Past

Scattering amplitudes computed at a fixed loop order, along with any other object computed in perturbative QFT, can be expressed as a linear combination of a finite basis of loop integrals. To compute loop amplitudes in practise, such a basis of integrals must be determined. In this talk I introduce a new algorithm for finding bases of loop integrals and discuss its implementation in the publically available package Azurite.

The bipolar filtration of Cochran, Harvey and Horn initiated the study of deeper structures of the smooth concordance group of the topologically slice knots. We show that the graded quotient of the bipolar filtration has infinite rank at each stage greater than one. To detect nontrivial elements in the quotient, the proof uses higher order amenable Cheeger-Gromov $L^2$ $\rho$-invariants and infinitely many Heegaard Floer correction term $d$-invariants simultaneously. This is joint work with Jae Choon Cha.

A bold conjecture of Boyer-Gorden-Watson and others posit that for any irreducible rational homology 3-sphere M the following three conditions are equivalent: (1) the fundamental group of M is left-orderable, (2) M has non-minimal Heegaard Floer homology, and (3) M admits a co-orientable taut foliation. Very recently, this conjecture was established for all graph manifolds by the combined work of Boyer-Clay and Hanselman-Rasmussen-Rasmussen-Watson. I will discuss a computational survey of these properties involving half a million hyperbolic 3-manifolds, including new or at least improved techniques for computing each of these properties.
 

The theory of metric measure spaces with Ricci curvature from below is growing very quickly, both in the "Riemannian" class RCD and the general  CD one. I will review some of the most recent results, by illustrating the key identification results and technical tools (at the level of calculus in metric measure spaces) underlying these results.
 

The last 500 million years of Earth’s history have been punctuated by numerous episodes of abrupt climate change, some of them coincident with mass extinction events. Many of these climate events have been associated with massive volcanism, occurring during the emplacement of so-called Large Igneous Provinces (LIPs). Because of the significant impact of small modern eruptions on the Earth’s climate, a link between LIP volcanism and past climate change has been strongly advocated. Geochemical investigations of the sedimentary records which record major climate changes can give a profound insight into the proposed interactions between volcanic activity and climate. Mercury is a trace-gas emitted by modern volcanoes, which are the main source of this metal to the atmosphere. Ultimately atmospheric mercury is deposited in sediments, thus if enrichments in mercury are observed in sediments of the same age across the globe, a volcanic cause of these enrichments might be inferred. Osmium isotopes can also be used as a fingerprint of volcanic activity, as primitive basalts are enriched in unradiogenic 188Os. However, the continental crust is enriched in radiogenic 187Os. Therefore, the 187Os/188Os ratio can change with either more volcanic activity, or increased continental weathering during climate change. Changes in sedimentary mercury content and osmium isotopes can thus be used as markers of volcanism or weathering during climate events. However, a possible future step would be to quantify the amount of volcanism and/or weathering on the basis of these sedimentary excursions. The final part of this talk will introduce some simple quantitative models which may represent a first step towards such quantification, with the aim of further elaborating these models in the future.

In this paper, we consider the pricing and hedging of a financial derivative for an insider trader, in a model-independent setting. In particular, we suppose that the insider wants to act in a way which is independent of any modelling assumptions, but that she observes market information in the form of the prices of vanilla call options on the asset. We also assume that both the insider’s information, which takes the form of a set of impossible paths, and the payoff of the derivative are time-invariant. This setup allows us to adapt recent work of Beiglboeck, Cox, and Huesmann [BCH16] to prove duality results and a monotonicity principle, which enables us to determine geometric properties of the optimal models. Moreover, we show that this setup is powerful, in that we are able to find analytic and numerical solutions to certain pricing and hedging problems. (Joint with B. Acciaio and M. Huesmann)

This talk will introduce the notion of categorical rigidity and the automorphism class group of a category. We will proceed with calculations for several important categories, hopefully illuminating the inverse relationship between the automorphisms of a category and the extent to which the structure of its objects is determined categorically. To conclude, some discussion of what progress there is on currently open/unknown cases.

I will present a new approach to study the RG flow in 3d N=4 gauge theories, based on an analysis of the Coulomb branch of vacua. The Coulomb branch is described as a complex algebraic variety and important information about the strongly coupled fixed points of the theory can be extracted from the study of its singularities. I will use this framework to study the fixed points of U(N) and Sp(N) gauge theories with fundamental matter, revealing some surprising scenarios at low amount of matter.

 In a recent paper Friedl, Zentner and Livingston asked when a sum of torus knots is concordant to an alternating knot. After a brief analysis of the problem in its full generality, I will describe some effective obstructions based on Floer type theories.

We count monic quartic polynomials with prescribed Galois group, by box height. Among other things, we obtain the order of magnitude for  quartics, and show that non-quartics are dominated by reducibles. Tools include the geometry of numbers, diophantine approximation, the invariant theory of binary forms, and the determinant method. Joint with Rainer Dietmann.

Inelastic surface growth associated with continuous creation of incompatibility on the boundary of an evolving body is behind a variety of both natural processes (embryonic development,  tree growth) and technological processes (dam construction, 3D printing). Despite the ubiquity of such processes, the mechanical aspects of surface growth are still not fully understood. In this talk we present  a new approach to surface growth that allows one to address inelastic effects,  path dependence of the growth process and the resulting geometric frustration. In particular, we show that incompatibility developed during deposition can be fine-tuned to ensure a particular behaviour of the system in physiological (or working) conditions. As an illustration, we compute an explicit deposition protocol aimed at "printing" arteries, that guarantees the attainment of desired stress distributions in physiological conditions. Another illustration is the growth starategy for explosive plants, allowing a complete release of residual elastic energy with a single cut.

In this talk, I consider the problem of pricing and (statically)
hedging short-term contingent claims written on illiquid or
non-tradable assets.
In a first part, I show how to find the best European payoff written
on a given set of underlying assets for hedging (under several
metrics) a given European payoff written on another set of underlying
assets -- some of them being illiquid or non-tradable. In particular,
I present new results in the case of the Expected Shortfall risk
measure. I also address the associated pricing problem by using
indifference pricing and its link with entropy.
In a second part, I consider the more classic case of hedging with a
finite set of simple payoffs/instruments and I address the associated
pricing problem. In particular, I show how entropic methods (Davis
pricing and indifference pricing à la Rouge-El Karoui) can be used in
conjunction with recent results of extreme value theory (in dimension
higher than 1) for pricing and hedging short-term out-of-the-money
options such as those involved in the definition of Daily Cliquet
Crash Puts.

We analyze a fully discrete numerical scheme for solving a parabolic PDE on a moving surface. The method is based on a diffuse interface approach that involves a level set description of the moving surface. Under suitable conditions on the spatial grid size, the time step and the interface width we obtain stability and error bounds with respect to natural norms. Test calculations are presented that confirm our analysis.

I will introduce a McKean—Vlasov problem arising from a simple mean-field model of interacting neurons. The equation is nonlinear and captures the positive feedback effect of neurons spiking. This leads to a phase transition in the regularity of the solution: if the interaction is too strong, then the system exhibits blow-up. We will cover the mathematical challenges in defining, constructing and proving uniqueness of solutions, as well as explaining the connection to PDEs, integral equations and mathematical finance.

 The aim of this talk is to describe the classification problem for Higgs bundles and to explain how a combination of classical and Non-Reductive Geometric Invariant Theory might be used to solve this classification problem.
 
I will start by defining Higgs bundles and their physical origins. Then, I will present the classification problem for Higgs bundles. This will involve introducing the "stack" of Higgs bundles, a purely formal object which allows us to consider all isomorphism classes of Higgs bundles at once. Finally, I will explain how the stack of Higgs bundles can be described geometrically. As we will see, the stack of Higgs bundles can be decomposed into disjoint strata, each consisting of Higgs bundles of a given "instability type". Both classical and Non-Reductive GIT can then be applied to obtain moduli spaces for each of the strata.

The “field with one element” is an interesting algebraic object that in some sense relates linear algebra with set theory. In a much deeper vein it is also expected to have a role in algebraic geometry that could potentially “lift" Deligne’s proof of the final Weil Conjecture for varieties over finite fields to a proof of the Riemann hypothesis for the Riemann zeta function. The only problem is that it doesn’t exist. In this highly speculative talk I will discuss some of these concepts, and focus mainly on zeta functions of algebraic varieties over finite fields. I will give a (very) brief sketch of how to interpret various zeta functions in a geometric context, and try to explain what goes wrong for the Riemann zeta function that makes this a difficult problem.

Oxford Mathematics Public Lectures - Andrew Wiles, 28th November, 6.30pm, Science Museum, London SW7 2DD

Oxford Mathematics in partnership with the Science Museum is delighted to announce its first Public Lecture in London. World-renowned mathematician Andrew Wiles will be our speaker. Andrew will be talking about his current work and will also be 'in conversation' with mathematician and broadcaster Hannah Fry after the lecture.

This lecture is now sold out, but it will be streamed live and recorded. https://livestream.com/oxuni/wiles
 

Roelcke precompact groups are exactly the topological groups that can be realized as automorphism groups of omega-categorical structures (in continuous logic). In this talk, I will discuss a model-theoretic framework for the study of those groups and their dynamical systems as well as two concrete applications. The talk is based on joint work with Itaï Ben Yaacov and Tomás Ibarlucía.

The specialization question for rationality is the following one: assume that very general fibers of a flat proper morphism are rational, does it imply that all fibers are rational? I will talk about recent solution of this question in characteristic zero due to myself and Nicaise, and Kontsevich-Tschinkel. The method relies on a construction of various specialization morphisms for the Grothendieck ring of varieties (stable rationality) and the Burnside ring of varieties (rationality), which in turn rely on the Weak Factorization and Semi-stable Reduction Theorems.

The purpose of this work is to introduce a new constraint functional for shape optimization problems, which enforces the constructibility by means of additive manufacturing processes, and helps in preventing the appearance of overhang features - large regions hanging over void which are notoriously difficult to assemble using such technologies. The proposed constraint relies on a simplified model for the construction process: it involves a continuum of shapes, namely the intermediate shapes corresponding to the stages of the construction process where the total, final shape is erected only up to a certain level. The shape differentiability of this constraint functional is analyzed - which is not a standard issue because of its peculiar structure. Several numerical strategies and examples are then presented. This is a joint work with G. Allaire, R. Estevez, A. Faure and G. Michailidis.

In this joint work with D. Ciubotaru, we introduce the notion of local and global indices of Dirac operators for a rational Cherednik algebra H, with underlying reflection group G. In the local theory, I will report on some relations between the (local) Dirac index of a simple module in category O, the graded G-character and the composition series polynomials for standard modules. In the global theory, we introduce an "integral-reflection" module over which we define and compute the index of a (global) Dirac operator and show that the index is independent of the parameters. If time permits, I will discuss some local-global relations.

A new generation of low cost 3D tomography systems is based on multiple emitters and sensors that partially convolve measurements. A successful approach to deconvolve the measurements is to use nonlinear compressed sensing models. We discuss such models, as well as algorithms for their solution.