Perturbation theory describes quantizations as small deformations of classical or free systems that are well understood. It produces power series that for example predict the particle interactions at high energies. The traditional techniques for such calculations are exhausted, and even with modern computers, progress is only possible through the development of new mathematics. This very active area of research at the cutting edge of theoretical physics and mathematics is transferring ideas between these disciplines and has already led to exciting new insights in both.

The long-term goal of my research is to solve the fundamental problems of QFT by advancing perturbation theory. My work [PW] with Wulkenhaar shows how high order perturbation series led to the first solution of an interacting theory in 4 dimensions [GHW]; I strive to achieve the same in physical theories. Over the next years, my contributions to this program will include:

- insights into large orders and improved resummation of 4 dimensional theories,
- methods to solve difference equations of Mellin transforms (
*integral reduction*), - large order asymptotics and Galois theory of Kontsevich's deformation quantization.

I developed a variant [P1] of Feynman integrals that assigns a rational function $\Hepp{G}$ to a Feynman diagram. It is defined by replacing the operations `+' and `·' of addition and multiplication by `max' and `+'. This is a drastic simplification, but the idea behind tropical geometry is that at least some interesting information is retained. In fact, it works extremely well; the integral $\Period{G}$ is predicted within 1% precision from $\Hepp{G}$ through a smooth interpolating curve. For the Feynman diagrams at 7 loops in $\phi^4$ theory, this is shown in the following figure:

This strong correlation shows that the tremendous transcendental complexity of Feynman integrals can be captured to high accuracy by their tropical analogues. This opens unprecedented opportunities to explore Feynman integrals at large orders and to study resummation.

**Goal 1.**Develop a full tropical version of scalar field theories.

The tropical perturbation series have rational coefficients and can be computed to arbitrary order. This will give valuable insights into the resummation problem.

**Goal 2.**Improve resummation methods for perturbation series in 4 dimensions.

In [KP] we showed that the large, uncontrolled uncertainties within these resummation procedures severely limit the utility of higher order perturbative corrections. With the tropical model, I can for the first time test the underlying assumptions, identify suitable resummation procedures and determine the errors. These improvements are crucial to obtain accurate predictions from perturbation theory, and to make contact, for example, with recent bootstrap calculations.

**Goal 3.**Approximate Feynman integrals at large order.

Remarkably, the tropical integral also detects all known identities of Feynman integrals. In fact, it suggests new identities not explained by any known relations [P1].

Symmetries are fundamental in physics, and understanding these new relations will reveal a hitherto missed symmetry.

**Goal 4.**Identify and prove the missing symmetries that explain all such identities.

To solve the reduction problem, I will determine the matrices representing the basic shifts $\vec{\ind} \mapsto \vec{\ind} + \vec{e}_i$ along coordinate directions. With these at hand, the reduction of an arbitrary shift $\Period{G}(\vec{\ind}+\vec{c})$ trivializes to a sequence of matrix multiplications. This proposal is radically different from established methods, which suffer from a severe increase in complexity with each additional shift. I already succeeded with the new approach in first examples, and I will develop the theory further and exploit the recursive structure of the face lattice of the Newton polytope of $P$.

**Goal 5.**Develop effective algorithms to reduce Mellin transforms.

A further implication of regulators is that the Mellin transform $\Period{G}(\vec{\ind})$ becomes a generating series for infinitely many periods, upon expanding $x_i^{\ind_i} = \sum_{k} (\ind_i \log x_i)^k/k!$. Each such coefficient is typically considered individually and gives ever more involved integrals with growing $k$. This inefficient approach obfuscates an important structure: The motivic Galois theory of the generating series as a whole admits a concise description, encoding information about its infinitely many coefficients [ABDG], [BD]. The techniques [P2] of fibrations by marked genus 0 curves extend to twisted cycles and can be applied directly to the entire generating series.

**Goal 6.**Extend iterated integration by fibrations to generating series of periods.

**Goal 7.**Prove convergence of canonical quantization for quadratic Poisson brackets.

Another direction of our project is the mysterious connection between the Grothendieck—Teichmüller group $\GRT$ and Brown's motivic Galois group $\Gmot$ of multiple zeta values: $\GRT$ acts on associators and quantizations by work of Dolgushev and Willwacher [W]; $\Gmot$ acts on the coefficients by our theorem. This suggests a new action of $\Gmot$ on quantizations.

**Goal 8.**Define Kontsevich weights motivically. Relate the $\Gmot$ and $\GRT$ actions.

**Goal 9.**Compute the Alekseev—Torossian and Rossi—Willwacher [RW] associators.

- [ABDG]
S. Abreu, R. Britto, C. Duhr and E. Gardi,
The algebraic structure of cut Feynman integrals and the diagrammatic coaction,
Phys. Rev. Lett.
**119**(July, 2017) p. 051601, arXiv:1703.05064. - [B]
M. Borinsky,
Tropical Monte Carlo quadrature for Feynman integrals,
Annales de l'Institut Henri Poincaré D
**10**(2023), no. 4, pp. 635—685, arXiv:2008.12310. - [BBKP]
T. Bitoun, C. Bogner, R. P. Klausen and E. Panzer,
Feynman integral relations from parametric annihilators,
Letters in Mathematical Physics
**109**, pp. 497—564 (2019), arXiv:1712.09215. - [BPP]
P. Banks, E. Panzer and B. Pym,
Multiple zeta values in deformation quantization,
Inventiones Mathematicae
**222**(2020), pp. 79—159, software here and here, arXiv:1812.11649. - [BD]
F. Brown and C. Dupont,
Lauricella hypergeometric functions, unipotent fundamental groups of the punctured Riemann sphere, and their motivic coactions, Nagoya Math. J.
**249**(2023), pp. 148—220, arXiv:1907.06603. - [EF]
I. Z. Emiris and V. Fisikopoulos,
Practical polytope volume approximation,
ACM Trans. Math. Softw.
**44**(June, 2018) pp. 38:1—38:21. - [GHW]
H. Grosse, A. Hock and R. Wulkenhaar,
Solution of the self-dual $\Phi^4$ QFT-model on four-dimensional Moyal space,
J. High Energ. Phys.
**2020**, 81 (2020). arXiv:1908.04543. - [KP] M. Kompaniets and E. Panzer, Minimally subtracted six loop renormalization of $O(n)$-symmetric $\phi^4$ theory and critical exponents, Phys. Rev. D 96:036016 (2017), arXiv:1705.06483.
- [P1]
E. Panzer,
Hepp's bound for Feynman integrals and matroids, Annales de l'Institut Henri Poincaré D
**10**(2023), no. 1, pp. 31—119, arXiv:1908.09820. - [P2]
E. Panzer,
Algorithms for the symbolic integration of hyperlogarithms with applications to Feynman integrals,
Computer Physics Communications
**188**(2015), pages 148—166, arXiv:1403.3385 [hep-th]. Latest version:*HyperInt*. - [PW]
E. Panzer and R. Wulkenhaar,
Lambert-W solves the noncommutative $\phi^4$-model,
Communications in Mathematical Physics
**374**, pp. 1935—1961 (2020), arXiv:1807.02945. - [RW] C. A. Rossi and T. Willwacher, P. Etingof's conjecture about Drinfeld associators. Apr., 2014, arXiv:1404.2047.
- [W]
T. Willwacher,
M. Kontsevich's graph complex and the Grothendieck—Teichmüller Lie algebra,
Inventiones mathematicae
**200**(June, 2015) pp. 671—760, arXiv:1009.1654.