Contact

Email
mdehling [at] gmail.com
Mathematisches Institut
Georg-August Universität
Bunsenstraße 3–5
D-37073 Göttingen
Deutschland
Office
Nebengebäude N.31

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?

Actually, they are everywhere. I will be more specific and list some examples as soon as I find the time. For now, see e.g. the above cited textbook.

Publications

Shifted $L_\infty$ 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(Master thesis, unpublished, 2011.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, Preprint, September 2011. [arxiv] Z. Chen, M. Stienon, P. Xu, Weak Lie 2-bialgebras, Preprint, September 2011. [arxiv]

Teaching

I am currently a teaching assistent for the course linear algebra II given by Preda Mihaelescu.

In the past I have been a teaching assistent for the following courses.

• 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

I plan to attend the following events.

• Higher structures in China IV, Lanzhou, 2013-08-14 – 08-16.
• Higher algebras and Lie-infinity homotopy theory, University of Luxembourg, 2013-06-25 – 06-28.

And these are some conferences and events I attended in the past.

• 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.