We characterise the embeddability of simply connected 2-dimensional simplicial complexes in 3-space in a way analogous to Kuratowski’s characterisation of graph planarity, by excluded minors. This answers questions of Lovász, Pardon and Wagner.

 

The Gyárfás-Sumner conjecture says that every graph with huge (enough) chromatic number and bounded clique number contains any given forest as an induced subgraph. The Erdős-Hajnal conjecture says that for every graph H, all graphs not containing H as an induced subgraph have a clique or stable set of polynomial size. This talk is about a third problem related to both of these, the following. Say an n-vertex graph is "c-coherent" if every vertex has degree <cn, and every two disjoint vertex subsets of size at least cn have an edge between them. To prove a given graph H satisfies the Erdős-Hajnal conjecture, it is enough to prove H satisfies the conjecture in all c-coherent graphs and their complements, where c>0 is fixed and as small as we like. But for some graphs H, all c-coherent graphs contain H if c is small enough, so half of the task is done for free. Which graphs H have this property? Paths do (a theorem of Bousquet, Lagoutte, and Thomassé), and non-forests don't. Maybe all forests do? In other words, do all c-coherent graphs with c small enough contain any given forest as an induced subgraph? That question is the topic of the talk. It looks much like the Gyárfás-Sumner conjecture, but it seems easier, and there are already several pretty results. For instance the conjecture is true for all subdivided caterpillars (which is more than we know for Gyárfás-Sumner), and all trees of radius two. Joint work with Maria Chudnovsky, Jacob Fox, Anita Liebenau, Marcin Pilipczuk, Alex Scott and Sophie Spirkl.

A famous theorem of Mantel from 1907 states that every n-vertex graph with more than n^2/4 edges contains at least one triangle. In the 50s, Erdős asked for a quantitative version of this statement: for every n and e, how many triangles must an n-vertex e-edge graph contain?

This question has received a great deal of attention, and a long series of partial results culminated in an asymptotic solution by Razborov, extended to larger cliques by Nikiforov and Reiher. Currently, an exact solution is only known for a small range of edge densities, due to Lovász and Simonovits. In this talk, I will discuss the history of the problem and recent work which gives an exact solution for almost the entire range of edge densities. This is joint work with Hong Liu and Oleg Pikhurko.

(Joint work with Siva Athreya & Leonid Mytnik).

It is well known from the literature that ordinary differential equations (ODEs) regularize in the presence of noise. Even if an ODE is “very bad” and has no solutions (or has multiple solutions), then the addition of a random noise leads almost surely to a “nice” ODE with a unique solution. The first part of the talk will be devoted to SDEs with distributional drift driven by alpha-stable noise. These equations are not well-posed in the classical sense. We define a natural notion of a solution to this equation and show its existence and uniqueness whenever the drift belongs to a certain negative Besov space. This generalizes results of E. Priola (2012) and extends to the context of stable processes the classical results of A. Zvonkin (1974) as well as the more recent results of R. Bass and Z.-Q. Chen (2001).

In the second part of the talk we investigate the same phenomenon for a 1D heat equation with an irregular drift. We prove existence and uniqueness of the flow of solutions and, as a byproduct of our proof, we also establish path-by-path uniqueness. This extends recent results of A. Davie (2007) to the context of stochastic partial differential equations.

[1] O. Butkovsky, L. Mytnik (2016). Regularization by noise and flows of solutions for a stochastic heat equation. arXiv 1610.02553. To appear in Annal. Probab.

[2] S. Athreya, O. Butkovsky, L. Mytnik (2018). Strong existence and uniqueness for stable stochastic differential equations with distributional drift. arXiv 1801.03473.

Smooth Gaussian functions appear naturally in many areas of mathematics. Most of the talk will be about two special cases: the random plane model and the Bargmann-Fock ensemble. Random plane wave are conjectured to be a universal model for high-energy eigenfunctions of the Laplace operator in a generic domain. The Bargmann-Fock ensemble appears in quantum mechanics and is the scaling limit of the Kostlan ensemble, which is a good model for a `typical' projective variety. It is believed that these models, despite very different origins have something in common: they have scaling limits that are described be the critical percolation model. This ties together ideas and methods from many different areas of mathematics: probability, analysis on manifolds, partial differential equation, projective geometry, number theory and mathematical physics. In the talk I will introduce all these models, explain the conjectures relating them, and will talk about recent progress in understanding these conjectures.

Forecasting a diagnosis of Alzheimer’s disease is a promising means of selection for clinical trials of Alzheimer’s disease therapies. A positive PET scan is commonly used as part of the inclusion criteria for clinical trials, but PET imaging is expensive, so when a positive scan is one of the trial inclusion criteria it is desirable to avoid screening failures. In this talk I will describe a scheme for pre-selecting participants using statistical learning methods, and investigate how brain regions change as the disease progresses.  As a means of generating features I apply the Chen path signature. This is a systematic way of providing feature sets for multimodal data that can probe the nonlinear interactions in the data as an extension of the usual linear features. While it can easily perform a traditional analysis, it can also probe second and higher order events for their predictive value. Combined with Lasso regularisation one can auto detect situations where the observed data has nonlinear information.

The latest CPUs by Intel and ARM support vectorised operations, where a single set of instructions (e.g. add, multiple, bit shift, XOR, etc.) are performed in parallel for small batches of data. This can provide great performance improvements if each parallel instruction performs the same operation, but carries the risk of performance loss if each needs to perform different tasks (e.g. if else conditions). I will present the work I have done so far looking into how to recover the full performance of the hardware, and some of the challenges faced when trading off between ever larger parallel tasks, risks of tasks diverging, and how certain coding styles might be modified for memory bandwidth limited applications. Examples will be taken from finance and Monte Carlo applications, inspecting some standard maths library functions and possibly random number generation.

Low-rank plus sparse matrices arise in many data-oriented applications, most notably in a foreground-background separation from a moving camera. It is known that low-rank matrix recovery from a few entries (low-rank matrix completion) requires low coherence (Candes et al 2009) as in the extreme cases when the low-rank matrix is also sparse, where matrix completion can miss information and be unrecoverable. However, the requirement of low coherence does not suffice in the low-rank plus sparse model, as the set of low-rank plus sparse matrices is not closed. We will discuss the relation of non-closedness of the low-rank plus sparse model to the notion of matrix rigidity function in complexity theory.

The object of this talk is a class of generalised Newtonian fluids with implicit constitutive law.
Both in the steady and the unsteady case, existence of weak solutions was proven by Bul\'\i{}\v{c}ek et al. (2009, 2012) and the main challenge is the small growth exponent qq and the implicit law.
I will discuss the application of a splitting and regularising strategy to show convergence of FEM approximations to weak solutions of the flow. 
In the steady case this allows to cover the full range of growth exponents and thus generalises existing work of Diening et al. (2013). If time permits, I will also address the unsteady case.
This is joint work with Endre Suli.

We consider the problem of localising non-negative point sources, namely finding their locations and amplitudes from noisy samples which consist of the convolution of the input signal with a known kernel (e.g. Gaussian). In contrast to the existing literature, which focuses on TV-norm minimisation, we analyse the feasibility problem. In the presence of noise, we show that the localised error is proportional to the level of noise and depends on the distance between each source and the closest samples. This is achieved using duality and considering the spectrum of the associated sampling matrix.