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Consider a collection of geodesic triangles on the unit sphere, or on a Euclidean plane or on the hyperbolic plane and identify their sides pairwise by isometries. This way you obtain a constant curvature surface with conical singularities, it's called spherical in the first case, Euclidean in the second and hyperbolic in the last. All such surfaces are naturally Riemannian surfaces (with marked points corresponding to conical singularities). An important question is the following: given a Riemann surface $S$ with marked points $x_1,\ldots,x_n$ and prescribed conical angles $2\pi\theta_1,\ldots,2\pi\theta_n$, does there exist on $S$ a conformal constant curvature metric with prescribed conical angles at points $x_i$? How many such metrics do we have? Naturally, these are questions about solutions to a certain non-linear PDE on $S$, and the Gauss-Bonnet formula says that in the case the metric exists, its curvature should have the same sign as $\chi(S)+\sum(\theta_i-1)$. Interestingly, the answer to the above two questions differs drastically according to the sing of the expression. If $\chi(S)+\sum(\theta_i-1)\le 0$ the metric exists and is essentially unique. If $\chi(S)+\sum(\theta_i-1)> 0$ very little is known, in particular neither uniqueness nor existence is guaranteed, we are in the realm of spherical surfaces. However, thinking of a spherical surface as glued from geodesic triangles permits one to avoid solving the PDE, since the solution to it is in your hands already. I will speak about some results obtained this way jointly with Gabriele Mondello and Alik Eremenko.

A random multiplicative function is a multiplicative function alpha(n) such that its values on primes, (alpha(p))_(p=2,3,5,...), are i.i.d. random variables. The simplest example is the Steinhaus function, which is a completely multiplicative function with alpha(p) uniformly distributed on the unit circle. A basic question in the field is finding the limiting distribution of the (normalized) sum of alpha(n) from n=1 to n=x, possibly restricted to a subset of integers of interest. This question is currently resolved only in a few cases. We shall describe ongoing work where we are able to find the limiting distribution in many new instances of interest. The distribution we find is not gaussian, in contrast to all previous works. This is joint work with Mo Dick Wong (Durham University).

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.

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 Möller (Frankfurt) and Martin Raum (Chalmers)

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)

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.