JVP seminars
8.30-9.15 Alexander Sahn Baumgartner
Codings for Dynamical Systems
The first part of the talk aims to provide a short introduction to the field of dynamical systems to a broad audience. A few examples are provided as well as ways to 'encode' their chaotic behaviors using mathematically tractable objects called subshifts of finite type. Time permitting, I will discuss how I applied generalisations of these techniques to certain flows with chaotic behavior.
9.15-10.00 Andrea Zanoni
Parameter estimation and uncertainty propagation for stochastic and deterministic systems
We introduce some examples of inverse and forward problems, and we consider challenges that may arise and possible solutions. Specifically, we focus on inference for stochastic models and uncertainty propagation for computationally expensive deterministic systems. First, using a maximum likelihood approach, we estimate unknown parameters in stochastic differential equations from observed trajectories. We address challenges like model misspecification, lack of information, and discrete time observations, proposing suitable estimators. Second, we approximate expectations of quantities of interest of expensive models, where Monte Carlo methods are unfeasible. Combining multifidelity approaches with dimensionality reduction techniques, we provide estimators with reduced variance without increasing the cost. The effectiveness of our methods is demonstrated through numerical experiments.
10:00-10.45 Andrea Bisterzo
Rigidity of an overdetermined heat equation and minimal helicoids in space-forms
In the seminal paper of 1995, J. C. C. Nitsche proved that if a domain of R^3 is uniformly dense in its boundary, then the boundary has to be a plane or a right helicoid, closing an open problem proposed by G. Cimmino in 1932. This result has since inspired a rich line of research on rigidity phenomena for overdetermined differential problems in possibly unbounded domains. The aim of this talk is to present an ongoing work in collaboration with Professor Alessandro Savo, in which we characterize embedded minimal helicoids and totally geodesic hypersurfaces in three-dimensional space-forms through the concept of "constant boundary temperature", an overdetermined condition involving the Cauchy problem for the heat equation.
10:45-11.15 Coffee break
11.15-12.00 George Cooper
Moduli Spaces in Algebraic Geometry: an Introduction
In this talk, aimed at a non-specialist audience, we introduce the concept of moduli in algebraic geometry, explain why algebraic geometers are interested in the study of moduli spaces, and begin to indicate the challenges involved in proving that moduli spaces exist in the first place. We also begin to explain how the notion of stability in algebraic geometry is related to the existence of moduli spaces, focusing in particular on the moduli space of semistable vector bundles on a smooth projective algebraic curve. At the end of the talk I will briefly report on work in progress, joint with L. Modin, about the existence of moduli spaces of unstable objects in Abelian categories, generalising results of Jackson, Hoskins—Jackson and Qiao concerning the existence of moduli spaces of unstable coherent sheaves on projective varieties, and of Hamilton concerning the existence of moduli spaces of unstable Higgs bundles on a smooth projective curve.
12:00-12:45 Liangjun Weng
The capillary Minkowski problem
The classical Minkowski problem is a fundamental inverse problem in convex geometry concerning the prescription of the surface area measure of a convex body, which was asked by Hermann Minkowski in 1897. This problem has been a key inspiration in the study of the fully nonlinear PDEs, as in the smooth setting, it reduces to a Monge-Ampère equation on the unit sphere. Through the seminal works of Nirenberg, Pogorelov, and Cheng-Yau, among many others, this problem has been fully resolved and was a milestone in global geometry. In this talk, we will discuss a boundary value problem of the classical Minkowski problem, which concerns the existence of a convex hypersurface with prescribed Gauss-Kronecker curvature and a capillary boundary supported on an obstacle. By formulating this as a Monge-Ampère equation with a Robin (or Neumann) boundary condition on a spherical cap, we establish the existence of smooth solutions to this problem.