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.

Buildings are geometric structures useful in understanding certain classes of groups. In a series of papers written during the 1980s, Ronan and Smith developed the theory of “presheaves on buildings”. By constructing a coefficient system consisting of kP-modules (where P is the stabiliser of a given simplex), and computing the sheaf homology, they proved several results relating the homology spaces with the irreducible G-modules. In this talk we discuss their methods as well as our implementation of the algorithms, which has allowed us to efficiently compute the irreducible representations of some groups of Lie type.

We consider the Plebanski-Demianski family of solutions of minimal gauged supergravity in D=4, which describes an accelerating, rotating and charged black-hole in AdS4. The 4d metric has conical singularities, but we show that it can uplifted to a completely regular solution of D=11 supergravity. We focus on the supersymmetric and extremal case, where the near-horizon geometry is AdS2x\Sigma, where \Sigma is a spindle, or weighted projective space. We argue that this is dual to a d=1, N=(2,0) SCFT which is the IR limit of a 3d SCFT compactified on a spindle. This, in turn, should be realized holographically by wrapping a stack of M2-branes on a spindle. Such construction displays two interesting features: 1) supersymmetry is realized in a novel way, which is not the topological twist, and 2) the R-symmetry of the d=1 SCFT mixes with the U(1) isometry of the spindle, even in the absence of rotation. A similar idea also applies to a class of AdS3x\Sigma solutions of minimal gauged supergravity in D=5.

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.

For some nonlocal PDEs, its steady states can be seen as critical points of an associated energy functional. Therefore, if one can construct perturbations around a function such that the energy decreases to first order along the perturbation, this function cannot be a steady state. In this talk, I will discuss how this simple variational approach has led to some recent progresses in the following equations, where the key is to carefully construct a suitable perturbation.

I will start with the aggregation-diffusion equation, which is a nonlocal PDE driven by two competing effects: nonlinear diffusion and long-range attraction. We show that all steady states are radially symmetric up to a translation (joint with Carrillo, Hittmeir and Volzone), and give some criteria on the uniqueness/non-uniqueness of steady states within the radial class (joint with Delgadino and Yan).

I will also discuss the 2D Euler equation, where we aim to understand under what condition must a stationary/uniformly-rotating solution be radially symmetric. Using a variational approach, we settle some open questions on the radial symmetry of rotating patches, and also show that any smooth stationary solution with compactly supported and nonnegative vorticity must be radial (joint with Gómez-Serrano, Park and Shi).

Symbol alphabets of n-particle amplitudes in N=4 super-Yang-Mills theory are known to contain certain cluster variables of Gr(4,n) as well as certain algebraic functions of cluster variables. In this talk we suggest an algorithm for computing these symbol alphabets from plabic graphs by solving matrix equations of the form C.Z = 0 to associate functions on Gr(m,n) to parameterizations of certain cells of Gr_+ (k,n) indexed by plabic graphs. For m=4 and n=8 we show that this association precisely reproduces the 18 algebraic symbol letters of the two-loop NMHV eight-point amplitude from four plabic graphs. We further show that it is possible to obtain all rational symbol letters (in fact all cluster variables) by solving C.Z = 0 if one allows C to be an arbitrary cluster parameterization of the top cell of Gr_+ (n-4,n).

There is a connection between certain smooth representations of a reductive p-adic group and the representations of the Iwahori-Hecke algebra of this p-adic group. This Iwahori-Hecke algebra is a specialisation of a more general affine Hecke algebra. In this talk, we will discuss affine Hecke algebras and graded Hecke algebras. We will state a result from Lusztig (1989) that relates the representation theory of an affine Hecke algebra and a particular graded Hecke algebra and we will present a simple example of this relation.

Nonlinear Fokker-Planck equations with measures as initial data and McKean-Vlasov equations This talk is about joint work with Viorel Barbu. We consider a class of nonlinear Fokker-Planck (- Kolmogorov) equations of type \begin{equation*} \frac{\partial}{\partial t} u(t,x) - \Delta_x\beta(u(t,x)) + \mathrm{div} \big(D(x)b(u(t,x))u(t,x)\big) = 0,\quad u(0,\cdot)=\mu, \end{equation*} where $(t,x) \in [0,\infty) \times \mathbb{R}^d$, $d \geq 3$ and $\mu$ is a signed Borel measure on $\mathbb{R}^d$ of bounded variation. In the first part of the talk we shall explain how to construct a solution to the above PDE based on classical nonlinear operator semigroup theory on $L^1(\mathbb{R}^d)$ and new results on $L^1- L^\infty$ regularization of the solution semigroups in our case. In the second part of the talk we shall present a general result about the correspondence of nonlinear Fokker-Planck equations (FPEs) and McKean-Vlasov type SDEs. In particular, it is shown that if one can solve the nonlinear FPE, then one can always construct a weak solution to the corresponding McKean-Vlasov SDE. We would like to emphasize that this, in particular, applies to the singular case, where the coefficients depend "Nemytski-type" on the time-marginal law of the solution process, hence the coefficients are not continuous in the measure-variable with respect to the weak topology on probability measures. This is in contrast to the literature in which the latter is standardly assumed. Hence we can cover nonlinear FPEs as the ones above, which are PDEs for the marginal law densities, realizing an old vision of McKean.

References V. Barbu, M. Röckner: From nonlinear Fokker-Planck equations to solutions of distribution dependent SDE, Ann. Prob. 48 (2020), no. 4, 1902-1920. V. Barbu, M. Röckner: Solutions for nonlinear Fokker-Planck equations with measures as initial data and McKean-Vlasov equations, J. Funct. Anal. 280 (2021), no. 7, 108926.