Week 06.03.2022 – 12.03.2022

We read the book on Quantum Field Theory in Curved Spacetime and Black Hole Thermodynamics by Robert M. Wald.

This is an online club held every Thursday 4:15pm-5:15pm on teams. Contact: george.papadopoulos@kcl.ac.uk

In these lectures, we will present two seemingly different theories. The first one is a theory of gravity in two dimensions, called Jackiw-Teitelboim (JT) gravity, that is relevant in the context of higher-dimensional, near-extremal black holes. The second one is a quantum mechanical theory of fermions — with no gravity — called the Sachdev, Ye and Kitaev (SYK) model. We will explore precisely how JT gravity emerges from the SYK model by studying their actions, correlation functions and thermodynamic properties. This constitutes the simplest toy model of what theoretical physicists now call the holographic principle.

Toric varieties are some of the simplest and best-studied objects in algebraic geometry. Many geometric notions have elegant pictorial and combinatorial descriptions in the toric world. I'll describe compactified cluster varieties as a generalization of toric varieties and explain how some of the pictorial and combinatorial gadgets of toric geometry can extend to the cluster setting. In particular, I plan to discuss extensions of following toric geometry topics: fans, polyhedra, convexity, Minkowski sums, and the Batyrev/ Batyrev-Borisov mirror constructions for Calabi-Yau subvarieties of Gorenstein toric Fanos. Some parts of this are worked out while others are still in the naïve guess stage. I'll indicate what we know, what we hope, and what causes complications. Based on collaborations with Lara Bossinger, Mandy Cheung, Juan Bosco Frías Medina, and Alfredo Nájera Chávez.

Speaker: Netan Dogra

Title: Rational points and p-adic integrals on families of curves.

Abstract: If X is a curve of genus >1 over a number field, then the set of rational points of X is finite. It is a big open problem to understand how this finite set varies with X. I will explain what this has to do with p-adic integration, and how a suitable notion of 'p-adic integration in families' enables us to say some new things.

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Speaker: Matthew Honnor

Title: Formulas for Brumer--Stark Units

Abstract: In the 1980's, Tate stated the Brumer--Stark conjecture which, for a totally real field $F$ with prime ideal $\mathfrak{p}$, conjectures the existence of a $\mathfrak{p}$-unit called the Brumer--Stark unit. This unit has $\mathfrak{P}$ order equal to the value of a partial zeta function at 0, for a prime $\mathfrak{P}$ above $\mathfrak{p}$. There have been three formulas conjectured for the Brumer--Stark unit by Dasgupta and Dasgupta--Spie\ss. In this talk, I will present forthcoming joint work with Dasgupta which shows that these three formulas are equivalent.