Past

I plan to present a brief introduction to optimal control theory (no background knowledge assumed), and discuss a fascinating and oft-forgotten family of problems where the optimal control behaves very strangely; it changes state infinitely often in finite time. This causes havoc in practice, and even more so in the literature.
 

"We study the hydrodynamic limit for the isothermal dynamics of an anharmonic chain under hyperbolic space-time scaling under varying tension. The temperature is kept constant by a contact with a heat bath, realised via a stochastic momentum-preserving noise added to the dynamics. The noise is designed to be large at the microscopic level, but vanishing in the macroscopic scale. Boundary conditions are also considered: one end of the chain is kept fixed, while a time-varying tension is applied to the other end. We show that the volume stretch and momentum converge to a weak solution of the isothermal Euler equations in Lagrangian coordinates with boundary conditions."

We will discuss a variant of Taubes’s Seiberg-Witten to Gromov theorem in the context of a 4-manifold with cylindrical ends, equipped with a nontrivial harmonic 2-form. This harmonic 2-form is allowed to be asymptotic to 0 on some (but not all) of its ends, and may have nondegenerate zeros along 1-submanifolds. Corollaries include various positivity results; some simple special cases of these constitute a key ingredient in Kutluhan-Lee-Taubes’s proof of HM = HF (Monopole Floer homology equals Heegaard Floer homology). The aforementioned general theorem is motivated by (potential) extensions of the HM = HF and Lee-Taubes’s HM = PFH (Periodic Floer homology) theorems.

What actually happens when you submit an article to a journal? How does refereeing work in practice? How can you keep editors happy as an author or referee? How does one become a referee or editor? What does 'publication' mean with the internet and arXiv?

In this panel we'll discuss what happens between finishing writing a mathematical paper and its final (?) publication, looking at the various roles that people play and how they work best.

Featuring Helen Byrne, Rama Cont and Jonathan Pila.

 

This talk will cover the connections of persistence with the topology of random structures. This includes an overview of various results from stochastic topology as well as the role persistence ideas  play in the analysis. This will include results on the maximally persistent classes and minimum spanning acycles/generalised trees.

This year sees the 100th anniversary of Emmy Noether receiving her Habilitation and thus becoming the first women to be granted the right to teach and lecture at a university in Prussia (now Germany).  Noether shaped modern algebra and her influence was felt in many other fields including topology.

We will start by exploring what algebraic topology is, how the subject was shaped by algebra (under the influence of Noether), before considering some current challenges and applications.

Despite their tiny size, microorganisms play a huge role in many biological, medical, and engineering phenomena. For example, massive plankton blooms are an integral part of the oceanic ecosystem. Algal cells incorporate carbon dioxide, which affects global warming. In industry, microorganisms are used in bioreactors to produce food and medicines and to treat sewage. The human body hosts hundreds of microorganism species, and the number of microorganisms in the human body is roughly double the number of cells in the body. In the intestine, approximately 1 kg of enterobacteria form a unique ecosystem, called the gut flora, which plays important roles in digestion and in relation to infection. Because of the considerable influence that microorganisms have on human life, the study of their behavior and function is important.

Recent research has demonstrated the importance of biomechanics in understanding the behavior and functions of microorganisms. For example, red tides can be induced by the interplay between the background flow and swimming cells. A dense suspension of bacteria can generate a coherent structure, which strongly enhances mass transport in a suspension. These phenomena show that the physical environments around cells alter their behavior and biological functions. Such a biomechanical understanding is still lacking in microbiology, and we believe that biomechanics can provide new perspectives on future microbiology.

In this talk, we first introduce some of our studies of the behavior of individual swimming microorganisms near surfaces. We show that hydrodynamic forces can trap cells at liquid–air or liquid–solid interfaces. We then introduce interactions between a pair of swimming microorganisms, because a two-body interaction is the simplest many-body interaction. We show that our mathematical models can describe the interactions between two nearby swimming microorganisms. Collective motions formed by a group of swimming microorganisms are also introduced. We show that some collective motions of microorganisms, such as coherent structures of bacterial suspensions, can be understood in terms of fluid mechanics. We then discuss how cellular-level phenomena can change the rheological and diffusion properties of a suspension. The macroscopic properties of a suspension are strongly affected by mesoscale flow structures, which in turn are strongly affected by the interactions between cells. Hence, a bottom-up strategy, i.e., from a cellular level to a continuum suspension level, represents a natural approach to the study of a suspension of swimming microorganisms. Finally, we discuss whether our understanding of biological functions can be strengthened by the application of biomechanics, and how we can contribute to the future of microbiology.

Berry's random wave conjecture is a heuristic that the eigenfunctions of a classically ergodic system ought to display Gaussian random behaviour, as though they were random waves, in the large eigenvalue limit. We discuss two manifestations of this conjecture for eigenfunctions of the Laplacian on the modular surface: Planck scale mass equidistribution, and an asymptotic for the fourth moment. We will highlight how the resolution of these two problems in this number-theoretic setting involves a delicate understanding of the behaviour of certain families of L-functions.

In the first part of the talk, we study risk-sharing equilibria where heterogenous agents trade subject to quadratic transaction costs. The corresponding equilibrium asset prices and trading strategies are characterised by a system of nonlinear, fully-coupled forward-backward stochastic differential equations. We show that a unique solution generally exists provided that the agents’ preferences are sufficiently similar. In a benchmark specification, the illiquidity discounts and liquidity premia observed empirically correspond to a positive relationship between transaction costs and volatility.
In the second part of the talk, we discuss how the model can be calibrated to time series of prices and the corresponding trading volume, and explain how extensions of the model with general transaction costs, for example, can be solved numerically using the deep learning approach of Han, Jentzen, and E (2018).
 (Based on joint works with Martin Herdegen and Dylan Possamai, as well as with Lukas Gonon and Xiaofei Shi)

Multiplicative preprojective algebras (MPAs) were originally defined by Crawley-Boevey and Shaw to encode solutions of the Deligne-Simpson problem as irreducible representations. 
MPAs have recently appeared in the literature from different perspectives including Fukaya categories of plumbed cotangent bundles (Etgü and Lekili) and, similarly, microlocal sheaves 
on rational curves (Bezrukavnikov and Kapronov.) After some motivation, I'll suggest a purely algebraic approach to study these algebras. Namely, I'll outline a proof that MPAs are 
2-Calabi-Yau if Q contains a cycle and an inductive argument to reduce to the case of the cycle itself.

Cracks in many soft solids behave very differently to the classical picture of fracture, where cracks are long and thin, with damage localised to a crack tip. In particular, small cracks in soft solids become highly rounded — almost circular — before they start to extend. However, despite being commonplace, this is still not well understood. We use a phase-separation technique in soft, stretched solids to controllably nucleate and grow small, nascent cracks. These give insight into the soft failure process. In particular, our results suggest fracture occurs in two regimes. When a crack is large, it obeys classical linear-elastic fracture mechanics, but when it is small it grows in a new, scale-free way at a constant driving stress.

Stochastic control problems are closely related to free boundary problems, where both the underlying fully nonlinear PDEs and the boundaries separating the action and waiting regions are integral parts of the problems. In this talk, we will propose a class of stochastic N-player games and show how the free boundary problems involve moving boundaries due to the additional game nature. We will provide explicit Nash equilibria by solving a sequence of Skorokhod problems. For the special cases of resource allocation problems, we will show how players change their strategies based on different network structures between players and resources. We will also talk about the insights from a sharing economy perspective. This talk is based on a joint work with Xin Guo (UC Berkeley) and Wenpin Tang (UCLA).

We present some regularity results for the gradient of solutions to very degenerate equations, which exhibit a great lack of ellipticity.
In particular we show that local weak solutions of the orthotropic p−harmonic equation are locally Lipschitz, for every $p\geq 2$ and in every dimension.
The results presented in this talk have been obtained in collaboration with Pierre Bousquet (Toulouse), Lorenzo Brasco (Ferrara) and Anna Verde (Napoli).
 

The Poincare Conjecture was first formulated over a century ago and states that there is only one closed simply connected 3-manifold, hinting at a link between 3-manifolds and their fundamental groups. This seemingly basic fact went unproven until the early 2000s when Perelman proved Thurston's much more powerful Geometrisation Conjecture, providing us with a powerful structure theorem for understanding all closed 3-manifolds.
In this talk I will introduce the results developed throughout the 20th century that lead to Thurston and Perelman's work. Then, using Geometrisation as a black box, I will present a proof of the Poincare Conjecture. Throughout we shall follow the crucial role that the fundamental group plays and hopefully demonstrate the geometric and group theoretical nature of much of the modern study of 3-manifolds.
As the title suggests, no prior understanding of 3-manifolds will be expected.
 

The Kronecker-Weber theorem states that every finite abelian extension of the rationals is contained in some cyclotomic field. I will present a proof that emphasizes the standard local-global philosophy by first proving it for the p-adics and then deducing the global case.

Many time-dependent PDEs -- KdV, Burgers, Gray-Scott, Allen-Cahn, Navier-Stokes and many others -- combine a higher-order linear term with a lower-order nonlinear term.  This talk will review the method of exponential integrators for solving such problems with better than 2nd-order accuracy in time.

A defect-based a posteriori error estimate for Krylov subspace approximations to the matrix exponential is introduced. This error estimate constitutes an upper norm bound on the error and can be computed during the construction of the Krylov subspace with nearly no computational effort. The matrix exponential function itself can be understood as a time propagation with restarts. In practice, we are interested in finding time steps for which the error of the Krylov subspace approximation is smaller than a given tolerance. Finding correct time steps is a simple task with our error estimate. Apart from step size control, the upper error bound can be used on the fly to test if the dimension of the Krylov subspace is already sufficiently large to solve the problem in a single time step with the required accuracy.