In this talk I will discuss my extension of the Koszul duality theory of Beilinson, Ginzburg, and Soergel to the more general setting of quasi-abelian categories. In particular, I will define the notions of Koszul monoids, and quadratic monoids and their duals. Schneiders' embedding of a quasi-abelian category into an abelian category, its left heart, allows us to prove an equivalence of derived categories for certain categories of modules over Koszul monoids and their duals. The key examples of categories for which this theory works are the categories of complete bornological spaces and the categories of inductive limits of Banach spaces. These categories frequently appear in derived analytic geometry.

Polynomial preconditioning for linear solvers is well-known but not frequently used in current scientific applications.  Furthermore, polynomial preconditioning has long been touted as well-suited for parallel computing; does this claim still hold in our new world of GPU-dominated machines?  We give details of the GMRES polynomial preconditioner and discuss its simple implementation, its properties such as eigenvalue remapping, and choices such as the starting vector and added roots.  We compare polynomial preconditioned GMRES to related methods such as FGMRES and full GMRES without restarting. We conclude with initial evaluations of the polynomial preconditioner for parallel and GPU computing, further discussing how polynomial preconditioning can become useful to real-word applications.

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One of the lasting puzzles in quantum gravity is whether the holographic description of a gravitational system is a single quantum mechanical theory or the disorder average of many. In the latter case, multiple copies of boundary observables do not factorize into a product, but rather have higher moments. These correlations are interpreted in the bulk as due to geometries involving spacetime wormholes which connect disjoint boundaries. 

I will talk about the question of factorization and the role of wormholes for supersymmetric observables, specifically the supersymmetric index. Working with the Euclidean gravitational path integral, I will start with a bulk prescription for computing the supersymmetric index, which agrees with the usual boundary definition. Concretely, I will focus on the setting of charged black holes in asymptotically flat four-dimensional N=2 ungauged supergravity. In this case, the gravitational index path integral has an infinite family of Kerr-Newman classical saddles with different angular velocities. However, fermionic zero-mode fluctuations annihilate the contribution of each saddle except for a single BPS one which yields the expected value of the index. I will then turn to non-perturbative corrections involving spacetime wormholes, and show that fermionic zero modes are present for all such geometries, making their contributions vanish. This mechanism works for both single- and multi-boundary path integrals. In particular, only disconnected geometries without wormholes contribute to the index path integral, and the factorization puzzle that plagues the black hole partition function is resolved for the supersymmetric index. I will also present all other single-centered geometries that yield non-perturbative contributions to the gravitational index of each boundary. Finally, I will discuss implications and expectations for factorization and the status of supersymmetric ensembles in AdS/CFT in further generality. Talk based on [2107.09062] with Luca Iliesiu and Joaquin Turiaci.

Blind deconvolution problems are ubiquitous in many areas of imaging and technology and have been the object of study for several decades. Recently, motivated by the theory of compressed sensing, a new viewpoint has been introduced, motivated by applications in wireless application, where a signal is transmitted through an unknown channel. Namely, the idea is to randomly embed the signal into a higher dimensional space before transmission. Due to the resulting redundancy, one can hope to recover both the signal and the channel parameters. In this talk we analyze convex approaches based on lifting as they have first been studied by Ahmed et al. (2014). We show that one encounters a fundamentally different geometric behavior as compared to generic bilinear measurements. Namely, for very small levels of deterministic noise, the error bounds based on common paradigms no longer scale linearly in the noise level, but one encounters dimensional constants or a sublinear scaling. For larger - arguably more realistic - noise levels, in contrast, the scaling is again near-linear.

This is joint work with Yulia Kostina (TUM) and Dominik Stöger (KU Eichstätt-Ingolstadt).

In this talk we cover recent work in collaboration with Diego Granziol and Steve Roberts where we study the effect of mini-batching on the loss landscape of deep neural networks using spiked, field-dependent random matrix theory. We demonstrate that the magnitude of the extremal values of the batch Hessian are larger than those of the empirical Hessian and derive an analytical expressions for the maximal learning rates as a function of batch size, informing practical training regimens for both stochastic gradient descent (linear scaling) and adaptive algorithms, such as Adam (square root scaling), for smooth, non-convex deep neural networks. Whilst the linear scaling for stochastic gradient descent has been derived under more restrictive conditions, which we generalise, the square root scaling rule for adaptive optimisers is, to our knowledge, completely novel. For stochastic second-order methods and adaptive methods, we derive that the minimal damping coefficient is proportional to the ratio of the learning rate to batch size. We validate our claims on the VGG/WideResNet architectures on the CIFAR-100 and ImageNet datasets.