# Michael Rothgang’s homepage

## About me

I’m Michael Rothgang, a mathematics PhD student with Prof. Chris Wendl at Humboldt-Universität zu Berlin. I work in symplectic geometry and am particularly interested in equivariant transversality problems. In my PhD thesis, I study the moduli space of equivariant closed holomorphic curves.

Recently, I also became interested in formalisation of mathematics, particularly in symplectic geometry and the underlying analysis (such as Sobolev spaces, elliptic regularity estimates or Fredholm operators). In October 2024, I will join Floris van Doorn's research group as a postdoc, focusing also on the formalisation of analysis and symplectic geometry.

I am a phase II student of Berlin Mathematical School (BMS) and also obtained my master's degree with the BMS. Before coming to Berlin, I obtained my undergraduate degree at Jacobs University Bremen.

## Research interests

In my PhD thesis, I study the moduli space of equivariant closed holomorphic curves: given a symplectic manifold with a symplectic action, I analyse holomorphic curves with respect to generic *equivariant* compatible almost complex structures. This leads to new challenges (after all, the standard approach to showing transversality requires genericity, which is generally incompatible with prescribing additional symmetries).
Generalising this to compact Lie groups and to punctured holomorphic curves is work in progress.

Recently, I have become interested in formalized mathematics, especially the Lean theorem prover and mathlib. I have formalised the reduction of Sard's theorem to normed spaces; the normed space version itself is active work in progress. A number of pre-requisites and related results have already been incorporated into mathlib; I anticipate that all results will eventually end up there. A detailed list of my contributions can be found here.

## Publications, preprints and theses

*Homotopy Hubbard trees for post-singularly finite exponential maps*(joint with David Pfrang and Dierk Schleicher).

Ergodic Theory and Dynamical Systems, volume 43(1), pages 253–298, October 2021 (open access).

This paper grew out of my bachelor’s thesis at Jacobs University Bremen. In his PhD thesis, David Pfrang generalised this result to all post-singularly finite transcendental functions.- My master’s thesis:
*Flexibility and symplectic fillability*(August 2019, last revised November 2019).

I have been told (by fellow students in Berlin and elsewhere) that my master’s thesis has been rather helpful as a learning resource, hence I am making it available here. This is a revised version, incorporating the feedback of my thesis readers. There were substantial changes to the overall structure, but no significant changes to the mathematical content.

If you are coming here for this reason, chapters 2 and 3 are probably the most useful. Chapter 2 introduces Liouville and Weinstein domains, speaking about loose Legendrians, handle attachments and flexible Weinstein domains also. Chapter 3 reviews Hamiltonian Floer homology and proceeds to basic properties of symplectic homology. Enjoy!

Here is a 3-page summary of my thesis, aimed at a more advanced audience. *Simulation and Visualization to Support Breast Surgery Planning*(joint with J. Georgii, T. Paetz, M. Harz, C. Stoecker, J. Colletta, K. Schilling, M. Schlooz-Vries, R. Mann, and H. K. Hahn). In: Proceedings of the 13th International Workshop on Breat Imaging (IWDM 2016), pages 257–264.

My undergraduate internship at the Fraunhofer MEVIS institute in Bremen was on an applied project: extending a deformation simulation for breast tissue. This has uses in breast surgery planning: breast position during surgery (patient lies on their back) differs substantially from its positioning in radiological images. (During a mammography, the breast is compressed; during MRI imaging the patient lies face down.) Hence, imaging information about the lesion must be adapted to the surgery scenario. MEVIS had developed a finite-element model to simulate the deformation of the breast tissue. My project dealt with extending the model to incorporate the influence of Cooper’s ligaments; this work led to the paper above.

## Research talks

- Upcoming:
*Equivariant transversality for holomorphic curves*, Dynamics Seminar, RWTH Aachen *An invitation to formalising mathematics*, BMS Student Conference, February 2024 (slides, handout version)*Equivariant transversality for holomorphic curves*, Dynamics seminar at Ruhr-Universität Bochum, November 14th, 2023.*Equivariant transversality for simple holomorphic curves*, Symplectic geometry seminar at Universität Heidelberg, June 28th, 2023.*Holomorphic curves and the restricted three-body problem*, Young researchers’ workshop Pseudoholomorphic Curves in Hamiltonian Dynamics, Heidelberg, March 2023.*A glimpse at symplectic geometry and pseudo-holomorphic curves*(slides), BMS Student Conference, February 2023.*Equivariant transversality for simple holomorphic curves*, Symplectic geometry seminar at HU Berlin, November 2022*Transversality for simple holomorphic curves and the slice theorem*, Symplectic geometry seminar at HU Berlin, November 2022*A glimpse at symplectic geometry and pseudo-holomorphic curves*, BMS-BGSMath Junior Meeting 2022, September 2022. Another talk to a general mathematical audience — leading up to the role of holomorphic curves in constructing symplectic invariants such as Hamiltonian Floer homology and symplectic homology (extended slides).*Bai-Swaminathan I (Taubes’ Gromov invariant in Calabi-Yau 6-folds)*, Symplectic geometry seminar at HU Berlin, November 2021*Isolated singularities, minimal discrepancy and exact fillability*(Slides), Symplectic geometry seminar at HU Berlin, February 2021*Symplectic cohomology of Liouville sectors, part 2: algebraic constructions*(Notes), Berlin-Hamburg-Augsburg symplectic geometry seminar, June 2020*Form follows function’s flow: what does pouring honey on a lifebuoy have to do with planetary motion?*, BMS Student conference 2020. A general audience introduction of Morse and Hamiltonian Floer homology. Slides (pdf).*The twisted bundle decomposition of Cauchy-Riemann operators with symmetry*, Symplectic geometry seminar at HU Berlin, December 2019*Teichmüller slices, Fredholm regularity and the implicit function theorem*, Symplectic geometry seminar at HU Berlin, October 2019*Hubbard trees for exponential maps*, BMS Student conference 2017. An expository talk about the Homotopy Hubbard trees paper above. Slides (pdf), video recording.

## Teaching

I have taught a variety of courses at various institutions, both in German and English and in various roles — ranging from a student assistant to a *Wissenschaftlicher Mitarbeiter*.

- In summer 2024, Yves Jäckele, Nicolas Weiss and I will organise a Lean student seminar: see here for further information.
*Algebra und number theory*(for math teacher students), summer term 2024, HU Berlin.

In German. Preparing and holding weekly tutorials, exam grading.*Geometrie*(axiomatic geometry, for math teacher students), winter term 2023/24, HU Berlin.

In German. Preparing and holding weekly tutorials, exam grading.*Linear algebra and analytic geometry II*, summer term 2020, HU Berlin.

In German. Preparing problem sheets, preparing and holding weekly tutorials, exam grading.*Linear algebra and analytic geometry I (for math teacher students)*, winter term 2019/20, HU Berlin.

In German. Preparing and holding weekly tutorials, exam grading.*Analysis II*, summer term 2018, FU Berlin. In German.

Preparing and holding weekly tutorials, grading homeworks.*Algebra and number theory*, winter term 2017/18, FU Berlin. In German.

Preparing and holding weekly tutorials, grading homeworks and exams.*Linear algebra (for computer science students)*, summer term 2017, FU Berlin. In German.

Preparing and holding weekly tutorials, grading homeworks and exams.*Stochastics (for math teacher students)*, winter term 2016/17, FU Berlin. In German.

Preparing and holding weekly tutorials, grading homeworks and exams.*Analysis II*, spring term 2016, Jacobs University Bremen. In English.

Student assistant: homework and exam grading, holding tutorials.*Analysis I*, autumn term 2015, Jacobs University Bremen. In English.

Student assistant: homework and exam grading, holding tutorials.*Modern Mathematics summer school*, July 2015. In English.

Teaching assistant: holding tutorials to lectures for Matthias Görner and Rebecca Waldecker.*Linear algebra I*, Jacobs University Bremen, autumn term 2014. In English.

Student assistant: grading of homework and holding office hours.

## How to contact me

*Mailing address*: Institut für Mathematik, Humboldt-Universität zu Berlin,
Unter den Linden 6, 10099 Berlin, Germany

*Office*: Rudower Chaussee 25 (John-von-Neumann-Haus), 12489 Berlin. Haus 1, Raum 305.

*Email*: `rothgami at math.hu-berlin.de`

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