I'll be talking about my masters' research in Quantum Gravity in a way that is accessible to mathematicians.
Ambitious mathematical models of highly complex natural phenomena are challenging to analyse, and more and more computationally expensive to evaluate. This is a particularly acute problem for many tasks of interest and numerical methods will tend to be slow, due to the complexity of the models, and potentially lead to sub-optimal solutions with high levels of uncertainty which needs to be accounted for and subsequently propagated in the statistical reasoning process. This talk will introduce our contributions to an emerging area of research defining a nexus of applied mathematics, statistical science and computer science, called "probabilistic numerics". The aim is to consider numerical problems from a statistical viewpoint, and as such provide numerical methods for which numerical error can be quantified and controlled in a probabilistic manner. This philosophy will be illustrated on problems ranging from predictive policing via crime modelling to computer vision, where probabilistic numerical methods provide a rich and essential quantification of the uncertainty associated with such models and their computation.
The cobordism hypothesis gives a correspondence between the
framed local topological field theories with values in C and a fully
dualizable objects in C. Changing framing gives an O(n) action on the
space of local TFTs, and hence by the cobordism hypothesis it gives a
(homotopy coherent) action of O(n) on the space of fully dualizable
objects in C. One example of this phenomenon is that O(3) acts on the
space of fusion categories. In fact, O(3) acts on the larger space of
finite tensor categories. I'll describe this action explicitly and
discuss its relationship to the double dual, Radford's theorem,
pivotal structures, and spherical structures. This is part of work in
progress joint with Chris Douglas and Chris Schommer-Pries.
Identifying correlations within multiple streams of high-volume time series is a general but challenging problem. A simple exact solution has cost that is linear in the dimensionality of the data, and quadratic in the number of streams. In this work, we use dimensionality reduction techniques (sketches), along with ideas derived from coding theory and fast matrix multiplication to allow fast (subquadratic) recovery of those pairs that display high correlation.
Joint work with Jacques Dark
Symplectic duality is an equivalence of mathematical structures associated to pairs of hyper-Kahler cones. All known examples arise as the `Higgs branch’ and `Coulomb branch' of a 3d superconformal quantum field theory. In particular, there is a rich class of examples where the Higgs branch is a Nakajima quiver variety and the Coulomb branch is a moduli spaceof singular magnetic monopoles. In this case, I will show that the equivariant cohomology of the moduli space of based quasi-maps to the Higgs branch transforms as a Verma module for the deformation quantisation of the Coulomb branch
Mirror symmetry predicts surprising geometric correspondences between distinct families of algebraic varieties. In some cases, these correspondences have arithmetic consequences. Among the arithmetic correspondences predicted by mirror symmetry are correspondences between point counts over finite fields, and more generally between factors of their Zeta functions. In particular, we will discuss our results on a common factor for Zeta functions alternate families of invertible polynomials. We will also explore closed formulas for the point counts for our alternate mirror families of K3 surfaces and their relation to their Picard–Fuchs equations. Finally, we will discuss how all of this relates to hypergeometric motives. This is joint work with: Charles Doran (University of Alberta, Canada), Tyler Kelly (University of Cambridge, UK), Steven Sperber (University of Minnesota, USA), John Voight (Dartmouth College, USA), and Ursula Whitcher (American Mathematical Society, USA).
Self-organization is observed in systems driven by the “social engagement” of agents with their local neighbors. Prototypical models are found in opinion dynamics, flocking, self-organization of biological organisms, and rendezvous in mobile networks.
We discuss the emergent behavior of such systems. Two natural questions arise in this context. The underlying issue of the first question is how different rules of engagement influence the formation of clusters, and in particular, the emergence of 'consensus'. Different paradigms of emergence yield different patterns, depending on the propagation of connectivity of the underlying graphs of communication. The second question involves different descriptions of self-organized dynamics when the number of agents tends to infinity. It lends itself to “social hydrodynamics”, driven by the corresponding tendency to move towards the local means.
We discuss the global regularity of social hydrodynamics for sub-critical initial configurations.
The Skorokhod embedding problem aims to represent a given probability measure on the real line as the distribution of Brownian motion stopped at a chosen stopping time. In this talk, we consider an extension of the weak formulation of the optimal Skorokhod embedding problem. Using the classical convex duality approach together with the optimal stopping theory, we establish some duality. Moreover, based on the duality, we provide an alternative proof of the monotonicity principle proved by Beiglbock, Cox and Huesmann.
In 1923, Dulac published a proof of the claim that every real analytic vector field on the plane has only finitely many limit cycles (now known as Dulac's Problem). In the mid-1990s, Ilyashenko completed Dulac's proof; his completion rests on the construction of a quasianalytic class of functions. Unfortunately, this class has very few known closure properties. For various reasons I will explain, we are interested in constructing a larger quasianalytic class that is also a Hardy field. This can be achieved using Ilyashenko's idea of superexact asymptotic expansion. (Joint work with Tobias Kaiser)
Hyperbolic groups were introduced by Gromov and generalize the fundamental groups of closed hyperbolic manifolds. Since a closed hyperbolic manifold is aspherical, it is a classifying space for its fundamental group, and a hyperbolic group will also admit a compact classifying space in the torsion-free case. After an introduction to this and other topological finiteness properties of hyperbolic groups and their subgroups, we will meet a construction of R. Kropholler, building on work of Brady and Lodha. The construction gives an infinite family of hyperbolic groups with finitely-presented subgroups which are non-hyperbolic by virtue of their finiteness properties. We conclude with progress towards determining minimal examples of the "sizeable" graphs which are needed as input to the construction.
The immune system finds very rare amounts of pathogens and responds against them in a timely and efficient manner. The time to find and respond against pathogens does not vary appreciably with the size of the host animal (scale invariant search and response). This is surprising since the search and response against pathogens is harder in larger animals.
The first part of the talk will focus on using techniques from computer science to solve problems in immunology, specifically how the immune system achieves scale invariant search and response. I use machine learning techniques, ordinary differential equation models and spatially explicit agent based models to understand the dynamics of the immune system. I will talk about Hierarchical Bayesian non-linear mixed effects models to simulate immune response in different species.
The second part of the talk will focus on taking inspiration from the immune system to solve problems in computer science. I will talk about a model that describes the optimal architecture of the immune system and then show how architectures and strategies inspired by the immune system can be used to create distributed systems with faster search and response characteristics.
I argue that techniques from computer science can be applied to the immune system and that the immune system can provide valuable inspiration for robust computing in human engineered distributed systems.
Let A be an abelian variety over a strictly henselian discretely valued field K. In his 1992 paper "Néron models and tame ramification", Edixhoven has constructed a filtration on the special fiber of the Néron model of A that measures the behaviour of the Néron model with respect to tamely ramified extensions of K. The filtration is indexed by rational numbers in [0,1], and if A is wildly ramified, it is an open problem whether the places where it jumps are always rational. I will explain how an interpretation of the filtration in terms of logarithmic geometry leads to explicit formulas for the jumps in the case where A is a Jacobian, which confirms in particular that they are rational. This is joint work with Dennis Eriksson and Lars Halvard Halle.
We propose a new flexible unified framework for studying the time consistency property suited for a large class of maps defined on the set of all cash flows and that are postulated to satisfy only two properties -- monotonicity and locality. This framework integrates the existing forms of time consistency for dynamic risk measures and dynamic performance measures (also known as acceptability indices). The time consistency is defined in terms of an update rule, a novel notion that would be discussed into details and illustrated through various examples. Finally, we will present some connections between existing popular forms of time consistency.
This is a joint work with Tomasz R. Bielecki and Marcin Pitera.
(joint work with Sergei Starchenko)
Let p:C^n ->A be the covering map of a complex abelian variety and let X be an algebraic variety of C^n, or more generally a definable set in an o-minimal expansion of the real field. Ullmo and Yafaev investigated the topological closure of p(X) in A in the above two settings and conjectured that the frontier of p(X) can be described, when X is algebraic as finitely many cosets of real sub tori of A, They proved the conjecture when dim X=1. They make a similar conjecture for X definable in an o-minimal structure.
In recent work we show that the above conjecture fails as stated, and prove a modified version, describing the frontier of p(X) as finitely many families of cosets of subtori. We prove a similar result when X is a definable set in an o-minimal structure and p:R^n-> T is the covering map of a real torus. The proofs use model theory of o-minimal structures as well as algebraically closed valued fields.
Tim Harford, Financial Times columnist and presenter of Radio 4's "More or Less", argues that politicians, businesses and even charities have been poisoning the value of statistics and data. Tim will argue that we need to defend the value of good data in public discourse, and will suggest how to lead the defence of statistical truth-telling.
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