Manifolds with G2 structure allow almost contact structures. In this talk I will discuss various aspects of such structures, and their effect on certain supersymmetric configurations in string and M-theory.

This is based on recent work with Xenia de la Ossa and Matthew Magill.

We discuss constraints from perturbative unitarity and crossing on the leading contributions of the higher-dimension operators to the four-graviton amplitude in four spacetime dimensions. We focus on the leading order effect due to exchange by massive degrees of freedom which makes the amplitudes of interest IR finite. To test the constraints we obtain nontrivial effective field theory data by computing and taking the large mass expansion of the one-loop minimally-coupled four-graviton amplitude with massive particles up to spin 2 circulating in the loop. Remarkably, the leading EFT corrections to Einstein gravity of physical theories, both string theory and QFT coupled to gravity, end up in minuscule islands which are much smaller than what is suggested by the generic bounds obtained from consistency of the 2-2 graviton scattering amplitude. We discuss the underlying mechanism for this phenomenon.

The observation that the Dirac operator on a spin manifold encodes both the Riemannian metric as well as the fundamental class in K-homology leads to the paradigm of noncommutative geometry: the viewpoint that spectral triples generalise Riemannian manifolds. To encode maps between Riemannian manifolds, one is naturally led to consider the unbounded picture of Kasparov's KK-theory. In this talk I will explain how smooth cycles in KK-theory give a natural notion of noncommutative fibration, encoding morphisms noncommutative geometry in manner compatible with index theory.

n this talk we consider a p-isometrisability property of discrete groups. If p=2 this property is equivalent to unitarisability. We prove that any group containing a non-abelian free subgroup is not p-isometrisable for any p∈(1,∞). We also discuss some open questions and possible relations of p-isometrisability with the recently introduced Littlewood exponent Lit(Γ).

The action of the automorphisms of a formal group on its deformation space is crucial to understanding periodic families in the homotopy groups of spheres and the unsolved Hecke orbit conjecture for unitary Shimura varieties. We can explicitly pin down this squirming action by geometrically modelling it as coming from an action on a moduli space, which we construct using inverse Galois theory and some representation theory (a Hurwitz space). I will show you pretty pictures.

A set of positive integers is primitive (or 1-primitive) if no member divides another. Erdős proved in 1935 that the weighted sum $\sum 1/(n\log n)$ for n ranging over a primitive set A is universally bounded over all choices for A. In 1988 he asked if this universal bound is attained by the set of prime numbers. One source of difficulty in this conjecture is that $\sum n^{-\lambda}$ over a primitive set is maximized by the primes if and only if $\lambda$ is at least the critical exponent $\tau_1\approx1.14$.
A set is $k$-primitive if no member divides any product of up to $k$ other distinct members. In joint work with C. Pomerance and T.H. Chan, we study the critical exponent $\tau_k$ for which the primes are maximal among $k$-primitive sets. In particular we prove that $\tau_2<0.8$, which directly implies the Erdős conjecture for 2-primitive sets.

Multiple zeta values, originally considered by Euler, generalise the Riemann zeta function to multiple variables. While values of the Riemann zeta function at odd positive integers are conjectured to be algebraically independent, multiple zeta values satisfy many algebraic and linear relations, even forming a Q-algebra. While families of well understood relations are known, such as the associator relations and double shuffle relations, they only conjecturally span all algebraic relations. As multiple zeta values arise as the periods of mixed Tate motives, we obtain further algebraic structures, which have been exploited to provide spanning sets by Brown. In this talk we will aim to define a new set of relations, known to be complete in low block degree.

To achieve this, we will first review the necessary algebraic set up, focusing particularly on the motivic Lie algebra associated to the thrice punctured projective line. We then introduce a new filtration on the algebra of (motivic) multiple zeta values, called the block filtration, based on the work of Charlton. By considering the associated graded algebra, we quickly obtain a new family of graded motivic relations, which can be shown to span all algebraic relations in low block degree. We will also touch on some conjectural ungraded `lifts' of these relations, and if we have time, compare to similar approaches using the depth filtration.

Over the past few decades, there has been a lot of interest in partial sums of Dirichlet characters. Montgomery and Vaughan showed that these character sums remain a constant size on average and, as a result, a lot of work has been done on the distribution of the maximum. In this talk, we will investigate the distribution of these character sums themselves, with the main goal being to describe the limiting distribution as the prime modulus approaches infinity. This is motivated by Kowalski and Sawin’s work on Kloosterman paths.
 

A semiprime is a natural number which can be written as the product of two primes. Using elementary methods, we'll explore an asymptotic expansion for the counting function of semiprimes $\pi_2(x)$, which generalises previous findings of Landau, Delange and Tenenbaum.  We'll also obtain an efficient way of computing the constants involved. In the end, we'll look towards possible generalisations for products of $k$ primes.