Talk

You might think that non-noncommutative geometry is just geometry. But what kind, really? In other words, what is the geometry that is generalised in noncommutative geometry (NCG)? I will try to answer this by giving a (slightly fake) history of NCG as originating from a combination of C*-algebra theory, spectral geometry (hearing the shape of a drum), and some K-theory. Depending on the time, I might then sketch some cool theorems in NCG and applications in mathematics and physics.

Derived differential geometry is the C-infinity counterpart of derived algebraic geometry. The role of affine schemes is played by derived manifolds, which are geometric objects constructed by taking iterative fibered products of smooth manifolds. This can be made precise via a simple universal property for the infinity category of derived manifolds, proposed by myself and Pelle Steffens. Steffens and I prove moreover that derived manifolds are equivalent to affine derived schemes of finite presentation with respect to the algebraic theory of C-infinity rings. At the same time, we have constructed a concrete model for derived manifolds using differential graded manifolds, which have various applications to mathematical physics. In this talk, we will explain the above results precisely, as well as sketching some of the main ideas behind the proofs.

A long-standing conjecture of Bergeron and Venkatesh predicts that in closed hyperbolic 3-manifolds, the amount of torsion in the first homology of finite-sheeted normal covers should grow exponentially with the degree of the cover as the covers become larger, at a rate reflecting the volume of the manifold. Yet no finitely presented residually finite group is known to exhibit such behaviour, and meaningful lower bounds on torsion growth are rare.

In this talk I will explain how a two-dimensional lens offers a clearer view of some of the underlying mechanisms that create homological torsion in finite covers, and how they might relate to its growth. If time allows, I will also discuss how these ideas connect to the question of profinite rigidity: how much information about a group is encoded in its finite quotients.

Martin conjectured that every non-negative integer can be expressed as the dimension of the space of newforms of weight 2 and level N for some positive integer N. Recently, Ross disproved this conjecture by showing that 67846 is not attained in this way and proposed a counter-conjecture that the set of such dimensions has density zero among the non-negative integers. We prove a general form of Ross’ conjecture. 

This is a joint work with Andrew R. Booker.

Title: Covering a curve

Abstract: Aimed at a general audience, the talk presents results and approaches to an unsolved problem in extremal geometry. 

Preceded and followed by musical contributions from the following artists:

Remy van Dobben de Bruyn (vocals)

Dr. Gotthelf (piano)

Eva-Maria Hekkelman (cello)

Davide Macera (piano)

Alexandre Maksoud (piano)

David Prinz (piano)

Wyatt Reeves (piano)

Lola Thompson (vocals)

Let $G_1 / K$ and $G / K$ be two semisimple, simply-connected groups defined over a global function field $K$, which are inner forms of each other. We give a short proof for the equality of the Tamagawa number, $\tau(G_1) = \tau(G)$. It is equal to $1$ (Gaitsgory-Lurie, 2019). It is done by interpreting the elements in double cosets $G(K) \backslash G(\mathbb{A}) / \mathcal{K}$ as points of certain moduli stacks $\mathcal{M}_G / \mathbb{F}_q$. We relate the Tamagawa number to traces of the Frobenius acting on $\ell$-adic cohomology groups. To construct these moduli stacks we use explicit formulas for $1$-cocycles which represent the relevant Galois-cohomology classes. These $1$-cocycles respect certain parahoric subgroups corresponding to points in the Bruhat-Tits building.

This is joint work with R. Bitan, G. Harder, R. Koehl and A. Zidani.

In 1922 Julia characterized the pairs of rational functions over the field of complex numbers which satisfy f(g) = g(f). The next year Ritt used his theory on decomposition of rational functions to give an alternative proof for a slightly refined form of Julia’s theorem. Several authors have extended the Julia-Ritt theorem for polynomials to wider classes of fields. In the lecture I shall report on joint work with Lajos Hajdu resulting into extensions of the Julia-Ritt for polynomials to equations of the form f(g(h)) = h(g(f)) and f(g) = g(h). Our proofs are based on a paper by Zieve and Mueller (2008).