Yaolong Cao: Gauge Theories on Geometric Spaces
In this talk, I will very briefly discuss gauge theories on various geometric spaces, including Riemann surfaces, 4-manifolds and manifolds with special or exceptional holonomy. More details on Calabi-Yau 4-folds will be mentioned, which are related to my research interests.

Doireann O'Kiely: Dynamic Wrinkling of Elastic Sheets
Our lives contain many scenarios in which thin structures wrinkle: a piece of tin foil or cling film crumples in our hand, and creases form in our skin as we age. In this talk I will discuss experimental and theoretical work by researchers in the Mathematical Institute on wrinkling of elastic sheets.
We study the impact of a solid onto an elastic sheet floating at a liquid-air interface. We observe a wave that is reminiscent of the ripples caused by dropping a stone in a pond, as well as spoke-like wrinkles, whose wavelength evolves in time. We describe these phenomena using a combination of asymptotic analysis, numerical simulations and scaling arguments.
 

In this talk I will discuss the upcoming REF2021 and its significance for early career researchers (research fellows and postdocs) including

• why it is so important to all UK maths departments
• why the timing of it could have important career consequences for ECRs
• publication issues such as quality versus quantity, and choice of journal
• the importance of Impact Case Studies
   

A panel discussion and Q&A, looking at some of the challenges and opportunities available for mathematicians outside universities. Featuring:

Madeleine Copin – North London Collegiate School
Josephine French – Health Data Insight, working in partnership with Public Health England
Martin Gould – Spotify
Dan Jones – Quadrature Capital
Adam Sardar – e-therapeutics

Wondering about how to organise your DPhil? How to make the most of your supervision meetings?

In this session we will explore these and other questions related to what makes a successful DPhil with help from faculty members, postdocs and DPhil students.

• In the first half of the session Dan Ciubotaru and Philip Maini will give short talks on their experiences as PhD students and supervisors.
• The second part of the session will be a panel discussion with final-year Dphil students and early postdocs.

The panel will consist of Thomas Wasserman, Renee Hoekzema, Jaroslav Fowkes and Carolina Matte Gregory. Senior faculty members will be kindly asked to leave the lecture theatre to ensure that students feel comfortable discussing their experiences with other students and postdocs without any senior faculty present.

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.