### About me

I am a researcher and work as an advanced fellow (which is a 5-year postdoc position) at the ETH-ITS in Zürich. My field of research is mathematical physics.

A lot of my research explores the interplay between *moduli spaces* and *quantum field theory*. I am fascinated by the connections between these two domains, due to their foundational significance in their respective field. I like to transform techniques that are established in the quantum field theory community into rigorous tools suitable for addressing problems in pure mathematics. During this process I also aim to improve the general understanding and the predictive power of quantum field theory.

Many objects that I work with can be labeled *discrete* or *tropical* geometries. Primarily, I use *algebraic topological* and *combinatorial* tools, complemented by methods from algebraic geometry and homological algebra.

Download my CV, view my or browse my articles on arXiv.

#### Contact

michael.borinsky@eth-its.ethz.ch

### News

During May 2023, I will be visiting the Institute of Advanced Study in Princeton, USA.

Leila Sloman from Quanta Magazine covered my research with Karen Vogtmann on Out(Fn) and graph moduli spaces using tools from quantum field theory.

During November 2022, I have been visiting the Perimeter Institute in Waterloo, Canada.

During October 2022, I have been visiting the Max-Planck-Institute for Mathematics in Bonn, Germany.

I organized a workshop on 'Tropical and Convex Geometry and Feynman integrals' 05 - 09 September 2022 at the ETH-ITS in Zürich.

### Publications

PhD and MSc student contributions are marked with ^{grad}.

#### Preprints

Weight 2 cohomology of graph complexes of cyclic operads and the handlebody group, preprint August 2023

with Benjamin Brück and Thomas Willwacher

The S

_{n}-equivariant Euler characteristic of the moduli space of graphs, preprint June 2023with Jos Vermaseren

The Euler characteristic of the moduli space of graphs, preprint January 2023, to appear in

*Advances in Mathematics*with Karen Vogtmann. This work is featured in a popular science piece by Leila Sloman in Quanta Magazine.

Computing Euler characteristics using quantum field theory, preprint February 2022, to appear in

*Geometric methods in group theory: papers dedicated to Ruth Charney*with Karen Vogtmann

Tropical Monte Carlo quadrature for Feynman integrals, preprint August 2020, to appear in

*Annales de l'Institut Henri Poincaré D*the proof-of-concept implementation of the algorithm for numerical Feynman integration is available on this page

#### Journal articles

Tropical Feynman integration in the Minkowski regime,

*Computer Physics Communications*, 292 (2023) 108874 (arXiv:2302.08955)with Henrik Munch

^{grad}and Felix Tellander^{grad}Try out the Feynman integration program feyntrop that we describe in the paper! It is fast and easy to use.

Flow-oriented perturbation theory,

*Journal of High Energy Physics*, 172 (2023) (arXiv:2210.05532)with Zeno Capatti

^{grad}, Eric Laenen, and Alexandre Salas-Bernárdez^{grad}Recursive computation of Feynman periods,

*Journal of High Energy Physics*, 291 (2022) (arXiv:2206.10460)with Oliver Schnetz

Bayesian Integrals on Toric Varieties,

*SIAM Journal on Applied Algebra and Geometry*7 (2023) 77-103, (arXiv:2204.06414)with Anna-Laura Sattelberger, Bernd Sturmfels, and Simon Telen

Resonant resurgent asymptotics from quantum field theory,

*Nuclear Physics B*, 981 (2022) 115861 (arXiv:2202.01513)with David Broadhurst

Graphical functions in even dimensions,

*Communications in Number Theory and Physics*, 16 (2022) 515-614 (arXiv:2105.05015)with Oliver Schnetz

Semiclassical Trans-Series from the Perturbative Hopf-Algebraic Dyson-Schwinger Equations: ϕ

^{3}QFT in 6 Dimensions,*SIGMA 17 (2021), 087*(arXiv:2104.00593)with Gerald Dunne and Max Meynig

^{grad}Five loop renormalization of ϕ

^{3}theory with applications to the Lee-Yang edge singularity and percolation theory,*Physical Review D*, Volume 103, June 2021, 116024, (*distinguished as Editors' suggestion*) (arXiv:2103.16224)with John Gracey, Mikhail Kompaniets and Oliver Schnetz

Non-Perturbative Completion of Hopf-Algebraic Dyson-Schwinger Equations,

*Nuclear Physics B*, Volume 957, August 2020, 115096 (arXiv:2005.04265)with Gerald Dunne

The Hopf algebra structure of the R

^{∗}-operation,*Journal of High Energy Physics*, Article 61, July 2020 (arXiv:2003.04301)with Robert Beekveldt

^{grad}and Franz HerzogThe Euler characteristic of Out(F

_{n}),*Commentarii Mathematici Helvetici*, 95 (2020), 703-748 (arXiv:1907.03543)with Karen Vogtmann

Renormalized asymptotic enumeration of Feynman diagrams,

*Annals of Physics*, Volume 385, October 2017, Pages 95-135 (arXiv:1703.00840)Generating asymptotics for factorially divergent sequences,

*Electronic Journal of Combinatorics*, Volume 25, Issue 4, October 2018, P4.1 (arXiv:1603.01236)The results of this article are summarized in this colorful poster for FPSAC 2017.

Algebraic lattices in QFT renormalization,

*Letters in Mathematical Physics*, Volume 106, Issue 7, July 2016, Pages 879-911 (arXiv:1509.01862).Featured in Edward Tufte's new book Seeing with fresh eyes.

Feynman graph generation and calculations in the Hopf algebra of Feynman graphs,

*Computer Physics Communications*, Volume 185, Issue 12, December 2014, Pages 3317-3330 (arXiv:1402.2613):Information and downloads to the two programs

*feyngen*and*feyncop*is available on this page.

#### Proceedings

Taming a resurgent ultra-violet renormalon,

*Proceedings contribution to Loops and Legs in Quantum Field Theory 2022*, to appear in PoS, (arXiv:2209.10586)with David Broadhurst

Generating Asymptotics for factorially divergent sequences,

*Séminaire Lotharingien de Combinatoire*, 78B.28 (2017), 12 pp.Proceedings contribution for FPSAC 2017.

Feynman diagrams and their algebraic lattices,

*Resurgence, Physics and Numbers*CRM Series - Edizioni della Normale, Volume 20, 2017, Pages 91-107 (arXiv:1512.05237)with Dirk Kreimer: Proceedings contribution for Resurgence, Physics and Numbers.

Computations and generation of elements on the Hopf algebra of Feynman graphs,

*Proceedings of 16th International workshop on Advanced Computing and Analysis Techniques in physics research (ACAT)*, J. Phys.: Conf. Ser. 608, 2015 (arXiv:1410.7324)Proceedings contribution for ACAT 2014.

#### Other

Graphs in perturbation theory: Algebraic structure and asymptotics, May 2018 (arXiv:1807.02046):

My PhD thesis - available as hardcover in the Springer Theses series (amazon)

Algorithmization of the Hopf algebra of Feynman graphs, August 2013:

My master's thesis

### Supervision of MSc and BSc theses

MSc thesis, 2023, by Andrea Favorito, ETH Zurich, on 'Tropical Feynman period integration'.

BSc thesis, 2022, by Jean-Luc Portner, ETH Zurich, on 'The forested graph complex'.

MSc thesis, 2019, by Robert Beekvelt, Universiteit van Amsterdam, on the 'Hopf algebra structure of the R

^{*}operation' (co-supervision with Franz Herzog).

Feel free to contact me if you are a MSc or BSc student (preferably at the ETH or the University Zurich) and interested in a thesis project along the lines of my research.

### Invited talks

#### 2022/2023

Amplitudes 2023, CERN - Switzerland, 8 August 2023

Talk on 'Tropical Feynman integration in Minkowski space' (slides/video)

Max-Planck-Institute for Physics, Munich - Germany, 18 July 2023

Talk on 'Tropical Feynman integration in the Minkowski regime'

SIAM Conference on Applied Algebraic Geometry, Eindhoven - The Netherlands, 13 July 2023

Talk on 'Generalized permutahedra tropical sampling' (slides)

Algebra/Topology Seminar, Copenhagen - Denmark, 23 June 2023

Talk on 'On the amount of top-weight cohomology in the moduli space of curves'

From perturbative to non-perturbative QFT, Muenster - Germany, 15 June 2023

Talk on 'Probing the non-perturbative regime with tropical Feynman integration' (video)

International Symposium on Radiative Corrections - RADCOR, Crieff - UK, 29 May 2023

Talk on 'Feynman amplitudes and the moduli space of graphs'

Geometry and Topology seminar, Warwick - UK, 25 May 2023

Talk on 'The commutative graph complex and the amount of top-weight cohomology in the moduli space of curves'

Institute of Advanced Studies - Special Topology Seminar, Princeton - US, 19 May 2023

Talk on 'Graphs, Curves, and their Moduli Spaces'

Institute of Advanced Studies - Amplitudes Pizza Seminar, Princeton - US, 12 May 2023

Talk on 'Greatest hits on tropical geometry and Feynman integrals'

Geometry seminar, ETH Zürich - Switzerland, 19 April 2023

Talk on 'On the amount of top-weight cohomology in the moduli space of curves'

Non-Linear Algebra Seminar, Leipzig - Germany, 07 April 2023

Talk on 'Graphs, Curves, and their Moduli Spaces'

Theory Seminar DESY Zeuthen, Zeuthen - Germany, 06 April 2023

Talk on 'Tropical Feynman integration in the Minkowski regime'

Nico van Kampen Colloquium in Theoretical Physics, Utrecht - The Netherlands, 15 March 2023

Colloquium talk on 'Feynman diagrams at the limits of perturbation theory'

Feynman Integrals - UZH workshop, Zürich - Switzerland, 22 February 2023

Talk on 'Tropical Feynman integration in the Minkowski regime'

Applicable resurgent asymptotics: summary workshop at the Newton Institute, Cambridge - UK, 14 December 2022

Talk on 'Tropical sampling and its application to large-loop Feynman integration' (video)

Algebraic and Enumerative Combinatorics Seminar, Waterloo - Canada, 24 November 2022

Talk on 'Asymptotics of the Euler characteristic of Kontsevich's commutative graph complex'

Perimeter Institute: Quantum Fields and Strings Seminar, Waterloo - Canada, 22 November 2022

Talk on 'Computability in QFT and tropical QFT' (video and slides)

Geometry and Topology Seminar, McMaster University - Canada, 17 November 2022

Talk on 'Euler characteristics of Out(Fn)'

SLAC Theoretical Physics Seminar, Stanford - US, 8 November 2022

Talk on 'Tropical sampling and fast Feynman integration'

Bethe Center for Theoretical Physics, Bonn - Germany, 18 October 2022

Talk on 'Tropical sampling and its applications to Feynman integrals'

Physical resurgence: On quantum, gauge, and stringy, Cambridge - UK, 13 September 2022

Talk on 'Resonant resurgent asymptotics from quantum field theory' (online talk) (video)

#### 2021/2022

Algebra, Topology and the Grothendieck-Teichmüller group, Les Diablerets - Switzerland, 30 August 2022

Talk on 'Asymptotics for graph complex Euler characteristics' (online talk) (video)

Joint Mathematical Congress AMS-EMS-SMF 2022, Grenoble - France, 20 July 2022

Talk on 'Euler characteristics of Out(Fn) and renormalized topological field theory' (slides)

Graph Complexes and Quantum Field Theory, Berlin - Germany, 28 March 2022

Talk on 'Asymptotics for graph complex Euler characteristics' (video)

Feynman integrals and nonlinear algebra, Leipzig - Germany, 10 March 2022

Talk on 'Tropical Geometries in Feynman Integrals'

Structure of Local Quantum Field Theories, Berlin - Germany, 21 February 2022

Talk on 'Resonant resurgent asymptotics from quantum field theory' (online talk)

QCD Seminar, CERN - Switzerland, 3 December 2021

Talk on 'Fast Feynman integration with tropical geometry' (slides)

Talks in mathematical physics, ETH Zurich - Switzerland, 4 November 2021

Talk on 'The Euler characteristic of Out(Fn) and renormalized topological field theory'

Algebra/Topology Seminar, Copenhagen - Denmark, 15 October 2021

Talk on 'The Euler characteristic of Out(Fn) and renormalized topological field theory'

Niels Bohr Institute - High Energy Physics Seminar, Copenhagen - Denmark, 14 October 2021

Talk on 'Tropical Feynman integration'

Humboldt University - Kolleg Mathematics and Physics Berlin Seminar (KMPB day), Berlin - Germany, 27 September 2021

Talk on 'Graphical functions in even dimensions' (online talk)

#### 2020/2021

Theory Seminar of the Max-Planck-Institute, Munich - Germany, 30 July 2021

Talk on 'Numerical Feynman integration: The tropical way' (online talk)

Geomplitudes III, UC Davis - US, 4 June 2021

Talk on 'Tropical Feynman integration' (online talk)

Nikhef Theory Day, Amsterdam - The Netherlands, 28 May 2021

Talk on 'Numerical Feynman integration: The tropical way' (online talk)

Radcor & Loopfest 2021, Tallahassee - US, 17 May 2021

Talk on 'The tropical approach to numerical Feynman integration' (online talk: video/slides)

IAS Informal Amplitudes Meeting, Princeton - US, 24 March 2021

Talk on 'Tropical QFT and its applications' (online talk)

Amplitudes Seminar - Higgs Centre, Edinburgh - UK, 10 February 2021

Talk on 'Tropical quantum field theory and its applications' (online talk: video/slides)

Physics@Veldhoven: The Magic of Physics, Veldhoven - The Netherlands, 20 January 2021

Talk on 'Magic numbers at the heart of physics' (online talk for a broad audience)

Structure of Local Quantum Field Theories, Berlin - Germany, 18 January 2021

Talk on 'Numerical Feynman integration: The tropical way' (online talk)

Algebraic Structures in Perturbative Quantum Field Theory, IHES - France, 16-20 November 2020

Talk on 'The Euler characteristic of Out(F_n) and the Hopf algebra of graphs' (online talk: video/slides)

ACPMS, Intereuropean virtual seminar, 18 September 2020

Talk on 'The Euler characteristic of Out(F_n) and the Hopf algebra of graphs' (online talk: video/slides)

~~Graph Complexes and Quantum Field Theory, Berlin - Germany, 21-25 September 2020~~~~Talk on 'Out(F_n), M_gn and renormalized topological field theory'~~*(postponed due to SARS-CoV-2)*

#### 2019/2020

Topology Seminar, Karlsruhe - Germany, 02 July 2020

Talk on 'The Euler characteristic of Out(Fn) and renormalized topological field theory'

*(online talk)*Structure of Local Quantum Field Theories, Berlin - Germany, 04 May 2020

Talk on 'Fast Feynman parametric integral evaluation via tropical sampling'

*(online talk)*~~Resurgence in string and gauge theory, AIM - San Jose - US, 05 Mai 2020~~~~Talk on 'The ring of factorially divergent power series and its applications'~~*(canceled due to SARS-CoV-2)*~~Algebraic Structures in Perturbative Quantum Field Theory, IHES - Paris - France, 30 March 2020~~~~Talk on 'The Hopf algebra structure of the R* operation'~~*(postponed due to SARS-CoV-2)*Theoretical Physics Seminar, Liverpool - UK, 26 February 2020

Talk on 'Graphical functions applied to phi^3 in D = 6' (slides)

Emmy-Noether-Seminar, Erlangen - Germany, 7 February 2020

Talk on 'The Euler characteristic of Out(F_n)' (slides)

Geometry Seminar, Nijmegen - Netherlands, 28 January 2020

Talk on 'Renormalized topological field theory and the Euler characteristic of Out(F_n)' (slides)

Algebraic Combinatorics Seminar, Waterloo - Canada, 12 December 2019

Talk on 'Out(F_n), M_gn and renormalized topological field theory'

Canadian Mathematical Society Winter Meeting, Toronto - Canada, 9 December 2019

Talk on 'Constrained Graph Counting' (slides)

Theory Seminar of the Max-Planck-Institute, Munich - Germany, 4 December 2019

Talk on 'Efficient Monte-Carlo on Feynman parametric integrals'

Combinatorics Seminar, Vienna - Austria, 21 November 2019

Talk on 'Renormalized topological field theory'

Mathematical Physics Seminar, Vienna - Austria, 19 November 2019

Talk on 'The graphical function method applied to phi^3 in D=6' (slides)

Topology Seminar, Oxford - UK, 4 November 2019

Talk on 'The Euler characteristic of Out(Fn) and renormalized topological field theory'

Emmy-Noether-Seminar, Erlangen - Germany, 18 September 2019

Talk on 'The graphical function method in 2n-dimensions' (slides)

RADCOR 2019, Avignon - France, 12 September 2019

Talk on 'Systematic integration of higher loop gauge theory amplitudes in position space' (slides)

#### 2018/2019

Structure of Local Quantum Field Theories, Berlin - Germany, 28 Januar 2019

Talk on 'The Euler characteristic of OutFn'

Emmy-Noether-Seminar, Erlangen - Germany, 16 November 2018

Talk on 'Deriving asymptotics for factorially divergent power series'

Higgs as Probe and Portal Meeting, Amsterdam - Netherlands, 21 September 2018

Talk on 'Asymptotics and algebraic structure in perturbation theory'

#### 2017/2018

Summer school on structures in local quantum field theory, Les Houches - France, 4 - 15 June 2018

Talk on 'Graphs in Perturbation Theory' (slides)

Combinatorial structures in perturbative QFT, Waterloo - Canada, 25 May 2018

Talk on 'Counting graphs without given edge-induced subgraphs' (slides)

University of Waterloo: Algebraic Combinatorics Seminar, Waterloo - Canada, 10 May 2018

Talk on 'Hopf algebra of graphs for restricted graphical enumeration'

Workshop on Enumerative Combinatorics, Vienna - Austria, 16 - 20 October, 2017

Talk on 'Hopf algebras and factorial divergent power series: Algebraic tools for graphical enumeration' (slides)

ALEA in Europe, Vienna - Austria, 9 - 13 October, 2017

Talk on 'Generating Asymptotics for factorially divergent sequences' (slides)

#### 2016/2017

Formal Power Series and Algebraic Combinatorics (FPSAC), London - United Kingdom, 9 - 13 July, 2017

Poster about 'Generating Asymptotics for factorially divergent sequences'

Proceedings contribution: Generating Asymptotics for factorially divergent sequences,

*Séminaire Lotharingien de Combinatoire*, 78B.28 (2017), 12 pp.

Algebraic Combinatorics, Resurgence, Moulds and Applications (CARMA), Luminy - France, 26 - 30 June, 2017

Talk on 'Flag decompositions of graphs and their Hopf algebraic structure'

International school of subnuclear physics, Erice - Italy, 14 - 23 June 2017

Talk on 'Bounds and estimates for Feynman-perturbative expansions' (slides)

University of Waterloo: Algebraic Combinatorics Seminar, Waterloo - Canada, 30 March 2017

Talk on 'Zero-dimensional quantum field theory or counting classes of graphs using Gaussian integrals and Hopf algebras'

#### 2015/2016

Resurgence in Gauge and String Theories, Lisbon - Portugal, 18 - 22 July 2016

Talk on 'Asymptotic Calculus for Combinatorial Dyson-Schwinger Equations' (slides)

Kolleg Mathematik Physik Berlin (KMPB-Day), Berlin - Germany, 20 June 2016

Talk on 'Factorially divergent expansions from a QFT perspective'

Combinatorial Structures in Perturbative Quantum Field Theory, Vancouver - Canada, 21 March 2016

Talk on 'Generating functions' asymptotics' generating functions' (slides)

Paths to, from and in renormalisation, Potsdam - Germany, 8 - 12 February 2016

Talk on 'Asymptotic expansions and combinatorial Dyson-Schwinger equations' (slides)

#### 2014/2015

Resurgence, Physics and Numbers, Pisa - Italy, 18 - 22 May 2015

Talk on 'Combinatorics of Feynman diagrams and the algebraic lattice structure in QFT' (slides)

Proceedings contribution with Dirk Kreimer: Feynman diagrams and their algebraic lattices,

*Resurgence, Physics and Numbers*CRM Series - Edizioni della Normale, Volume 20, 2017, Pages 91-107 (arXiv:1512.05237)Dyson-Schwinger Equations in Modern Mathematics & Physics, Trento - Italy, 22 - 26 September 2014

Talk on 'The Hopf-Algebra of Feynman graphs as the foundation of Dyson-Schwinger equations'

Parallel session - Track 3, 16th International workshop on Advanced Computing and Analysis Techniques in physics research (ACAT), Prague - Czech Republic, 1 - 5 September 2014

Talk on 'Computations and generation of elements on the Hopf algebra of Feynman graphs'

Proceedings contribution: Computations and generation of elements on the Hopf algebra of Feynman graphs,

*Proceedings of 16th International workshop on Advanced Computing and Analysis Techniques in physics research (ACAT)*, J. Phys.: Conf. Ser. 608, 2015 (arXiv:1410.7324).

#### 2013/2014

Summer school on structures in local quantum field theory, Les Houches - France, 9 - 20 June 2014

Talk on 'The Hopf algebra of Feynman graphs as foundation for Dyson Schwinger equations'

#### 2012/2013

CERN Summer Student Lecture Programme, CERN - Switzerland, Summer 2012