaching

I am currently the teaching assistent for the 4th semester *symplectic geometry* course taught by
Prof. Chenchang Zhu. In previous semesters I have held teaching positions for the following courses.

*Algebra*, 2015/16.*Functional analysis*, 2014.*Analysis I*, 2013/14.*Linear algebra II*, 2013.*Complex analysis I*, 2012.*Differential geometry I*, 2011/12.*Numbers and number theory*, 2011.*Analysis III*, 2010/11.*Analysis II*, 2010.*Analysis I*, 2009/10.*Precourse in mathematics*, 2009.

## Conferences

This is a list of some of the conferences and events I attended in the past.

*Infinite-dimensional structures in higher geometry and representation theory*, Center for Mathematical Physics, Hamburg, 2015-02-16 – 02-20.*Young topologists meeting*, Lausanne, 2013-07-08 – 07-12.*Higher structures 2013*, Newton Institute, Cambridge, 2013-04-02 – 04-05.*Grothendieck-Teichmüller groups, deformation and operads*, Newton Institute, Cambridge, 2013-02-04 – 02-28.*Higher geometric structures along the lower Rhine II*, Radboud university, Nijmegen, 2012-12-06 – 12-07.*Higher structures in China III*, Jilin University, Changchun, 2012-08-13 – 08-19.*Koszul duality for operads*, Centre for Symmetry and Deformation, Copenhagen, 2012-04-16 – 04-27.*Logic and interactions: algebra and computation*, Centre International de Rencontres Mathématiques, Luminy, 2012-02-27 – 03-02.*Higher geometric structures along the lower Rhine*, Max Planck Institute for Mathematics, Bonn, 2012-01-12 – 01-13.*Higher structures in mathematics and physics*, University of Göttingen, 2011-11-28 – 12-02.