Local-to-global principles are a key tool in arithmetic geometry. Through a theorem of Siegel on representations of totally positive numbers as sums of squares in number fields we give a concrete introduction to the Hasse principle, and briefly talk about other local-to-global principles. No prerequisites from algebraic number theory are assumed, although some familiarity is helpful for context.

In this session, Ben Fehrman and Markus Upmeier will give their thoughts on how to deliver a good talk for a conference or a seminar and tips for what to do and what to avoid. There will be a particular emphasis on how to give a good talk online. 

In this talk I describe a holographic perspective to study field spaces that arise in string compactifications. The constructions are motivated by a general description of the asymptotic, near-boundary regions in complex structure moduli spaces of Calabi-Yau manifolds using Hodge theory. For real two-dimensional field spaces, I introduce an auxiliary bulk theory and describe aspects of an associated sl(2) boundary theory. The classical bulk reconstruction is provided by the sl(2)-orbit theorem, which is a famous and general result in Hodge theory. I then apply this correspondence to the flux landscape of Calabi-Yau fourfold compactifications and discuss how this allows us to prove that the number of self-dual flux vacua is finite. I will point out how the finiteness result for supersymmetric fluxes relates to the Hodge conjecture.

Phase transitions are present in a wide array of systems ranging from traffic to machine learning algorithms. In this talk, we will relate the concept of phase transitions to the convexity properties of the associated thermodynamic energy. Motivated by noisy stochastic gradient descent in supervised learning, we will consider the problem of understanding the thermodynamic limit of exchangeable weakly interacting diffusions (AKA propagation of chaos) from an energetic perspective. The strategy will be to exploit the 2-Wasserstein gradient flow structure associated with the thermodynamic energy in the infinite particle setting. Using this perspective, we will show how the convexity properties of the thermodynamic energy affects the homogenization limit or the stability of the log-Sobolev inequality.

Mathematicians need to talk and write about their mathematics.  This includes undergraduates and MSc students, who might be writing a dissertation or project report, preparing a presentation on a summer research project, or preparing for a job interview.  It can be helpful to think of this as a form of storytelling, as this can lead to more effective communication.  For a story to be engaging you also need to know your audience.  In this interactive session, we'll discuss what we mean by telling a mathematical story, give you some top tips from our experience, and give you a chance to think about how you might put this into practice.

We consider a nonlinear flow on simplicial complexes related to the simplicial Laplacian, and show that it is a generalization of various consensus and synchronization models commonly studied on networks. In particular, our model allows us to formulate flows on simplices of any dimension, so that it includes edge flows, triangle flows, etc. We show that the system can be represented as the gradient flow of an energy functional, and use this to deduce the stability of various steady states of the model. Finally, we demonstrate that our model contains higher-dimensional analogues of structures seen in related network models.

arXiv link: https://arxiv.org/abs/2010.07421

 We generalize the recent holographic correspondence between an ensemble average of free bosons in two dimensions, and a Chern-Simons-like theory of gravity in three dimensions, by Afkhami-Jeddi et al and Maloney and Witten. We find that the correspondence also works for toroidal orbifolds, but we run into difficulties generalizing to K3 and Calabi-Yau sigma models. For the case of toroidal orbifolds, we extend the holographic correspondence to averages of correlation functions of twist operators by using properties of rational tangles in three-dimensional balls and their covering spaces. Based on work to appear with C. Keller, H. Ooguri, and I. Zadeh.