Events

The stability of topological indices of condensed matter systems under the introduction of interactions is expected to fail in some examples. In this talk, I will present a Z_2-valued index for strongly interacting fermions on infinite lattices and prove its topological stability. I will show that it generalizes the well-known Fu-Kane-Mele index of time reversal invariant topological insulators, thereby proving its stability. This is joint work with Alex Bols and Mahsa Rahnama.

The stability of topological indices of condensed matter systems in the presence of interactions is not expected to hold universally. In this colloquium, I will first discuss the mathematical setup of the classification of interacting phases. I will then focus on a new Z_2-valued index for time-reversal invariant interacting fermions on infinite lattices and prove its topological stability. I will show that it generalizes the well-known Fu-Kane-Mele index of topological insulators, thereby proving its stability under large perturbations. This is joint work with Alex Bols and Mahsa Rahnama.

(Pseudo)spectral methods are popular for solving a wide variety of differential equations and generic optimization problems. Due to favourable approximation properties, such as rapid con-vergence for smooth functions, they are particularly popular and effective for solving time-independent Schrödinger equations. For example, in the domain of molecular quantum phy-sics, spectral and pseudospectral methods are the building blocks for a variety of variational techniques to solve nuclear Schrödinger equations. Despite their many favourable approxima-tion properties, these methods suffer from the curse of dimensionality and have slow conver-gence rates for highly-oscillatory functions. This limits their applicability in a wide variety of fields. Moreover, their effectiveness is highly dependent on the initial choice of the basis. In this talk I propose increasing the expressivity of (pseudo)spectral methods by composing a chosen orthonormal basis with an optimizable measurable mapping. This gives rise to an in-duced sequence. I characterize necessary and sufficient conditions for this sequence to inherit the completeness of the underlying orthonormal basis. Here, it is shown that the invertibility of the mapping is a necessary condition. Subsequently, I discuss the approximation of Schwartz functions in the linear span of Hermite functions that are composed with invertible mappings. To this end, I derive convergence guarantees and characterise the convergence order. Finally, I show numerical simulations for computing the vibrational spectra of polyatomic molecules. In these simulations, the invertible mapping was modelled using a normalizing flow, i.e., an inver-tible neural network to augment the expressivity of a given basis. Comparisons against the use of standard bases demonstrate orders-of-magnitude increased accuracy when using normali-zing flows.

Reinforcement learning (RL) is a version of stochastic control in which the system dynamics are unknown (up to the type of dynamics such as Markov chains or diffusion processes). There has been an upsurge of interest in RL for (continuous-time) controlled diffusions in recent years. In this talk I will highlight the latest developments on theory and algorithms arising from this study, including entropy regularized exploratory formulation, policy evaluation, policy gradient, q-learning, and regret analysis. Time permitting, I will also discuss applications to mathematical finance and generative AI.