L4

The chromatic polynomial \chi(Q) can be defined for any graph, such that for Q integer it counts the number of colourings. It has many remarkable properties, and I describe several that are derived easily by using fusion categories, familiar from topological quantum field theory. In particular, I define the chromatic algebra, a planar algebra whose evaluation gives the chromatic polynomial. Linear identities of the chromatic polynomial at certain values of Q then follow from the Jones-Wenzl projector of the associated category. An unusual non-linear one called Tutte's golden identity relates \chi(\phi+2) for planar triangulations to the square of \chi(\phi+1), where \phi is the golden mean. Tutte's original proof is purely combinatorial. I will give here an elementary proof by manipulations of a topological invariant related to the Jones polynomial. Time permitting, I will also mention analogous identities for graphs on more general surfaces. Based on work with Slava Krushkal.

I will talk about some current work with Jesse Jaasaari and Steve Lester where we investigate the analogue of the Quantum Unique Ergodicity (QUE) conjecture in the weight aspect for Siegel cusp forms of degree 2 and full level. Assuming the Generalized Riemann Hypothesis (GRH) we establish QUE for Saito–Kurokawa lifts as the weight tends to infinity. As an application, we prove the equidistribution of zero divisors.

Given a finite set A of integer lattice points in d dimensions, let NA denote the N-fold iterated sumset (i.e. the set comprising all sums of N elements from A). In 1992 Khovanskii observed that there is a fixed polynomial P(N), depending on A, such that the size of the sumset NA equals P(N) exactly (once N is sufficiently large, that is). In addition to this 'size stability', there is a related 'structural stability' property for the sumset NA, which Granville and Shakan recently showed also holds for sufficiently large N. But what does 'sufficiently large' mean in practice? In this talk I will discuss some perspectives on these questions, and explain joint work with Granville and Shakan which proves the first explicit bounds for all sets A. I will also discuss current work with Granville, which gives a tight bound 'up to logarithmic factors' for one of these properties. 

Assuming the Riemann Hypothesis, Soundararajan established almost sharp upper bounds for all positive moments of the Riemann zeta-function. This result was later improved by Harper, who proved upper bounds of the right order of magnitude. I will describe some of the ideas in their proofs, and then discuss recent joint work with Alexandra Florea, where we consider negative moments of the Riemann zeta-function. For example, we can obtain asymptotic formulas for negative moments when the shift in the zeta function is large enough, confirming a conjecture of Gonek.  We also obtain an upper bound for the average of the generalised Mobius function.

Conditional on finiteness of relevant Shafarevich--Tate groups, Harpaz and Skorobogatov established the Hasse principle for Kummer varieties associated to a 2-covering of a principally polarised abelian variety A, under certain large image assumptions on the Galois action on A[2]. However, their method stops short of treating the case where the image is the full symplectic group, due to the possible failure of the Shafarevich--Tate group to have square order in this setting. I will explain work in progress which overcomes this obstruction by combining second descent ideas of Harpaz with new results on the 2-parity conjecture. 

I will present results on short sums of arithmetic functions (in particular the von Mangoldt and divisor functions) twisted by polynomial exponential phases or more general nilsequence phases. These results imply the Gowers uniformity of suitably normalised versions of these functions in intervals of length X^c around X for suitable values of c (depending on the function and on whether one considers all or almost all short sums). I will also discuss an application to an averaged form of the Hardy-Littlewood conjecture. This is based on joint works with Kaisa Matomäki, Maksym Radziwiłł, Xuancheng Shao and Terence Tao.

Wiles' modularity lifting theorem was the central argument in his proof of modularity of (semistable) elliptic curves over Q, and hence of Fermat's Last Theorem. His proof relied on two key components: his "patching" argument (developed in collaboration with Taylor) and his numerical isomorphism criterion.

In the time since Wiles' proof, the patching argument has been generalized extensively to prove a wide variety of modularity lifting results. In particular Calegari and Geraghty have found a way to generalize it to prove potential modularity of elliptic curves over imaginary quadratic fields (contingent on some standard conjectures). The numerical criterion on the other hand has proved far more difficult to generalize, although in situations where it can be used it can prove stronger results than what can be proven purely via patching.

In this talk I will present joint work with Srikanth Iyengar and Chandrashekhar Khare which proves a generalization of the numerical criterion to the context considered by Calegari and Geraghty (and contingent on the same conjectures). This allows us to prove integral "R=T" theorems at non-minimal levels over imaginary quadratic fields, which are inaccessible by Calegari and Geraghty's method. The results provide new evidence in favor of a torsion analog of the classical Langlands correspondence.

Mathematicians frequently present their own work in a diachronic fashion, e.g. by comparing their "modern" methods to those supposedly of the "Ancients," or by situating their latest theories as an "abstract" counterpart to more "classical" ones. The construction of such contrasts entangle mathematical labour and cultural life writ large. Indeed, it involves on the part of mathematicians the shaping up of correspondences between their technical achievements and intellectual discussions taking place on a much broader stage, such as those surrounding the concept of modernity, its relation to an imagined ancient past, or the characterisation of scientific progress as an increase in abstraction. This talk will explore the creation and use of such mathematical diachronies, the focus being on the works of Felix Klein, Hieronymus Zeuthen, and Hermann Schubert.

My talk will attempt to capture the imperfect state of the art in high-resolution ice sheet modelling, aiming to expose the core performance-limiting issues.  The essential equations for modeling ice flow in a changing climate will be presented, assuming no prior knowledge of the problem.  These geophysical/climate problems are of both free-boundary and algebraic-equation-constrained character.  Current-technology models usually solve non-linear Stokes equations, or approximations thereof, at every explicit time-step.  Scale analysis shows why this current design paradigm is expensive, but building significantly faster high-resolution ice sheet models requires new techniques.  I'll survey some recently-arrived tools, some near-term improvements, and sketch some new ideas.

We consider the varifolds associated to phase transitions whose first variation of Allen-Cahn energy is $L^p$ integrable with respect to the energy measure. We can see that the Dirichlet and potential part of the energy are almost equidistributed. After passing to the phase field limit, one can obtain an integer rectifiable varifold with bounded $L^p$ mean curvature. This is joint work with Huy Nguyen.