The hard Lefschetz theorem is a fundamental statement about the symmetry of the cohomology of algebraic varieties. In nearly all cases that we systematically understand it, it comes with a geometric meaning, often in form of Hodge structures and signature data for the Hodge-Riemann bilinear form.

Nevertheless, similar to the role the standard conjectures play in number theory, several intriguing combinatorial problems can be reduced to hard Lefschetz properties, though in extreme cases without much geometric meaning, lacking any existence of, for instance, an ample cone to do Hodge theory with.

I will present a way to prove the hard Lefschetz theorem in such a situation, by introducing biased pairing and perturbation theory for intersection rings. The price we pay is that the underlying variety, in a precise sense, has itself to be sufficiently generic. For instance, we shall see that any quasismooth, but perhaps nonprojective toric variety can be "perturbed" to a toric variety with the same equivariant cohomology, and that has the hard Lefschetz property.

Finally, I will discuss how this applies to prove some interesting theorems in geometry, topology and combinatorics. In particular, we shall see a generalization of a classical result due to Descartes and Euler: We prove that if a simplicial complex embeds into euclidean 2d-space, the number of d-simplices in it can exceed the number of (d-1)-simplices by a factor of at most d+2.

# Forthcoming Seminars

Please note that the list below only shows forthcoming events, which may not include regular events that have not yet been entered for the forthcoming term. Please see the past events page for a list of all seminar series that the department has on offer.

It is a long-standing open problem whether the ring of integers Z has an existential first-order definition in Q, the field of rational numbers. A few years ago, Jochen Koenigsmann proved that Z has a universal first-order definition in Q, building on earlier work by Bjorn Poonen. This result was later generalised to number fields by Jennifer Park and to global function fields of odd characteristic by Kirsten Eisenträger and Travis Morrison, who used classical machinery from number theory and class field theory related to the behaviour of quaternion algebras over global and local fields.

In this talk, I will sketch a variation on the techniques used to obtain the aforementioned results. It allows for a relatively short and uniform treatment of global fields of all characteristics that is significantly less dependent on class field theory. Instead, a central role is played by Hilbert's Reciprocity Law for quaternion algebras. I will conclude with an example of a non-global set-up where the existence of a reciprocity law similarly yields universal definitions of certain subrings.

We present some regularity estimates for viscosity solutions to a class of possible degenerate and singular integro-differential equations whose leading operator switches between two different types of fractional elliptic phases, according to the zero set of a modulating coefficient a = a(·, ·). The model case is driven by the following nonlocal double phase operator,

$$\int \frac{|u(x) − u(y)|^{p−2} (u(x) − u(y))} {|x − y|^{n+sp}} dy+ \int a(x, y) \frac{|u(x) − u(y)|^{ q−2} (u(x) − u(y))} {|x − y|^{n+tq}} dy$$

where $q ≥ p$ and $a(·, ·) = 0$. Our results do also apply for inhomogeneous equations, for very general classes of measurable kernels. By simply assuming the boundedness of the modulating coefficient, we are able to prove that the solutions are Hölder continuous, whereas similar sharp results for the classical local case do require a to be Hölder continuous. To our knowledge, this is the first (regularity) result for nonlocal double phase problems.

I will present an analysis of a continuous version of the compressed sensing problem, where the l^1 norm is replaced by the total variation of measures, and one aims to recover the positions and amplitudes of Dirac masses. We show that provided that the Diracs are sufficiently separated under a Fisher metric (which accounts for the geometry of the problem), stable recovery can be achieved when the number of random samples scales linearly with sparsity (up to log factors). This is joint work with Nicolas Keriven and Gabriel Peyre.

Would you like to meet some of your fellow students, and some graduate students and postdocs, in an informal and relaxed atmosphere, while building your communication skills? In this Friday@2 session, you'll be able to play a selection of board games, meet new people, and practise working together. What better way to spend the final Friday afternoon of term?! We'll play the games in the south Mezzanine area of the Andrew Wiles Building, outside L3.