N3.12

This talk is a brief introduction to the analysis of Donaldson theory, a branch of gauge theory. Roughly, this is an area of differential topology that aims to extract smooth structure invariants from the geometry of the space of solutions (moduli space) to a system of partial differential equations: the Yang-Mills equations.

I will start by discussing the differential geometric background required to talk about Yang-Mills connections. This will involve introducing the concepts of principal fibre bundles, connections and curvature. In the second half of the talk I will attempt to convey the flavour of the mathematics used to address technical issues in gauge theory. I plan to do this by presenting a sketch of the proof of Uhlenbeck's compactness theorem, the main technical tool involved in the compactification of the moduli space.

This talk will cover the connections of persistence with the topology of random structures. This includes an overview of various results from stochastic topology as well as the role persistence ideas  play in the analysis. This will include results on the maximally persistent classes and minimum spanning acycles/generalised trees.

One of the key properties of 1-parameter persistent homology is that its output can entirely encoded in a purely combinatorial way via persistence diagrams or barcodes.  However, many applications of topological data analysis naturally present themselves with more than 1 parameter. Multiparameter persistence suggests itself as the natural invariant to use, but the problem here is that the moduli space of multiparameter persistence diagrams has a much more complicated structure and we lack a combinatorial diagrammatic description.  An alternative approach was suggested by work of Giansiracusa-Moon-Lazar, where they investigated calculating a series of 1-parameter persistence diagrams as the other parameter is varied. In this talk I will discuss work in progress to produce a refinement of their perspective, making use the Algebraic Stability Theorem for persistent homology and work of Bauer-Lesnick on induced matchings.

The class of sheaf Laplacians can be characterized as the convex closure of a certain set of sparse semidefinite matrices. From this viewpoint, the study of sheaf Laplacians becomes a question of linear algebra on sparse matrices. I will discuss the applications of this perspective to the problems of approximating, sparsifying, and learning sheaves.

KW states that every finite abelian extension of the rationals is contained in a cyclotomic extension. In a previous talk, this was reduced to considering cyclic extensions of the local fields Q_p of prime power order l^r. When l\neq p, general theory is sufficient, however for l=p, more specific (although not necessarily more abstruse) descriptions become necessary.
I will focus on the simple structure of Q_p's extensions to obstruct the remaining obstructions to KW (and hopefully provoke some interest in local fields in those less familiar). Time-permitting, I will talk about this theorem in the context of class field theory and/or Hilbert's 12th problem.

The Kronecker-Weber theorem states that every finite abelian extension of the rationals is contained in some cyclotomic field. I will present a proof that emphasizes the standard local-global philosophy by first proving it for the p-adics and then deducing the global case.

The vulnerability of a host population to a specific disease measures how likely pathogen introduction will lead to an epidemic outbreak, and how hard it is to contain or eliminate an ongoing one. Predicting vulnerability is thus key to designing risk-reduction strategies that limit disease burden on public health and economic development. To do that, highly-resolved data tracking contacts and mobility of the host population need to integrate into detailed models of disease dynamics. This represents a twofold challenge. Firstly, we need theoretical frameworks that turn data feeds into predictors of epidemic risk, and can identify which of the structural features of the host population drive its vulnerability. Secondly, we need new ways to access, analyze, and share the relevant contact and mobility data: a necessary step to make our predictions realistic and reliable. In my talk, I will address both issues. I will show how to analytically derive the conditions that discriminate between epidemic regime and quick pathogen extinction, by representing empirically measured contacts as time-evolving complex networks. The analytical core of this theory leads to a broad range of applications. At the same time, its data-driven nature prompts context-specific predictions that can inform policymaking, as I will show in two case studies: reorganizing nurse scheduling to reduce the risk of spread of healthcare-associated infections; linking the features of livestock trade movements to the spatial spread of cattle diseases. The latter application is also an example of how limited access and incomplete data collection represent a big hurdle to predictive vulnerability analysis. To overcome this, I will present a collaborative platform for analyzing and comparing trade networks coming from several European countries. Using a bring code to the data approach, our platform surmounts the strict regulations preventing data sharing, and builds an algorithm that predicts vulnerability even in situations when limited data on cattle trade are available. The ultimate goal of all these theoretical and numerical developments is to inform strategies that reduce the vulnerability of the host population by restructuring its contacts. However, such restructuring may entail a feedback effect, acting as selective pressure on the pathogen itself. In the last part of my talk, I will extend the developed formalism to modeling evolutionary pathways that maximize the invasion potential of the pathogen, given the observed host population structure. Specifically, I will link the emergence of exotic replication behaviors in plant-infecting viruses to historical changes in plant distribution patterns.

In this talk I want to outline the proofs our of main results, i.e. the localisation theorem and the identification of the homotopy type of Grothendieck-Witt theory in terms of K- and L-theory.
Finally, as a small application I want to present a refinement and extension of certain maps relating certain Madsen-Tillmann spectra and orthogonal/symplectic algebraic K-theory spectra of the integers.

All original material is joint work with B.Calmès, E.Dotto, Y.Harpaz, M.Land, K.Moi, D.Nardin, T.Nikolaus and W.Steimle.
 

I will start by briefly reviewing the Tate construction and in particular, the Tate diagonal. Using these I will then illustrate Lurie’s notion of Poincaré categories by considering Poincaré structures on module categories over a ring (spectrum) in detail. In particular, I will describe the somewhat subtle genuine Poincaré structure on the category of perfect complexes of an ordinary ring, which conjecturally links the classical notion of the Grothendieck-Witt spectrum to our derived version. Finally, I will compute its associated L-groups.

I will present some basic concepts in Large Cardinal theory. A Large Cardinal axiom is the assertion of the existence of a cardinal so large that it entails the existence of set-sized models of ZFC, something which we know ZFC alone does not do. Large Cardinal axioms are therefore strengthenings of ZFC. We believe them to be consistent with ZFC, but this is a touchy subject. Nevertheless, Large Cardinal axioms have become an essential tool in set theory, providing insight into the fine structure of the set theoretic universe. In my talk, I will focus on inaccessible and measurable cardinals, and, if time permits, I will discuss supercompact cardinals.