A classical result in spectral theory is that the space of square integrable functions on the modular surface $X = SL(2,\mathbb Z) \backslash SL(2,\mathbb R)$ can be decomposed as the space of Eisenstein series and its orthogonal complements, the cusp forms. The former space corresponds to the spectral projection on the continuous spectrum of the Laplacian on X, and the cusp forms to the projection on the point spectrum. This result is relevant in the geometry of numbers and in dynamics because the modular surface can parameterise the space of all unimodular lattices (and, thus, also the space of all unit area flat tori).

In this talk, I will explain how to extend these ideas to the study of spaces of flat surfaces of higher genus with singularities. We replace the Eisenstein series with the range of the Siegel���Veech transform and in some specific cases can also identify precisely the cusp forms. I will focus on the case of marked flat tori, this space corresponding to the space of affine lattices. In this situation, we can also identify an operator\DSEMIC which is not the Laplacian but a foliated Laplacian\DSEMIC where the natural decomposition corresponds to its spectrum.

This is joint work with Jayadev S. Athreya (Washington), Martin M��ller (Frankfurt) and Martin Raum (Chalmers)

I will explain how free resolutions of ideals can be used to systematically formulate invariant theory for several moduli spaces of varieties that are of interest in arithmetic statistics and computational number theory. In particular, we extend the classical invariant theory formulas for the Jacobian of a genus one curve of degree n=2,3,4,5 to curves of arbitrary degree, generalizing the work on genus one models of Cremona, Fisher and Stoll, and in a joint work with Tom Fisher, we compute structure constants for a rank n ring from the free resolution of its associated set of n points in projective space, generalizing the previously known constructions of Levi-Delone-Faddeev and Bhargava. Time permitting I will talk about an ongoing project to extend these results to abelian varieties of higher dimension.

At thermal equilibrium, chiral molecules form a range of liquid-crystalline phases, such as the cholesteric which presents a helical structure of the molecular orientation. Chirality, though essential to the construction of the cholesteric, is totally absent in its long-wavelength hydrodynamics, which is identical to that of the achiral smectic-A liquid crystal. This cloaking of chirality, however, relies on the existence of an energy function for the dynamics. I will talk about how macroscopic mechanics of active layered phases carry striking chiral signatures. Thanks to the mix of solid and liquid-like directions, the chiral active stresses create a force density tangent to contours of constant mean curvature of the layers. This non-dissipative force in a fluid direction ��� odder than odd elasticity ��� leads, in the presence of an active instability, to spontaneous vortical flows arranged in a two-dimensional array with vorticity aligned along the pitch axis and alternating in sign in the plane.

In addition, I will discuss how odd elasticity, an effect that is attracting much current attention, is naturally realised in polar and chiral columnar systems. The resulting oscillatory mode, thanks to the Stokesian hydrodynamic interaction, has a nonzero frequency on macroscopic scales, set by the ratio of the coefficient of chiral and polar active stress and the viscosity. A bulk columnar phase undergoes a spontaneous buckling instability due to extensile activity. If the active units composing the columnar state are, in addition, chiral, the twisted columns host large-scale shear flows due to a new form of odd elasticity.

In a very general setting, entropy quantifies the amount of
information about a system that an observer has access to. However, in contrast to quantum mechanics, in quantum field theory naive measures of entropy are divergent. To obtain finite results, one needs to consider measures such as relative entropy, which can be computed from the modular Hamiltonian using Tomita--Takesaki theory.

In this talk, I will give a short introduction to Tomita--Takesaki modular theory and present examples of modular Hamiltonians. Using these, I will give results for therelative entropy between the de Sitter vacuum state and a coherent excitation thereof in diamond and wedge regions, and show explicitly that the result satisfies the expected properties for a relative entropy. Finally, I will use local thermodynamic laws to determine the local temperature that is measured by an observer, and consider the backreaction of the quantum state on the geometry to prove an entropy-area law for de Sitter spacetime.

Based on arXiv:arXiv:2308.14797, 2310.12185, 2311.13990 and 2312.04629.

'Sums of Hecke Eigenvalues'

Understanding the distribution of sums of arithmetic functions is a classical problem in analytic number theory. In this talk, we investigate sums of Hecke eigenvalues attached to cusp forms, on average over forms of large weight. We find some interesting transitions in the behaviour of the sums as their length varies in relation to the weight.

One of the most important problems of modern science is that of emergence. How do laws of motion emerge at large scales of space and time, from much different laws at small scales? A foremost example is the theory of hydrodynamics. Take molecules in air, which simply follow Newton���s equations. When there are very many of them, these equations becomes untractable\DSEMIC seeking the knowledge of each molecule���s individual trajectory is completely impractical. Happily it is also unnecessary. At our human scale, new, different equations emerge for aggregate quantities: those of hydrodynamics. And these are apparently all we need to know in order to understand the weather! Despite its conceptual significance, the passage from microscopic dynamics to hydrodynamics remains a notorious open problem of mathematical physics. This goes much beyond molecules in air: similar principles hold very generally, such as in quantum gases and spin lattices, where the resulting equations themselves can be very different. In particular, integrable models, where an extensive mathematical structure allows us to make progress, admit an entirely new universality class of hydrodynamic equations. In this talk, I will discuss in a pedagogical and mathematically precise fashion the general problem and principles of hydrodynamics as an emergent theory, and some recent advances in our understanding, including those obtained in integrable models

The description of subspaces invariant under the Volterra operator goes back to a problem of Gelfand from 1938. Invariant subspaces for differentiation on $C^{\infty}$ were studied much later by Aleman and Korenblum and continued by Aleman, Baranov and Belov. Both problems contain a wealth of interesting ideas and have several interesting connections to exponential systems, among other things. I intend to give a review of some of these results and then continue with a more abstract setting consisting of an unbounded operator D with a compact quasi-nilpotent right inverse V. It turns out that under certain general conditions one can prove similar results for a large class of examples (for D) containing Schr��dinger operators, Dirac operators and other Canonical systems of differential equations. This is a report about recent joint work with Alexandru Aleman.