Algebraic geometry and combinatorics have a long-established culture of producing and interrogating classification datasets. These datasets can be at the limit of current computing resources, e.g., the Kreuzer-Skarke classification has almost half-a-billion entries. The time is ripe to explore the landscape of algebra-geometric datasets by means of data science techniques. In this talk, I will report on initial joint work with Bao, He, Hirst, Kasprzyk, and Majumder investigating the effectiveness of machine learning (ML) methods for predicting properties of Hilbert series and lattice polytopes. Our results support the idea that in many cases a mathematical theorem underlies ML predictions of high accuracy. Furthermore, our observations show that ML can also provide hints to new exciting and unexpected relations.

Feynman rules, 1-particle irreducible graphs, effective action and Legendre transform.

We give a new interpretation of the shifted Littlewood-Richardson coefficients $f_{\lambda\mu}^\nu$ ($\lambda,\mu,\nu$ are strict partitions). The coefficients $g_{\lambda\mu}$ which appear in the decomposition of Schur $Q$-function $Q_\lambda$ into the sum of Schur functions $Q_\lambda = 2^{l(\lambda)}\sum\limits_{\mu}g_{\lambda\mu}s_\mu$ can be considered as a special case of $f_{\lambda\mu}^\nu$ (here $\lambda$ is a strict partition of length $l(\lambda)$). We also give another description for $g_{\lambda\mu}$ as the cardinal of a subset of a set that counts Littlewood-Richardson coefficients $c_{\mu^t\mu}^{\tilde{\lambda}}$. This new point of view allows us to establish connections between $g_{\lambda\mu}$ and $c_{\mu^t \mu}^{\tilde{\lambda}}$. More precisely, we prove that $g_{\lambda\mu}=g_{\lambda\mu^t}$, and $g_{\lambda\mu} \leq c_{\mu^t\mu}^{\tilde{\lambda}}$. We conjecture that $g_{\lambda\mu}^2 \leq c^{\tilde{\lambda}}_{\mu^t\mu}$ and formulate some conjectures on our combinatorial models which would imply this inequality if it is valid. We present an approach using Fomin diagrams and Viennot's geometric construction for RSK correspondence to attack the conjecture.