Past events

We show the primes have level of distribution 66/107 using triply well-factorable weights. This gives the highest level of distribution for primes in any setting, improving on the prior record level 3/5 of Maynard. We also extend this level to 5/8, assuming Selberg's eigenvalue conjecture. As a result, we obtain new upper bounds for twin primes and for Goldbach representations of even numbers $a$. For the Goldbach problem, this is the first use of a level of distribution beyond the 'square-root barrier', and leads to the greatest improvement on the problem since Bombieri--Davenport from 1966.

We will discuss the physics of M-theory compactifications onto G2-orbifolds of the type that can be desingularised via the method of Joyce and Karigiannis i.e. orbifolds where one has a singular locus of A1 singularities that admits a nowhere-vanishing (Z2-twisted) harmonic 1-form. Interestingly, there are topologically distinct desingularisations of such orbifolds which we show can be physically interpreted as different branches of the 4d vacuum moduli space of the arising gauge theories: Coulomb and Higgs branches. The results suggest generalisations of the results of Joyce and Karigiannis to G2-orbifolds with more diverse ADE singularities and higher order twists. As a bonus, we also get an isomorphism between the moduli space of flat connections on flat compact 3-manifolds and the moduli space of Ricci flat metrics on the G2-orbifolds. We will briefly discuss this. Based on 2309.12869 and 2312.12311.

As machine learning algorithms get integrated into the decision-making process of companies and organizations, insurance products are being developed to protect their providers from liability risk. Algorithmic liability differs from human liability since it is based on data-driven models compared to multiple heterogeneous decision-makers and its performance is known a priori for a given set of data. Traditional actuarial tools for human liability do not consider these properties, primarily focusing on the distribution of historical claims. We propose, for the first time, a quantitative framework to estimate the risk exposure of insurance contracts for machine-driven liability, introducing the concept of algorithmic insurance. Our work provides ML model developers and insurance providers with a comprehensive risk evaluation approach for this new class of products. Thus, we set the foundations of a niche area of research at the intersection of the literature in operations, risk management, and actuarial science. Specifically, we present an optimization formulation to estimate the risk exposure of a binary classification model given a pre-defined range of premiums. Our approach outlines how properties of the model, such as discrimination performance, interpretability, and generalizability, can influence the insurance contract evaluation. To showcase a practical implementation of the proposed framework, we present a case study of medical malpractice in the context of breast cancer detection. Our analysis focuses on measuring the effect of the model parameters on the expected financial loss and identifying the aspects of algorithmic performance that predominantly affect the risk of the contract.

Paper Reference: Bertsimas, D. and Orfanoudaki, A., 2021. Pricing algorithmic insurance. arXiv preprint arXiv:2106.00839.

Paper link: https://arxiv.org/pdf/2106.00839.pdf

Graph braid groups are similar to braid groups, except that they are defined as ‘braids’ on a graph, rather than the real plane. We can think of graph braid groups in terms of the discrete configuration space of a graph, which is a CW-complex. One can compute a presentation of a graph braid group using Morse theory. In this talk I will give a few examples on how to compute these presentations in terms of generating circuits of the graph. I will then go through a detailed example of a graph that gives a one-ended braid group.

We introduce a new class of groups with Thompson-like group properties. In the surface case, the asymptotic mapping class group contains mapping class groups of finite type surfaces with boundary. In dimension three, it contains automorphism groups of all finite rank free groups. I will explain how asymptotic mapping class groups act on a CAT(0) cube complex which allows us to show that they are of type F_infinity. 

This is joint work with Javier Aramayona, Kai-Uwe Bux, Jonas Flechsig and Xaolei Wu.

Brown-Erdős-Sós initiated the study of the maximum number of edges in an $n$-vertex $r$-graph such that no $k$ edges span at most $s$ vertices. If $s=rk-2k+2$ then this function is quadratic in $n$ and its asymptotic was previously known for $k=2,3,4$. I will present joint work with Stefan Glock, Jaehoon Kim, Lyuben Lichev and Shumin Sun where we resolve the cases $k=5,6,7$.

Porta and Yue Yu's model of derived analytic geometry takes as its category of basic, or affine, objects the category opposite to simplicial algebras over the entire functional calculus Lawvere theory. This is analogous to Lurie's approach to derived algebraic geometry where the Lawvere theory is the one governing simplicial commutative rings, and Spivak's derived smooth geometry, using the Lawvere theory of C-infinity-rings. Although there have been numerous important applications including GAGA, base-change, and Riemann-Hilbert theorems, these methods are still missing some crucial ingredients. For example, they do not naturally beget a good definition of quasi-coherent sheaves satisfying descent. On the other hand, the Toen-Vezzosi-Deligne approach of geometry relative to a symmetric monoidal category naturally provides a definition of a category of quasi-coherent sheaves, and in two such approaches to analytic geometry using the categories of bornological and condensed abelian groups respectively, these categories do satisfy descent.  In this talk I will explain how to compare the Porta and Yue Yu model of derived analytic geometry with the bornological one. More generally we give conditions on a Lawvere theory such that its simplicial algebras embed fully faithfully into commutative bornological algebras. Time permitting I will show how the Grothendieck topologies on both sides match up, allowing us to extend the embedding to stacks.

This is based on joint work with Oren Ben-Bassat and Kobi Kremnitzer, and follows work of Kremnitzer and Dennis Borisov.

The S-Matrix in flat space is a naturally holographic observable. S-Matrix elements thus contain valuable information about the putative dual CFT. In this talk, I will first introduce some basic aspects of Celestial Holography and then explain how these can be inferred directly from scattering amplitudes. I will then focus on how the singularity structure of amplitudes interplays with traditional CFT structures particularly in the context of the operator product expansion (OPE) of the dual CFT. I will conclude with some discussion about the role played by supersymmetry in simplifying the putative dual CFT.
 

First we will show that the continuum counterpart of the discrete individual-based mechanical model that describes the dynamics of two contiguous cell populations is given by a free-boundary problem for the cell densities.  Then, in addition to interactions, we will consider the microscopic movement of cells and derive a fractional cross-diffusion system as the many-particle limit of a multi-species system of moderately interacting particles.

First we will show that the continuum counterpart of the discrete individual-based mechanical model that describes the dynamics of two contiguous cell populations is given by a free-boundary problem for the cell densities.  Then, in addition to interactions, we will consider the microscopic movement of cells and derive a fractional cross-diffusion system as the many-particle limit of a multi-species system of moderately interacting particles.

Let $E$ be an elliptic curve over a number field $K$ and $n \geq 2$ an integer. We recall that elements of the $n$-Selmer group of $E/K$ can be explicitly written in terms of certain equations for $n$-coverings of $E/K$. Writing the elements in this way is called conducting an explicit $n$-descent. One of the applications of explicit $n$-descent is in finding generators of large height for $E(K)$ and from this point of view one would like to be able to take $n$ as large as possible. General algorithms for explicit $n$-descent exist but become computationally challenging already for $n \geq 5$. In this talk we discuss combining $n$- and $(n+1)$-descents to $n(n+1)$-descent and the role that invariant theory plays in this procedure.

The handlebody group is defined to be the mapping class group of a handelbody (rel. boundary). It is a subgroup of the mapping class group of the surface of the handlebody, and maps onto the outer automorphism group of its fundamental group (the free group of rank equal to its genus). 

Recently Hainaut and Petersen described a subspace of moduli space forming an orbifold classifying space for the handlebody group, and combined this with work of Chan-Galatius-Payne to construct cohomology classes in the group. I will talk about how one can build on their ideas to define a cocompact EG for the handlebody group inside Teichmüller space. This is a manifold with boundary and comes with a filtration by labelled disk systems which we call the `RGB (red-green-blue) disk complex.' I will describe this filtration, use it to describe the boundary of the manifold, and speculate about potential applications to duality results. Based on work-in-progress with Dan Petersen.

This presentation focuses on the study of the regulartiy of random wavelet series. We first study their belonging to certain functional spaces and we compare these results with long-established results related to random Fourier series. Next, we show how the study of random wavelet series leads to precise pointwise regularity properties of processes like fractional Brownian motion. Additionally, we explore how these series helps create Gaussian processes  with random Hölder exponents.

On homogeneous spaces, solutions to the prescribed Ricci curvature equation coincide with the critical points of the scalar curvature functional subject to a constraint. We provide a complete description of Palais--Smale sequences for this functional. As an application, we obtain new existence results for the prescribed Ricci curvature equation, which enables us to observe previously unseen phenomena. Joint work with Wolfgang Ziller (University of Pennsylvania).

The tremendous success of the Machine Learning paradigm heavily relies on the development of powerful optimization methods, and the canonical algorithm for training learning models is SGD (Stochastic Gradient Descent). Nevertheless, the latter is quite different from Gradient Descent (GD) which is its noiseless counterpart. Concretely, SGD requires a careful choice of the learning rate, which relies on the properties of the noise as well as the quality of initialization.

 It further requires the use of a test set to estimate the generalization error throughout its run. In this talk, we will present a new SGD variant that obtains the same optimal rates as SGD, while using noiseless machinery as in GD. Concretely, it enables to use the same fixed learning rate as GD and does not require to employ a test/validation set. Curiously, our results rely on a novel gradient estimate that combines two recent mechanisms which are related to the notion of momentum.

Finally, as much as time permits, I will discuss several applications where our method can be extended.

Oxford University Innovation, the University’s commercialisation team, will explain the support they can give to Maths researchers who want to generate commercial impact from their work and expertise. In addition to an overview of consulting, this talk will explain how mathematical techniques and software can be protected and commercialised.

In this possibly speculative talk I will try to outline a way to define analytic D-modules, using (higher) category theory and the ``six operations" on quasicoherent sheaves as the main tools. The aim is to follow the successful approach of Andy Jiang in the algebraic setting, who obtained such a theory without using stacks or formal schemes (as in Gaitsgory-Rozenblyum's approach). By using local cohomology, Jiang was able to avoid enlarging the category of algebras beyond the usual ones. We believe that an analytic variant of local cohomology can be used to recover the Ardakov-Wadsley theory of D-cap modules ``on the nose". (Work in progress).

Kaufman constructed a family of Fourier-decaying measures on the set of badly approximable numbers. Pollington and Velani used these to show that Littlewood’s conjecture holds for a full-dimensional set of pairs of badly approximable numbers. We construct analogous measures that have implications for inhomogeneous diophantine approximation. In joint work with Agamemnon Zafeiropoulos and Evgeniy Zorin, our idea is to shift the continued fraction and Ostrowski expansions simultaneously.

Junior Strings is a seminar series where DPhil students present topics of common interest that do not necessarily overlap with their own research area. This is primarily aimed at PhD students and post-docs but everyone is welcome.

Global scale ocean currents are strongly constrained by the Earth’s rotation, while this effect is generally negligible at small scales. In between, motions with scales from 1-10km are marginally affected by the Earth’s rotation. These intermediate scales, collectively termed the ocean submesoscale, have been hidden from view until recent years. Evidence from field measurements, numerical models, and satellite data have shown that submesoscales play a particularly important role in the upper ocean where they help to control the transport of material between the ocean surface and interior. In this talk I will review some recent work on submesoscale dynamics and their influence on biogeochemistry and accumulation of microplastics in the surface waters.