In this talk, I will give a short introduction to symplectic geometry and define symplectic capacities. I will then discuss the following question, which I looked at in my thesis: What information about a given class of symplectic manifolds can be recovered from the capacities?
If time permits, I will also talk about subsets of symplectic manifolds that can be arbitrarily squeezed, meaning that they can be embedded into arbitrarily small balls.
Supervisors:Dr. Álvaro del Pino Gómez and Dr. Johan Commelin
Abstract
Thistalk presents the formalisation of differential forms as sections of thecontinuous alternating bundle in the functional programming language andtheorem prover Lean. We start with a brief introduction to Lean and itsmathematical library mathlib. Building on the existing formalisations ofmanifolds and bundles in mathlib we then construct differential forms step bystep. We discuss core functionalities such as multilinear and alternating maps,alternating forms and the continuous alternating bundle to get to ourformalisation of differential forms on manifolds. We also cover theimplementation of standard operations on differential forms, including theexterior derivative and the wedge product, highlighting the incrementalstructure needed to support these features in Lean.
Beyondsummarising the current state of the formalisation, we aim to provide adiscussion of the design choices that we made and the challenges that we faced.In particular, we for example discuss the choice for constructing differentialforms using the bundle of continuous alternating maps instead of the morecommon exterior algebra approach.
A common question for geometric structures is that of rigidity, i.e. given two geometric structures sufficiently close, are they equivalent? In this talk we discuss this question for regular foliations on a closed manifold.
In the first part we give an overview of the current state of the art and highlight its relation with the rigidity for group actions. We will see that several rigidity results for foliations require the leaves to be compact.
In an attempt to understand whether such results can be extended to Riemannian foliations, we seek examples of such foliations which are infinitesimally rigid with non-compact leaves. In the second part, we use the relation to group actions to construct infinitesimally rigid Lie foliation with non-compact, dense leaves. This is based on joint work in progress with Stephane Geudens.
This talk will be accessible for both physicists and mathematicians who are interested in the geometric construction of topological string theories and mirror symmetry. For the mathematicians, it will be beneficial, but not necessary, to have a conceptual understanding of quantum field theory.
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.