Oxford

The construction of permutation functions of a finite field is a task of great interest in cryptography and coding theory. In this talk we describe a method which combines Chebotarev density theorem with elementary group theory to produce permutation rational functions over a finite field F_q. Our method is entirely constructive and as a corollary we get the classification of permutation polynomials up to degree 4 over any finite field of odd characteristic.

This is a joint work with Andrea Ferraguti.
 

We study the winding mode sector of recently discovered string theories, which were, until now, believed to describe only conventional field theories in target space. We discover that upon compactification winding modes allows the string to acquire an oscillator spectrum giving rise to an infinite tower of massive higher-spin modes. We study the spectra, S-matrices, T-duality and high-energy behaviour of the bosonic and supersymmetric models. In the tensionless limit, we obtain formulae for amplitudes based on the scattering equations. The windings decouple from the scattering equations but remain in the integrands. The existence of this winding sector shows that these new theories do have stringy aspects and describe non-conventional field theories.  This talk is based on https://arxiv.org/abs/1710.01241.

Let $\mathbb K$ be a field, and $\mathcal M$ be the “projective linear" moduli stack of objects in a suitable $\mathbb K$-linear abelian category  $\mathcal A$ (such as the coherent sheaves coh($X$) on a smooth projective $\mathbb K$-scheme $X$) or triangulated category $\mathcal T$ (such as the derived category $D^b$coh($X$)). I will explain how to define a Lie bracket [ , ] on the homology $H_*({\mathcal M})$ (with a nonstandard grading), making $H_*({\mathcal M})$ into a graded Lie algebra. This is a new variation on the idea of Ringel-Hall algebra.
 There is also a differential-geometric version of this: if $X$ is a compact manifold with a geometric structure giving instanton-type equations (e.g. oriented Riemannian 4-manifold, $G_2$-manifold, Spin(7)-manifold) then we can define Lie brackets both on the homology of the moduli spaces of all $U(n)$ or $SU(n)$ connections on $X$ for all $n$, and on the homology of the moduli spaces of instanton $U(n)$ or $SU(n)$ connections on $X$ for all $n$.
 All this is (at least conjecturally) related to enumerative invariants, virtual cycles, and wall-crossing formulae under change of stability condition.
 Several important classes of invariants in algebraic and differential geometry — (higher rank) Donaldson invariants of 4-manifolds (in particular with $b^2_+=1$), Mochizuki invariants counting semistable coherent sheaves on surfaces, Donaldson-Thomas type invariants for Fano 3-folds and CY 4-folds — are defined by forming virtual classes for moduli spaces of “semistable” objects, and integrating some cohomology classes over them. The virtual classes live in the homology of the “projective linear" moduli stack. Yuuji Tanaka and I are working on a way to define virtual classes counting strictly semistables, as well as just stables / stable pairs. 
 I conjecture that in all these theories, the virtual classes transform under change of stability condition by a universal wall-crossing formula (from my previous work on motivic invariants) in the Lie algebra $(H_*({\mathcal M}), [ , ])$. 

M-theory on K3 surfaces and Heterotic Strings on T^3 give rise to dual theories in 7 dimensions. Applying this duality fibre-wise is expected to connect G2 manifolds with Calabi-Yau threefolds (together with vector bundles). We make these ideas explicit for a class of G2 manifolds realized as twisted connected sums and prove the equivalence of the spectra of the dual theories. This naturally gives us examples of singular TCS G2 manifolds realizing non-abelian gauge theories with non-chiral matter.

A famous theorem of Mantel from 1907 states that every n-vertex graph with more than n^2/4 edges contains at least one triangle. In the 50s, Erdős asked for a quantitative version of this statement: for every n and e, how many triangles must an n-vertex e-edge graph contain?

This question has received a great deal of attention, and a long series of partial results culminated in an asymptotic solution by Razborov, extended to larger cliques by Nikiforov and Reiher. Currently, an exact solution is only known for a small range of edge densities, due to Lovász and Simonovits. In this talk, I will discuss the history of the problem and recent work which gives an exact solution for almost the entire range of edge densities. This is joint work with Hong Liu and Oleg Pikhurko.

One of the fundamental themes of geometric group theory is to
view finitely generated groups as geometric objects in their own right,
and to then understand to what extent the geometry of a group determines
its algebra. A theorem of Stallings says that a finitely generated group
has more than one end if and only if it splits over a finite subgroup.
In this talk, I will explain an analogous geometric characterisation of
when a group admits a splitting over certain classes of infinite subgroups.

The Z/2-equivariant Heegaard Floer cohomlogy of the double cover of S^3 along a knot, defined by Lipshitz, Hendricks, and Sarkar, 
is an isomorphism class of F_2[\theta]-modules. In this talk, we show that this invariant is natural, and is functorial under based cobordisms. 
Given a transverse knot K in the standard contact 3-sphere, we define an element of the Z/2-equivariant Heegaard Floer cohomology 
that depends only on the tranverse isotopy class of K, and is functorial under certain symplectic cobordisms.

Symplectic reduction is the natural quotient construction for symplectic manifolds. Given a free and proper action of a Lie group G on a symplectic manifold M, this process produces a new symplectic manifold of dimension dim(M) - 2 dim(G). For non-free actions, however, the result is usually fairly singular. But Sjamaar-Lerman (1991) showed that the singularities can be understood quite precisely: symplectic reductions by non-free actions are partitioned into smooth symplectic manifolds, and these manifolds fit nicely together in the sense that they form a stratification.

Symplectic reduction has an analogue in hyperkähler geometry, which has been a very important tool for constructing new examples of these special manifolds. In this talk, I will explain how Sjamaar-Lerman’s results can be extended to this setting, namely, hyperkähler quotients by non-free actions are stratified
spaces whose strata are hyperkähler.

I will talk about recent work, joint with M. Gröchenig and D. Wyss, on two related results involving the cohomology of moduli spaces of Higgs bundles. The first is a positive answer to a conjecture of Hausel and Thaddeus which predicts the equality of suitably defined Hodge numbers of moduli spaces of Higgs bundles with SL(n)- and PGL(n)-structure. The second is a new proof of Ngô's geometric stabilization theorem which appears in the proof of the fundamental lemma. I will give an introduction to these theorems and outline our argument, which, inspired by work of Batyrev, proceeds by comparing the number of points of these moduli spaces over finite fields via p-adic integration.

 The basic algebra-geometry dictionary for finitely generated k-algebras is one of the triumphs of 19th and early 20th century mathematics.  However, classes of related rings, such as their k-subalgebras, lack clean general properties or organizing principles, even when they arise naturally in problems of smooth projective geometry.  “Stabilization” in smooth topology and symplectic geometry, achieved by products with Euclidean space, substantially simplifies many
problems.  We discuss an analog in the more rigid setting of algebraic and arithmetic geometry, which, among other things (e.g., applications to counting rational points), gives some structure to the study of k-subalgebras.  We focus on the case of the moduli space of stable rational n-pointed curves to illustrate.