Conjectures hold a special status in mathematics. Good conjectures epitomise milestones in mathematical discovery, and have historically inspired new mathematics and shaped progress in theoretical physics. Hilbert’s list of 23 problems and André Weil’s conjectures oversaw major developments in mathematics for decades. Crafting conjectures can often be understood as a problem in pattern recognition, for which Machine Learning is tailor-made. In this talk, I will propose a framework that allows a principled study of a space of mathematical conjectures. Using this framework and exploiting domain knowledge and machine learning, we generate a number of conjectures in number theory and group theory. I will present evidence in support of some of the resulting conjectures and present a new theorem. I will lay out a vision for this endeavour, and conclude by posing some general questions about the pipeline.

Superconformal ‘type B’ quantum mechanical sigma models arise in a variety of interesting contexts, such as the description of D-brane bound states in an $AdS_2$ decoupling limit. Focusing on $N= 2B$ models, we study superconformal indices which count short multiplets and provide an alternative to the standard Witten index, as the latter suffers from infrared issues. We show that the basic index receives contributions from lowest Landau level states in an effective magnetic field and that, due to the noncompactness of the target space, it is typically divergent. Fortunately, the models of interest possess an additional target space isometry which allows for the definition of a well-behaved refined index. We compute this index using localization of the functional integral and find that the result agrees with a naive application of the Atiyah-Bott fixed point formula outside of it’s starting assumptions. In the simplest examples, this formula can also be directly verified by explicitly computing the short multiplet spectrum.

It has been recently suggested that the strong Emergence Proposal is realized in equi-dimensional M-theory limits by integrating out all light towers of states with a typical mass scale not larger than the species scale, i.e the eleventh dimensional Planck mass. Within the BPS sector, these are transverse M2- and M5-branes, that can be wrapped and particle-like, carrying Kaluza-Klein momentum along the compact directions. We provide additional evidence for this picture by revisiting and investigating further the computation of $R^4$-interactions in M-theory à la Green-Gutperle-Vanhove. A central aspect is a novel UV-regularization of Schwinger-like integrals, whose actual meaning and power we clarify by first applying it to string perturbation theory. We consider then toroidal compactifications of M-theory and provide evidence that integrating out all light towers of states via Schwinger-like integrals thus regularized yields the complete result for $R^4$-interactions. In particular, this includes terms that are tree-level, one-loop and space-time instanton corrections from the weakly coupled point of view. Finally, we comment on the conceptual difference of our approach to earlier closely related work by Kiritsis-Pioline and Obers-Pioline, and conjecture a correspondence between two types of constrained Eisenstein series.

We will present a MCMC quantum algorithm to sample ground states of Hamiltonian systems.

We study transitions from chaotic to integrable Hamiltonians in the double scaled SYK and $p$-spin systems. The dynamics of our models is described by chord diagrams with two species. We begin by developing a path integral formalism of coarse graining chord diagrams with a single species of chords, which has the same equations of motion as the bi-local Liouville action, yet appears otherwise to be different and in particular well defined. We then develop a similar formalism for two types of chords, allowing us to study different types of deformations of double scaled SYK and in particular a deformation by an integrable Hamiltonian. The system has two distinct thermodynamic phases: one is continuously connected to the chaotic SYK Hamiltonian, the other is continuously connected to the integrable Hamiltonian, separated at low temperature by a first order phase transition.