Abstract: Given a smooth projective and geometrically irreducible curve C and a Mori Dream Space X, we present a general parametrisation of morphisms from C to X which allows us to express the Grothendieck motive of Hom (C,X) as a motivic function defined on some power of the scheme of effective divisors of C, generalising previous works of Bourqui. Such a parametrisation should be understood as lifting our morphisms to the universal torsor of X. 

As an application, we prove a motivic analogue of a variant of Manin’s conjecture for Campana curves on smooth projective split toric varieties.

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Minimal geodesic flows on closed hyperbolic manifolds (manifolds which support a negative curvature metric) exhibit a strong rigidity: their flows are topologically conjugate. Slightly more generally, for any geodesic flow on a hyperbolic manifold, it has a non-empty set of globally minimal geodesics, which always shadow geodesics of a given negative curvature metric.

  For Reeb flows on star-shaped contact surfaces, it is impossible to be minimal when the surface is strictly not convex fiberwise. However, for a slight generalization of minimality, called coarse minimality, we can produce examples of non-convex hypersurfaces with coarsely minimal Reeb flows. Additionally, there is a semi-conjugacy coarsely minimal flows and the geodesic flow a given negative curvature metric.

In this talk we will continue to explore the world of topological strings, specifically the A-twisted nonlinear sigma model (NLSM) as introduced by Jesse Straat on May 16th. The aim will be to actually compute the observables of a NLSM with a (smooth) toric variety as target. The trick will be to look at a so-called gauged linear sigma model (GLSM) whose low-energy limit is the NLSM we want. When studying the observables, we encounter the quantum cohomology ring, an extension of the ordinary cohomology, as a way of combining this information into a ring with product. It turns out that these observables are closely related to the Gromov–Witten invariants of classes corresponding to the operators we consider.

If time permits, we will look at a different formulation of quantum cohomology using the Gromov–Witten invariants directly.

Note that the talk will be accessible for both physicists and mathematicians that want to see such an approach to quantum cohomology.

Institute meeting for full professors, associate professors and assistant professors, as well as support staff of the MI.

Please let us know whether you'll attend.

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Host: Wioletta Ruszel 
Title: Extremes in dynamical systems: max-stable and max-semistable laws

Abstract:
Extreme value theory for chaotic, deterministic dynamical systems is a rapidly expanding area of research. Given a dynamical system and a real-valued observable defined on its state space, extreme value theory studies the limit probabilistic laws for asymptotically large values attained by the observable along orbits of the system. Under suitable mixing conditions the extreme value laws are the same as those for stochastic processes of i.i.d. random variables. 

Max-stable laws typically arise for probability distributions with regularly varying tails. However, in the context of dynamical systems, where the underlying invariant measure can be irregular, max-semistable distributions also have a natural place in studying extremal behaviour. In this talk I will first discuss a family of autoregressive processes with marginal distributions resembling the Cantor function. The resulting extreme value law can be proven to be a max-semistable distribution. Alternatively, we can describe the autoregressive process in terms of an iterated map with an invariant measure. Further examples of extreme value laws in dynamical systems are discussed as well.

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