L4

We give a description of the category of ordinary K3 surfaces over a finite field in terms of linear algebra data over Z. This gives an analogue for K3 surfaces of Deligne's description of the category of ordinary abelian varieties over a finite field, and refines earlier work by N.O. Nygaard and J.-D. Yu. Two important ingredients in the proof are integral p-adic Hodge theory, and a description of CM points on Shimura stacks in terms of associated Galois representations. References: arXiv:1711.09225, arXiv:1707.01236.

For energy functionals composed of competing short- and long-range interactions, minimizers are often conjectured to form essentially periodic patterns on some intermediate lengthscale. However,  not many detailed structural results or proofs of periodicity are known in dimensions larger than 1. We study a functional composed of  the attractive, local interfacial energy of charges concentrated on a hyperplane and the energy of the electric field generated by these charges in the full space, which can be interpreted as a repulsive, non-local functional of the charges. We follow the approach of Alberti-Choksi-Otto and prove that the energy of minimizers of this functional is uniformly distributed  on cubes intersecting the hyperplane, which are sufficiently large with respect to the intrinsic lengthscale.

This is a joint work with A. Julia and F. Otto.

Some natural moduli problems give rise to stacks with infinite stabilizers. I will report on recent work with Dan Edidin where we give a canonical sequence of saturated blow-ups that makes the stabilizers finite. This generalizes earlier work in GIT by Kirwan and Reichstein, and on toric stacks by Edidin-More. Time permitting, I will also mention a recent application to generalized Donaldson-Thomas invariants by Kiem-Li-Savvas.

 A law for a group G is a non-trivial equation satisfied by all tuples of elements in G. We study the length of the shortest law holding in a finite group. We produce new short laws holding (a) in finite simple groups of Lie type and (b) simultaneously in all finite groups of small order. As an application of the latter we obtain a new lower bound on the residual finiteness growth of free groups. This talk is based on joint work with Andreas Thom.

Post-quantum cryptography has been one of the most active subfields of
cryptography in the last few years. This is especially true today as
standardization efforts are currently underway, with no less than 69
candidate cryptographic schemes proposed.

In this talk, I will present one of these schemes: Falcon, a signature
scheme based on the NTRU class of structured lattices. I will focus on
mathematical aspects of Falcon: for example how we take advantage of the
algebraic structure to speed up some operations, or how relying on the
most adequate probability divergence can go a long way in getting more
efficient parameters "for free". The talk will be concluded with a few
open problems.

Let n be a positive integer. In this talk we provide a recipe to 
construct full orbit sequences in the affine n-dimensional space over a 
finite field. For n=1 our construction covers the case of the well 
studied pseudorandom number generator ICG.

This is a joint work with Federico Amadio Guidi.

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 $(H_*({\mathcal M}), [ , ])$.