We introduce meta-factorization, a theory that describes matrix decompositions as solutions of linear matrix equations: the projector and the reconstruction equation. Meta-factorization reconstructs known factorizations, reveals their internal structures, and allows for introducing modifications, as illustrated with SVD, QR, and UTV factorizations. The prospect of meta-factorization also provides insights into computational aspects of generalized matrix inverses and randomized linear algebra algorithms. The relations between the Moore-Penrose pseudoinverse, generalized Nyström method, and the CUR decomposition are revealed here as an illustration. Finally, meta-factorization offers hints on the structure of new factorizations and provides the potential of creating them. 

In commutative algebra localizing a ring and its modules is a fundamental technique. In the general case of a Grothendieck abelian category or even a triangulated category with small direct sums this is replaced by forming the quotient category by a localizing subcategory. Therefore the classification of these localizing subcategories becomes an important problem. I will begin by recalling the case of the (derived) module category of a commutative noetherian ring due to Gabriel and Hopkins/Neeman, respectively, in order to give an idea how such a classification can look like.

The case of interest in this talk is the derived category D(G) of smooth representation in characteristic p of a p-adic Lie group G. This is motivated by the emerging p-adic Langlands program. In joint work with C. Heyer we have some modest initial results if G is compact pro-p or abelian. which I will present.

1pm Jonathan Tam: Markov decision processes with observation costs

We present a framework for a controlled Markov chain where the state of the chain is only given at chosen observation times and of a cost. Optimal strategies therefore involve the choice of observation times as well as the subsequent control values. We show that the corresponding value function satisfies a dynamic programming principle, which leads to a system of quasi-variational inequalities (QVIs). Next, we give an extension where the model parameters are not known a priori but are inferred from the costly observations by Bayesian updates. We then prove a comparison principle for a larger class of QVIs, which implies uniqueness of solutions to our proposed problem. We utilise penalty methods to obtain arbitrarily accurate solutions. Finally, we perform numerical experiments on three applications which illustrate our framework.

Preprint at https://arxiv.org/abs/2201.07908

1.45pm Remy Messadene: signature asymptotics, empirical processes, and optimal transport

Rough path theory provides one with the notion of signature, a graded family of tensors which characterise, up to a negligible equivalence class, and ordered stream of vector-valued data. In the last few years, use of the signature has gained traction in time-series analysis, machine learning, deep learning and more recently in kernel methods. In this work, we lay down the theoretical foundations for a connection between signature asymptotics, the theory of empirical processes, and Wasserstein distances, opening up the landscape and toolkit of the second and third in the study of the first. Our main contribution is to show that the Hambly-Lyons limit can be reinterpreted as a statement about the asymptotic behaviour of Wasserstein distances between two independent empirical measures of samples from the same underlying distribution. In the setting studied here, these measures are derived from samples from a probability distribution which is determined by geometrical properties of the underlying path.

2.30-3.00 Tea & coffee in the mezzananie

3-4pm Julien Berestycki: Extremal point process of the branching Brownian motion

Holographic correlation functions are under good analytic control when none of the single trace operators live in long multiplets. This is famously the case for SCFTs with sixteen supercharges but it is also possible to construct examples with eight supercharges by exploiting space filling branes in AdS. In particular, one can study 4d N=2 theories which are related to each other by an S-fold in much the same way that N=3 theories are related to N=4 Super Yang-Mills. I will describe how modern methods provide a window into their correlation functions with an emphasis on anomalous dimensions. To compare the different S-folds we will need to go to one loop, and to go to one loop we will need to account for operator mixing. This provides an example of resolving degeneracy by resolving degeneracy.

The infamous Ramanujan tau-function is the starting point for many mysterious conjectures and difficult open problems within the realm of modular forms. In this talk, I will discuss some of our recent results pertaining to odd values of the Ramanujan tau-function. We use a combination of tools which include the Primitive Divisor Theorem of Bilu, Hanrot and Voutier, bounds for solutions to Thue–Mahler equations due to Bugeaud and Gyory, and the modular approach via Galois representations of Frey-Hellegouarch elliptic curves. This is joint work with Mike Bennett (UBC), Adela Gherga (Warwick) and Samir Siksek (Warwick).

A loop of Hamiltonian diffeomorphisms of a symplectic manifold $X$ defines, by clutching, a symplectic fibration over the two-sphere with fibre $X$.  We prove that the integral cohomology of the total space splits additively, answering a question of McDuff, and extending the rational cohomology analogue proved by Lalonde-McDuff-Polterovich in the late 1990’s. The proof uses a virtual fundamental class of moduli spaces of sections of the fibration in Morava K-theory. This talk reports on joint work with Mohammed Abouzaid and Mark McLean.

Bacteria represent the major component of the world’s biomass. A number of these bacteria are motile and swim with the use of flagellar filaments, which are slender helical appendages attached to a cell body by a flexible hook. Low Reynolds number hydrodynamics is the key for flagella to generate propulsion at a microscale [1]. In this talk I will discuss two projects related to swimming of a model bacterium Escherichia coli (E. coli).

E. coli has many flagellar filaments that are wrapped in a bundle and rotate in a counterclockwise fashion (if viewed from behind the cell) during the so-called ‘runs’, wherein the cell moves steadily forward. In between runs, the cell undergoes quick ‘tumble’ events, during which at least one flagellum reverses its rotation direction and separates from the bundle, resulting in erratic motion in place. Alternating between runs and tumbles allows cells to sample space by stochastically changing their propulsion direction after each tumble. In the first part of the talk, I will discuss how cells reorient during tumble and the mechanical forces at play and show the predominant role of hydrodynamics in setting the reorientation angle [2].

In the second part, I will talk about hydrodynamics of bacteria near walls in visco-elastic fluids. Flagellar motility next to surfaces in such fluids is crucial for bacterial transport and biofilm formation. In Newtonian fluids, bacteria are known to accumulate near walls where they swim in circles [3,4], while experimental results from our collaborators at the Wu Lab (Chinese University of Hong Kong) show that in polymeric liquids this accumulation is significantly reduced. We use a combination of analytical and numerical models to propose that this reduction is due to a viscoelastic lift directed away from the plane wall induced by flagellar rotation. This viscoelastic lift force weakens hydrodynamic interaction between flagellated swimmers and nearby surfaces, which results in a decrease in surface accumulation for the cells. 

References

[1] Lauga, Eric. "Bacterial hydrodynamics." Annual Review of Fluid Mechanics 48 (2016): 105-130.

[2] Dvoriashyna, Mariia, and Eric Lauga. "Hydrodynamics and direction change of tumbling bacteria." Plos one 16.7 (2021): e0254551.

[3] Berke, Allison P., et al. "Hydrodynamic attraction of swimming microorganisms by surfaces." Physical Review Letters 101.3 (2008): 038102.

[4] Lauga, Eric, et al. "Swimming in circles: motion of bacteria near solid boundaries." Biophysical journal 90.2 (2006): 400-412.

TBC

In this talk, I will explain how to obtain the space of local operators of a 4d N=2 gauge theory using the category of line operators in the Kapustin twist (holomorphic topological twist). This category is given a precise definition by Cautis-Williams, as the category of equivariant coherent sheaves on the space of Braverman-Finkelberg-Nakajima. We compute the derived endomorphism of the monoidal unit in this category, and show that it coincides with the vacuum module of the Poisson vertex algebra of Oh-Yagi and Butson. The Euler character of this space reproduces the Schur index. I will also explain how to obtain the space of local operators at the junction of minimal Wilson-t’Hooft line operators. Its Euler character can be compared to the index formula of Cordova-Gaiotto-Shao. This is based on arXiv: 2112.12164.