### About me

I am a researcher currently working as an Advanced Fellow at the ETH-ITS in Zürich. In February 2025, I will join the Perimeter Institute as a junior faculty member. My field of research is mathematical physics.

Much of my research explores the interplay between *moduli spaces* and *quantum field theory*. Moduli spaces are pivotal objects in mathematics, while quantum field theory is the language of physics at the elementary particle level. I am fascinated by the connections between these two domains due to their foundational significance in their corresponding fields.

This article in Quanta Magazine gives a popular scientific account of parts of my research.

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

#### Contact

michael.borinsky@eth-its.ethz.ch

### Upcoming events and trips

18-22 November: In Princeton, US at the Combinatorics of Fundamental Physics Workshop.

04-06 December: In Odense, Denmark at the Centre for Quantum Mathematics.

Feel free to email me if you are nearby and would like to discuss!

### Publications

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

#### Preprints

Bivariate exponential integrals and edge-bicolored graphs, preprint September 2024

with Chiara Meroni and Maximilian Wiesmann (further illustrations and computer code)

Tree-tubings and the combinatorics of resurgent Dyson-Schwinger equations, preprint August 2024

with Gerald Dunne and Karen Yeats

On the Euler characteristic of the commutative graph complex and the top weight cohomology of M

_{g}, preprint May 2024with an appendix by Don Zagier

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

with Benjamin Brück and Thomas Willwacher

#### Journal articles

The S

_{n}-equivariant Euler characteristic of the moduli space of graphs, preprint June 2023, to appear in*Algebraic & Geometric Topology*with Jos Vermaseren

The Euler characteristic of the moduli space of graphs,

*Advances in Mathematics*, 432 (2023) 109290 (arXiv:2301.01121)with Karen Vogtmann. This work is featured in a popular science piece by Leila Sloman in Quanta Magazine.

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}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

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 (further illustrations and computer code)

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

Tropical Monte Carlo quadrature for Feynman integrals,

*Ann. Inst. Henri Poincaré Comb. Phys. Interact.*, Volume 10 (2023), no. 4, pp. 635–685, (arXiv:2008.12310)the proof-of-concept implementation of the algorithm for numerical Feynman integration is available on this page

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

Tropical Feynman integration in the physical region,

*Proceedings contribution to EPS-HEP2023*, PoS 449, (arXiv:2310.19890)with Henrik Munch and Felix Tellander

Flow Oriented Perturbation Theory,

*Proceedings contribution to RADCOR2023*, PoS 432, (arXiv:2310.09708)with Alexandre Salas-Bernárdez, Zeno Capatti and Eric Laenen

Taming a resurgent ultra-violet renormalon,

*Proceedings contribution to Loops and Legs in Quantum Field Theory 2022*, PoS 416, (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

### Organization of scientific events

9-13 September 2024: With Karen Vogtmann, I organized a workshop on Moduli spaces of curves, moduli spaces of graphs and graph complexes at the ETH-ITS in Zürich, Switzerland.

19-23 August 2024: With Christian Blohmann, Ralph Kaufmann, Dirk Kreimer, and Karen Yeats, I organized a workshop on Combinatorics, Resurgence and Algebraic Geometry in Quantum Field Theory at the Max Planck Institute for Mathematics in Bonn, Germany.

05-09 September 2022: With Erik Panzer, I organized a workshop on Tropical and Convex Geometry and Feynman integrals at the ETH-ITS in Zürich, Switzerland.

### 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 interested in a BSc, MSc or PhD project along the lines of my research.

### Teaching activities

13-17 May 2024, Max Planck Institute for Mathematics in the Sciences, Leipzig - Germany

Invited 5-part lecture series on 'Counting graphs using quantum field theory'

17-21 June 2024, Institute for Advanced Study (IAS), Princeton - US

Invited 4-part lecture series on 'The tropical and discrete geometry of Feynman integrals'

26-27 October 2021, ETH Zürich, Switzerland

Mini-course From Feynman graphs to Calabi-Yau manifolds with Johannes Brödel and Albrecht Klemm

Summer 2019, Universiteit van Amsterdam, The Netherlands

Field Theory in Particle Physics, exercise sessions for course on particle physics

Winter 2016/2017, Humboldt Universität zu Berlin, Germany

Winter 2015/2016, Humboldt Universität zu Berlin, Germany

Proseminar on Infrared and collinear divergences in perturbative QFT

### Invited talks

#### 2024/2025

Hamburg University - ZMP Colloquium, Hamburg - Germany, 24 October 2024

Talk on 'On the Dark-Matter Problem of the Moduli Space of Curves'

DESY - Theory Seminar, Hamburg - Germany, 21 October 2024

Talk on 'Tropical (Feynman) Sampling'

#### 2023/2024

Max Planck Institute for Mathematics, Bonn - Germany, 7 August 2024

Talk 'On the Euler characteristic of the commutative graph complex'

Center Of Mathematical Sciences And Applications (CMSA), Cambridge - US, 21 May 2024

Talk 'On the Euler characteristic of the commutative graph complex and the top weight cohomology of M

_{g}'A panorama of moduli spaces, Frankfurt - Germany, 28 February 2024

Talk 'On the top-weight cohomology of the moduli space of curves' (video)

Perimeter Institute Colloquium, Waterloo - Canada, 20 February 2024

Talk on 'Algebraic geometry at the limits of perturbative QFT and dark matter in moduli spaces' (video)

Particle Physics Seminar, Torino - Italy, 22 November 2023

Talk on 'Tropical Feynman integration'

Purdue Topology Seminar, Lafayette - US, 15 November 2023

Talk 'On the top-weight cohomology of the moduli space of curves' (online)

IAS HEP Seminar, Princeton - US, 6 October 2023

Talk on 'The Dark-Matter Problem of the Moduli Space of Curves and its (partial) Tropical Solution' (video)

#### 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