Past

Will a large economy be stable? In this talk, I will present a model for a network economy where firms' productions are interdependent, and study the conditions under which such input-output networks admit a competitive economic equilibrium, where markets clear and profits are zero. Insights from random matrix theory allow to understand some of the emergent properties of this equilibrium and to provide a classification for the different types of crises it can be subject to. After this, I will endow the model with dynamics, and present results with strong links to generalised Lotka-Volterra models in theoretical ecology, where inter-species interactions are modelled with random matrices and where the system naturally self-organises into a critical state. In both cases, the stationary points must consist of positive species populations/prices/outputs. Building on these ideas, I will show the key concepts behind an economic agent-based model that can exhibit convergence to equilibrium, limit cycles and chaotic dynamics, as well as a phase of spontaneous crises whose origin can be understood using "semi-linear" dynamics.

Successful navigation of the Covid-19 pandemic is predicated on public cooperation with safety measures and appropriate perception of risk, in which emotion and attention play important roles. Signatures of public emotion and attention are present in social media data, thus natural language analysis of this text enables near-to-real-time monitoring of indicators of public risk perception. We compare key epidemiological indicators of the progression of the pandemic with indicators of the public perception of the pandemic constructed from ∼20 million unique Covid-19-related tweets from 12 countries posted between 10th March and 14th June 2020. We find evidence of psychophysical numbing: Twitter users increasingly fixate on mortality, but in a decreasingly emotional and increasingly analytic tone. Semantic network analysis based on word co-occurrences reveals changes in the emotional framing of Covid-19 casualties that are consistent with this hypothesis. We also find that the average attention afforded to national Covid-19 mortality rates is modelled accurately with the Weber–Fechner and power law functions of sensory perception. Our parameter estimates for these models are consistent with estimates from psychological experiments, and indicate that users in this dataset exhibit differential sensitivity by country to the national Covid-19 death rates. Our work illustrates the potential utility of social media for monitoring public risk perception and guiding public communication during crisis scenarios.

We investigate a graph theoretic analog of geodesic geometry. In a graph $G=(V,E)$ we consider a system of paths $P=\{P_{u,v}| u,v\in V\}$ where $P_{u,v}$ connects vertices $u$ and $v$. This system is consistent in that if vertices $y,z$ are in $P_{u,v}$, then the sub-path of $P_{u,v}$ between them coincides with $P_{y,z}$. A map $w:E\to(0,\infty)$ is said to induce $P$ if for every $u,v\in V$ the path $P_{u,v}$ is $w$-geodesic. We say that $G$ is metrizable if every consistent path system is induced by some such $w$. As we show, metrizable graphs are very rare, whereas there exist infinitely many 2-connected metrizable graphs.
 

We consider a chain of oscillators with one particle in contact with a thermostat at temperature T. The thermostat is modeled by a Langevin dynamics or a renewal of the velocity with a gaussian random variable with variance T. The dynamics of the oscillators is perturbed by a random exchange on velocities between nearest neighbor particles.
The (thermal) energy has a macroscopic superdiffusive behavior governed by a fractional heat equation (i.e. with a fractional Laplacian). The microscopic thermostat impose a particular boundary condition to the fractional Laplacian, corresponding to certain probabilities of transmission/reflection/absorption/creation for the corresponding superdiffusive Levy process.
This is from a series of works in collaboration with Tomazs Komorowski, Lenya Ryzhik, Herbert Spohn.

We will survey an analogy between random integers and random permutations, which goes back to works of Erdős and Kac and of Billingsley.
This analogy inspired results and proofs about permutations, originating in the setting of integers, and vice versa.
Extensions of this analogy will be described, involving the generalized Ewens measure on permutations, based on joint work with D. Elboim.
If time permits, an analogous analogy, this time between random polynomials over a finite field and random permutations, will be discussed and formalized, with some applications.
 

For an ordinary commutative Noetherian ring R we would define the singularity category to be the quotient of the (derived category of) finitely generated modules modulo the (derived category of) fg projective modules [``the bounded derived category modulo compact objects’’]. For a ring spectrum like C^*(BG) (coefficients in a field of characteristic p) it is easy to define the module category and the compact objects, but finitely generated objects need a new definition. The talk will describe the definition and show that the singularity category is trivial exactly when G is p-nilpotent. We will go on to describe the singularity category for groups with cyclic Sylow p-subgroup.

The idea of assigning weights to local coordinate functions is used in many areas of mathematics, such as singularity theory, microlocal analysis, sub-Riemannian geometry, or the theory of hypo-elliptic operators, under various terminologies. In this talk, I will describe some differential-geometric aspects of weightings along submanifolds. This includes a coordinate-free definition, and the construction of weighted normal bundles and weighted blow-ups. As an application, I will describe a canonical local model for isotropic embeddings in symplectic manifolds. (Based on joint work with Yiannis Loizides.)

In this session, Ben Fehrman and Markus Upmeier will give their thoughts on how to deliver a good talk for a conference or a seminar and tips for what to do and what to avoid. There will be a particular emphasis on how to give a good talk online. 

I will discuss a class of generalized global symmetries, which we call “Chern-Weil global symmetries,” that arise ubiquitously in gauge theories. The Noether currents of these Chern-Weil global symmetries are given by wedge products of gauge field strengths and their conservation follows from Bianchi identities, so they are not easy to break. However, exact global symmetries should not be allowed in a consistent theory of quantum gravity. I will explain how these symmetries are typically gauged or broken in string theory. Interestingly, many familiar phenomena in string theory, such as axions, Chern-Simons terms, worldvolume degrees of freedom, and branes ending on or dissolving in other branes, can be interpreted as consequences of the absence of Chern-Weil symmetries in quantum gravity, suggesting that they might be general features of quantum gravity.

Crystal Structure Prediction aims to reveal the properties that stable crystalline arrangements of a molecule have without stepping foot in a laboratory, consequently speeding up the discovery of new functional materials. Since it involves producing large datasets that themselves have little structure, an appropriate classification of crystals could add structure to these datasets and further streamline the process. We focus on geometric invariants, in particular introducing the density fingerprint of a crystal. After exploring its computations via Brillouin zones, we go on to show how it is invariant under isometries, stable under perturbations and complete at least for an open and dense space of crystal structures.

Human life expectancy has been increasing steadily over the last century but this has resulted in an increasing incidence of age-related chronic diseases. Over 60% of people over the age of 65 will suffer from more than one disease at the same time (multimorbidity) and 25-50% of those over 80 years old develop frailty, defined as an accumulation of deficits and loss of reserve. Multimorbidity and frailty have complex medical needs and are strongly associated with disability and hospitalization. However, current treatments are suboptimal with problems of polypharmacy due to the fact that each disease is treated individually. Geroprotectors target fundamental mechanisms of ageing common to multiple age-related diseases and shows promise in delaying the onset of multimorbidity and frailty in animal models. However, their clinical testing in patients has been challenging due to the high level of complexity in the mode of action of geroprotectors and in the way multimorbidity and frailty develop.

 The talk will give an overview of these problems and make the case for the use of AI approaches to solve some of those complex issues with a view of designing appropriate clinical trials with geroprotectors to prevent age-related multimorbidity and frailty and extend healthspan.

The interplay between fluid flows and fractures is ubiquitous in Nature and technology, from hydraulic fracturing in the shale formation to supraglacial lake drainage in Greenland and hydrofracture on Antarctic ice shelves.

In this talk I will discuss the above three examples, focusing on the scaling laws and their agreement with lab experiments and field observations. As climate warms, the meltwater on Antarctic ice shelves could threaten their structural integrity through propagation of water-driven fractures. We used a combination of machine learning and fracture mechanics to understand the stability of fractures on ice shelves. Our result also indicates that as meltwater inundates the surface of ice shelves in a warm climate, their collapse driven by hydrofracture could significantly influence the flow of the Antarctic Ice Sheets. 

The well-known Schur-Weyl duality provides a link between the representation theories of the general linear group $GL_n$ and the symmetric group $S_r$ by studying tensor space $(\mathbb{C}^n)^{\otimes r}$ as a ${(GL_n,S_r)}$-bimodule. We will discuss a few variations of this idea which replace $GL_n$ with some other interesting algebraic object (e.g. O$_n$ or $S_n$) and $S_r$ with a so-called diagram algebra. If time permits, we will also briefly look at how this can be used to define Deligne's category which 'interpolates' Rep($S_t$) for any complex number $t \in \mathbb{C}$.

Recent advances in computing power and the potential to make more realistic assumptions due to increased flexibility have led to the increased prevalence of simulation models in economics. While models of this class, and particularly agent-based models, are able to replicate a number of empirically-observed stylised facts not easily recovered by more traditional alternatives, such models remain notoriously difficult to estimate due to their lack of tractable likelihood functions. While the estimation literature continues to grow, existing attempts have approached the problem primarily from a frequentist perspective, with the Bayesian estimation literature remaining comparatively less developed. For this reason, we introduce a widely-applicable Bayesian estimation protocol that makes use of deep neural networks to construct an approximation to the likelihood, which we then benchmark against a prominent alternative from the existing literature.
 

Computer-based simulation of partial differential equations (PDEs) involves approximating the unknowns and relies on suitable description of geometrical entities such as the computational domain and its properties. The Finite Element Method (FEM) is by large the most popular technique for the computer-based simulation of PDEs and hinges on the assumption that discretized domain and unknown fields are both represented by piecewise polynomials, on tetrahedral or hexahedral partitions. In reality, the simulation of PDEs is a brick within a workflow where, at the beginning, the geometrical entities are created, described and manipulated with a geometry processor, often through Computer-Aided Design systems (CAD), and then used for the simulation of the mechanical behaviour of the designed object. This workflow is often repeated many times as part of a shape optimisation loop. Within this loop, the use of FEM on CAD geometries (which are mainly represented through their boundaries) calls then for (re-) meshing and re-interpolation techniques that often require human intervention and result in inaccurate solutions and lack of robustness of the whole process. In my talk, I will present the mathematical counterpart of this problem, I will discuss the mismatch in the mathematical representations of geometries and PDEs unknowns and introduce a promising framework where geometric objects and PDEs unknowns are represented in a compatible way. Within this framework, the challenges to be addressed in order to construct robust PDE solvers are many and I will discuss some of them. Mathematical results will besupported by numerical validation.

This talk will be three short stories on the general theme of elastic
instabilities in soft solids. First I will discuss the inflation of a
cylindrical cavity through a bulk soft solid, and show that such a
channel ultimately becomes unstable to a finite wavelength peristaltic
undulation. Secondly, I will introduce the elastic Rayleigh Plateau
instability, and explain that it is simply 1-D phase separation, much
like the inflationary instability of a cylindrical party balloon. I will
then construct a universal near-critical analytic solution for such 1-D
elastic instabilities, that is strongly reminiscent of the
Ginzberg-Landau theory of magnetism. Thirdly, and finally, I will
discuss pattern formation in layer-substrate buckling under equi-biaxial
compression, and argue, on symmetry grounds, that such buckling will
inevitably produce patterns of hexagonal dents near threshold.

Totally geodesic submanifolds are perhaps one of the easiest types of submanifolds of Riemannian manifolds one can study, since a maximal totally geodesic submanifold is completely determined by any one of its points and the tangent space at that point. It comes as a bit of a surprise then that classification of such submanifolds — up to an ambient isometry — is a nightmarish and widely open question, even on such a manageable and well-understood class of Riemannian manifolds as symmetric spaces.

We will discuss the theory of totally geodesic submanifolds of symmetric spaces and see that any maximal such submanifold is homogeneous and thus can be completely encoded by some Lie algebraic data called a 'Lie triple'. We will then talk about the duality between symmetric spaces of compact and noncompact type and discover that there is a one-to-one correspondence between totally geodesic submanifolds of a symmetric space and its dual. Finally, we will touch on the known classification in rank one symmetric spaces, namely in spheres and projective/hyperbolic spaces over real normed division algebras. Time permitting, I will demonstrate how all this business comes in handy in other geometric problems on symmetric spaces, e. g. in classification of isometric cohomogeneity one actions.

Link: https://teams.microsoft.com/l/meetup-join/19%3ameeting_ZGRiMTM1ZjQtZWNi…

The course covers the standard material on nonlinear wave equations, including local existence, breakdown criterion, global existence for small data for semi-linear equations, and Strichartz estimate if time allows.

We will discuss a generalisation of hyperbolic groups, from the group actions point of view. By studying torsion, we will see how this can help to answer questions about ordinary hyperbolic groups.

I will introduce product structure theory of graphs and show how families of graphs that have such a structure admit short adjacency labeling scheme and small induced universal graphs. Time permitting, I will talk about another recent application of product structure theory, namely vertex ranking (coloring).

A quantum circuit defines a discrete-time evolution for a set of quantum spins/qubits, via a sequence of unitary 'gates’ coupling nearby spins. I will describe how random quantum circuits, where each gate is a random unitary matrix, serve as minimal models for various universal features of many-body dynamics. These include the dynamical generation of entanglement between distant spatial regions, and the quantum "butterfly effect". I will give a very schematic overview of mappings that relate averages in random circuits to the classical statistical mechanics of random paths. Time permitting, I will describe a new phase transition in the dynamics of a many-body wavefunction, due to repeated measurements by an external observer.

A wide variety of fixed-point iterative methods for the solution of nonlinear operator equations in Hilbert spaces exists. In many cases, such schemes can be interpreted as iterative local linearisation methods, which can be obtained by applying a suitable preconditioning operator to the original (nonlinear) equation. Based on this observation, we will derive a unified abstract framework which recovers some prominent iterative methods. It will be shown that for strongly monotone operators this unified iteration scheme satisfies an energy contraction property. Consequently, the generated sequence converges to a solution of the original problem.

--

A link for this talk will be sent to our mailing list a day or two in advance.  If you are not on the list and wish to be sent a link, please contact @email.