Seminars from until

We study the discrete-time evolution of a recombination transformation in population genetics acting on the set of measures on genetic sequences. The evolution can be described by a Markov chain on the set of partitions that converges to the finest partition. We describe the geometric decay rate to the limit and the quasi-stationary behavior when conditioned that the chain has not hit the limit.

Policy optimization, which learns the policy of interest by maximizing the value function via large-scale optimization techniques, lies at the heart of modern reinforcement learning (RL). In addition to value maximization, other practical considerations arise commonly as well, including the need of encouraging exploration, and that of ensuring certain structural properties of the learned policy due to safety, resource and operational constraints. These considerations can often be accounted for by resorting to regularized RL, which augments the target value function with a structure-promoting regularization term, such as Shannon entropy, Tsallis entropy, and log-barrier functions. Focusing on an infinite-horizon discounted Markov decision process, this talk first shows that entropy-regularized natural policy gradient methods converge globally at a linear convergence that is near independent of the dimension of the state-action space. Next, a generalized policy mirror descent algorithm is proposed to accommodate a general class of convex regularizers beyond Shannon entropy. Encouragingly, this general algorithm inherits similar convergence guarantees, even when the regularizer lacks strong convexity and smoothness. Our results accommodate a wide range of learning rates, and shed light upon the role of regularization in enabling fast convergence in RL.

This is a report on joint work in progress with L. Katzarkov, M. Kontsevich, and P. Pandit. The Homological Mirror Symmetry conjecture is stated as an equivalence of triangulated categories, one coming from algebraic geometry and the other from symplectic topology. An enhancement of the conjecture also identifies stability conditions (in the sense of Bridgeland) on these categories. We adopt the point of view that triangulated (DG/A-infinity) categories are non-commutative spaces of an algebraic nature. A stability condition can then be thought of as the analog of a Kähler class or polarization. Many, often still conjectural, constructions of stability conditions hint at a richer structure which we think of as analogous to a Kähler metric. For instance, a type of Donaldson and Uhlenbeck-Yau theorem is expected to hold. I will discuss these examples and common features among them, leading to a tentative definition.

The "sign problem" (SP) is the fundamental limitation to simulations of strongly correlated materials in condensed matter physics, solving quantum chromodynamics at finite baryon density, and computational studies of nuclear matter. It is often argued that the SP is not intrinsic to the physics of particular Hamiltonians, since the details of how it onsets, and its eventual occurrence, can be altered by the choice of algorithm or many-particle basis. Despite that, I plan to show in this talk that the SP in determinant quantum Monte Carlo (DQMC) is quantitatively linked to quantum critical behavior. This demonstration is done via simulations of a number of fundamental models of condensed matter physics, all of whose critical properties are relatively well understood.

One of the most surprising properties of deep neural networks (DNNs) is that they perform best in the overparameterized regime. We are taught early on that having more parameters than data points is a terrible idea. So why do DNNs work so well in a regime where classical learning theory predicts they should heavily overfit? By adapting the coding theorem from algorithmic information theory (which every physicist should learn about!) we show that DNNs are exponentially biased at initialisation to functions that have low descriptional (Kolmogorov) complexity. In other words, DNNs have an inbuilt Occam's razor, a bias towards simple functions. We next show that stochastic gradient descent (SGD), the most popular optimisation method for DNNs, follows the same bias, and so does not itself explain the good generalisation of DNNs. Our approach naturally leads to a marginal-likelihood PAC-Bayes generalisation bound which performs better than any other bounds on the market. Finally, we discuss why this bias towards simplicity allows DNNs to perform so well, and speculate on what this may tell us about the natural world.

We consider a class of holographic quantum error-correcting codes, built from perfect tensors in network configurations dual to Bruhat-Tits trees and their quotients by Schottky groups corresponding to BTZ black holes. The resulting holographic states can be constructed in the limit of infinite network size. We obtain a $p$-adic version of entropy which obeys a Ryu-Takayanagi like formula for bipartite entanglement of connected or disconnected regions, in both genus-zero and genus-one $p$-adic backgrounds, along with a Bekenstein-Hawking-type formula for black hole entropy. We prove entropy inequalities obeyed by such tensor networks, such as subadditivity, strong subadditivity, and monogamy of mutual information (which is always saturated).

I will begin by defining a canonical family of perturbations of the Dirac equation. These perturbations are complex anti-linear, thus ground states only form a real vector space. A special case of this theory is known as the Jackiw–Rossi theory, which models surface excitations on the surface of a topological insulator placed in proximity to an s-wave superconductor. While the physics of the theory is relatively well-understood, the mathematical side has not been studied, even on surfaces, not to mention its generalizations to higher dimensional and on nontrivial manifolds. I call these equations the ``generalized Jackiw–Rossi equations''.

After the definitions and connections to physics, I will present my current work on the generalized Jackiw–Rossi equations. My main result is a concentration phenomenon which proves the physical expectation that such Majorana fermions concentrate around the vortices of the superconducting order parameter. Moreover, I provide approximate solutions that are exponentially sharp in the large coupling limit.

If time permits, then I will show how these Majorana fermions define a bundle of projective spaces over the ``simple'' part of vortex moduli spaces. The holonomies of such bundles give rise to projective representations of (surface) braid groups, and thus, speculatively, can be of interest to quantum information theorists.

Neural network in machine learning has interesting similarity with quiver representation theory. In this talk, I will build an algebro-geometric formulation of a `computing machine', which is well-defined over the moduli space of representations. The main algebraic ingredient is to extend noncommutative geometry of Connes, Cuntz-Quillen, Ginzburg to near-rings, which capture the non-linear activation functions in neural network. I will also explain a uniformization between spherical, Euclidean and hyperbolic moduli of framed quiver representations.

A common form of extreme value modelling involves modelling excesses of a threshold by a generalised Pareto (GP) distribution. The GP model arises by considering the possible limiting distributions of excesses as the threshold increased. Selecting too low a threshold leads to bias from model misspecification; raising the threshold increases the variance of estimators: a bias-variance trade-off. Some threshold selection methods do not address this trade-off directly, but rather aim to select the lowest threshold above which the GP model is judged to hold approximately. We use Bayesian cross-validation to address the trade-off by comparing thresholds based on predictive ability at extreme levels. Extremal inferences can be sensitive to the choice of a single threshold. We use Bayesian model averaging to combine inferences from many thresholds, thereby reducing sensitivity to the choice of a single threshold. The methodology is illustrated using significant wave height datasets from the North Sea and from the Gulf of Mexico.

A common form of extreme value modelling involves modelling excesses of a threshold by a generalised Pareto (GP) distribution. The GP model arises by considering the possible limiting distributions of excesses as the threshold increased. Selecting too low a threshold leads to bias from model misspecification; raising the threshold increases the variance of estimators: a bias-variance trade-off. Some threshold selection methods do not address this trade-off directly, but rather aim to select the lowest threshold above which the GP model is judged to hold approximately. We use Bayesian cross-validation to address the trade-off by comparing thresholds based on predictive ability at extreme levels. Extremal inferences can be sensitive to the choice of a single threshold. We use Bayesian model averaging to combine inferences from many thresholds, thereby reducing sensitivity to the choice of a single threshold. The methodology is illustrated using significant wave height datasets from the North Sea and from the Gulf of Mexico.

Using the tensor Radon transform and related numerical methods, we study how bulk geometries can be explicitly reconstructed from boundary entanglement entropies in the specific case of AdS$_3$/CFT$_2$. We find that, given the boundary entanglement entropies of a 2d CFT, this framework provides a quantitative measure that detects whether the bulk dual is geometric in the perturbative (near AdS) limit. In the case where a well-defined bulk geometry exists, we explicitly reconstruct the unique bulk metric tensor once a gauge choice is made. We then examine the emergent bulk geometries for static and dynamical scenarios in holography and in many-body systems. Apart from the physics results, our work demonstrates that numerical methods are feasible and effective in the study of bulk reconstruction in AdS/CFT.

Recent years have seen much progress in the mathematical understanding of quantum charge transport under slow driving, in the presence of strong interactions between the charge carriers. I will give an overview of recent results, starting with the adiabatic theorem in an interacting setting, and continuing to topological transport where quantization can be shown to be valid beyond the linear response setting.