Week 20.05.2024 – 26.05.2024

'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.

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.

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.

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, which is not the Laplacian but a foliated Laplacian, where the natural decomposition corresponds to its spectrum.

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

This talk will discuss Bayesian methods of inference and develop flexible models for financial applications. One approach to flexible modeling is Bayesian nonparametric methods which use an infinite mixture model. A Dirichlet process mixture and an infinite hidden Markov model, a time-dependent version of the former, will be reviewed. Another important feature of financial data is heteroskedasticity. A popular class of specifications for the evolution of the conditional covariance of asset returns is the multivariate generalized autoregressive conditional heteroskedasticity (MGARCH) model. We will discuss an approach to combine an infinite mixture model with MGARCH dynamics suitable to capture the complex distribution of financial data. The talk will conclude with applications of these models to portfolio choice problems to evaluate their usefulness.