Time: 1.30pm-2.30pm

Speaker: Valerio Lucarini (University of Reading)

Title: A new mathematical framework for Atmospheric Blocking Events

Abstract: We use a simple yet Earth-like atmospheric model to propose a new framework for understanding the mathematics of blocking events, which are associated with low frequency, large scale waves in the atmosphere. Analysing error growth rates along a very long model trajectory, we show that blockings are associated with conditions of anomalously high instability of the atmosphere. Additionally, the lifetime of a blocking is positively correlated with the intensity of such an anomaly, against intuition. In the case of Atlantic blockings, predictability is especially reduced at the onset and decay of the blocking, while a relative increase of predictability is found in the mature phase, while the opposite holds for Pacific blockings, for which predictability is lowest in the mature phase. We associate blockings to a specific class of unstable periodic orbits (UPOs), natural modes of variability that cover the attractor of the system. The UPOs differ substantially in terms of instability, which explains the diversity of the atmosphere in terms predictability. The UPOs associated to blockings are indeed anomalously unstable, which leads to them being rarely visited. The onset of a blocking takes place when the trajectory of the system hops into the neighbourhood of one of these special UPOs. The decay takes place when the trajectory hops back to the neighbourhood of usual, less unstable UPOs associated with zonal flow. This justifies the classical Markov chains-based analysis of transitions between weather regimes. The existence of UPOs differing in the dimensionality of their unstable manifold indicates a very strong violation of hyperbolicity in the model, which leads to a lack of structural stability. We propose that this is could be a generic feature of atmospheric models and might be a fundamental cause behind difficulties in representing blockings for the current climate and uncertainties in predicting how their statistics will change as a result of climate change.

Time: 2.30pm - 3.30pm

Speaker: Yonatan Berman (London Mathematical Laboratory)

Title: The Ergodicity Problem in Economics

Abstract: The ergodicity problem queries the equality or inequality of time averages and expectation values. I will trace its curious history, beginning with the origins of formal probability theory in the context of gambling and economic problems in the 17th century. This is long before ergodicity was a word or a known concept, which led to an implicit assumption of ergodicity in the foundations of economic theory. Over the past decade we have asked what happens to foundational problems in economic theory if we export what is known about the ergodicity problem in physics and mathematics back to economics. Many problems can be resolved. Following an overview of our theoretical and conceptual progress, I will report on a recent experiment that strongly supports our view that human economic behavior is better described as optimizing time-average growth rates of wealth than as optimizing expectation values of wealth or utility of wealth.

Time: 3.30pm - 4.30pm

Speaker: Morgan Craig (University of Montreal)

Title: Exploiting heterogeneity to personalize oncolytic virus immunotherapy

Abstract: Cancer is an intrinsically heterogeneous disease distinguished by disparate outcomes based on cancer types, patient-specific characteristics, and treatment modalities. The classic approach to most cancer treatments is to administer cyclic chemotherapy at the maximal tolerable dose to reduce the tumour burden. Unfortunately, this is also generally accompanied by a host of secondary toxicity events, as cytotoxic chemotherapy also destroys healthy cells, primarily white blood cells. Instead, cancer immunotherapies aim to leverage the body’s own immune defence mechanisms against the tumour. Oncolytic viruses (OVs) are a type of immunotherapy that preferentially target cancerous cells, infecting and destroying them and simultaneously bolstering immune responses. Certain OVs are already clinically-available, however a barrier to their development is the heterogenous response to treatment observed in trials. It is therefore crucial to characterize heterogeneity within and around the tumour, and to quantify the effects that neighbouring tumour and immune cells have on therapeutic success. Here I will discuss quantitative approaches my lab uses to provide a way to test and optimize OV protocols before they are used in patients, ultimately reducing bottlenecks along the drug development pipeline, rationalizing therapeutic scheduling, and improving patient outcomes.

Abstract: I will discuss the construction of continuous solutions to the incompressible Euler equations that exhibit local dissipation of energy and the surrounding motivations. A significant open question, which represents a strong form of the Onsager conjecture, is whether such solutions exist that locally dissipate energy while having the maximal possible regularity of being 1/3-Hölder continuous.

Abstract: The main goal of the talk will be to present some recent results concerning the regularity of stable solutions to the semilinear problem $(-\Delta)^s u = f(u)$ in a bounded domain $\Omega\subset \mathbb R^n$ and its associated Dirichlet problem with zero exterior data.

The regularity of minimizers and, more generally, stable solutions to reaction-diffusion equations, has been a central problem in Calculus of Variations and PDEs in the last century. A phenomenon that is recurrent in these type of problems is that stable solutions are regular in low dimensions while singularities may appear in high dimensions. Our aim is to find the precise threshold dimension (depending on the parameter $s$) for which stable solutions to $(-\Delta)^s u = f(u)$ are bounded.

Despite the problem for $s=1$ involving the classical Laplacian is, nowadays, fully solved, there are quite few results in the case of the fractional Laplacian and the problem is still largely open. In this talk we will present some recent developments towards the answer of the previous question on the boundedness of stable solutions, and we will discuss some open questions.

Abstract: Given a bounded domain $\Omega$, the harmonic measure $\omega$ is a probability measure on $\partial\Omega$ and it characterizes where a Brownian traveller moving in $\Omega$ is likely to exit the domain from. It also enables a representation of solutions to elliptic PDEs by the Dirichlet boundary data. The elliptic measure is a non-homogenous variant of harmonic measure.

In the case of harmonic measures, there has been a lot of fruitful work in recent years that characterizes their relationship with the boundary surface measure. Some of the work uses the tool of Riesz transform, which is not readily available to the study of elliptic measures. In this talk I will discuss the analogous question for elliptic measures for a general class of elliptic operators, and show that quantitative absolute continuity of the elliptic measure with respect to the surface measure implies the uniform rectifiability of the boundary. We first use a compactness argument to prove the small constant case (i.e. when the elliptic operator is close to a constant-coefficient operator); then we use an extrapolation argument to bootstrap to the general case.

Abstract: In the last decades, it has been understood that many difficult optimization problems involving quadratic polynomial constraints could be approximated with semidefinite programs (that is, linear minimization problems over the set of positive semidefinite matrices). Unfortunately, solving general high-dimensional semidefinite programs is slow. Several heuristics have thus been developed to speed up the solving by exploiting the specific properties of semidefinite programs which appear in applications.

This talk will discuss one such heuristic, the Burer-Monteiro factorization. We will define it, explain which correctness guarantees have been rigorously established for it and see that, although numerical experiments seem to suggest that they are overly pessimistic, these guarantees are essentially optimal.

As a future application, we will consider phase retrieval problems, where one wants to recover a vector with complex coordinates from the modulus of linear measurements. We will explain how and why these problems can be approximated with semidefinite programs, and what the Burer-Monteiro heuristic could bring in this context.

(Joint work with Alden Waters)

Abstract: How well can you approximate real numbers by rationals with denominators coming from a given set? Although this old question has applications in many areas, in general this question seems impossibly hard - we don’t even know whether e+pi is rational or not! If you allow for a tiny number of bad exceptions, then a beautiful dichotomy occurs - either almost everything can be approximated or almost nothing! I’ll talk about this problem and recent joint work with Dimitris Koukoulopoulos which classifies when these options occur, answering an old question of Duffin and Schaeffer. This relies on a fun blend of different ideas, including ergodic theory, analytic number theory and graph theory.

Abstract: The goal of the talk is to show that every $L^2$ initial datum admits an a.e. smooth solution of the dissipative surface quasigeostrophic equation (SGQ); more precisely, we prove that those solutions are smooth outside a compact set (away from t=0) of quantifiable Hausdorff dimension. We will start the talk by introducing this PDE arising in meteorology, which shares many of the essential difficulties of the Euler/Navier-Stokes equations, for instance in terms of the question of finite time blow up. We draw analogies between SQG and the latter equations in several aspects, including the partial regularity results, and underline some extra structure that SQG enjoys. This is a joint work with Silja Haffter (EPFL).

Abstract: I will talk about recent work with Steve Shkoller, and Vlad Vicol, regarding shock wave formation for the compressible Euler equations.

Abstract: The main goal of this talk is to discuss the state-of-the-art in understanding the phenomena of phase transitions for a range of nonlinear Fokker-Planck equations with linear and nonlinear diffusion. They appear as natural macroscopic PDE descriptions of the collective behavior of particles such as Cucker-Smale models for consensus, the Keller Segel model for chemotaxis, and the Kuramoto model for synchronization. We will show the existence of phase transitions in a variety of these models using the natural free energy of the system and their interpretation as natural gradient flow structure with respect to the Wasserstein distance in probability measures. We will discuss both theoretical aspects as well as numerical schemes and simulations keeping those properties at the discrete level. This talk is based on several works in collaboration with A. Barbaro, J.A. Canizo, X. Chen, Y.-P. Choi, P. Degond, R.S. Gvalani, L. Pareschi, G.A. Pavliotis, A. Schlichting, Q. Wang, Z. Wang, L. Zhang.

Abstract: In this talk, I will discuss a Liouville type theorem for stationary or uniformly-rotating solutions of 2D Euler and other similar equations. The main question we want to address is whether every stationary/uniformly-rotating solution must be radially symmetric. Based on joint work with Jaemin Park, Jia Shi and Yao Yao.

Abstract: The BGK equation is a non-linear integro differential equation which models the behaviour of a dilute gas. We look at this equation posed on an interval where each end of the interval acts as a thermostatted wall with a given temperature. I will discuss how thermostatted walls can produce non-equilibrium steady states for spatially inhomogeneous kinetic equations and the mathematical challenges associated with this. I will also discuss our proof of existence of a steady state in a simple case. This is a joint work with Angeliki Menegaki.

Abstract: In this talk, we consider a Gross-Pitaevskii equation describing a dipolar Bose-Einstein condensate. We will show that solutions arising from initial data with energy below the energy of the Ground State, and that do not scatter, blow-up in finite time. The proof is based on localization properties for the fourth power of the Riesz transforms, that we prove by means of some decay estimates of the heat kernel associated to the parabolic biharmonic equation, and pointwise estimates for the square of the Riesz transforms.