L4

In this talk we discuss the Type I blow up and the related problems in the 3D Euler equations. We say a solution $v$ to the Euler equations satisfies Type I condition at possible blow up time $T_*$ if $\lim\sup_{t\nearrow T_*} (T_*-t) \|\nabla v(t)\|_{L^\infty} <+\infty$. The scenario of Type I blow up is a natural generalization of the self-similar(or discretely self-similar) blow up. We present some recent progresses of our study regarding this. We first localize previous result that ``small Type I blow up'' is absent. After that we show that the atomic concentration of energy is excluded under the Type I condition. This result, in particular, solves the problem of removing discretely self-similar blow up in the energy conserving scale, since one point energy concentration is necessarily accompanied with such blow up. We also localize the Beale-Kato-Majda type blow up criterion. Using similar local blow up criterion for the 2D Boussinesq equations, we can show that Type I and some of Type II blow up in a region off the axis can be excluded in the axisymmetric Euler equations. These are joint works with J. Wolf.

For measuring families of curves, or, more generally, of measures, $M_p$-modulus is traditionally used. More recent studies use so-called plans on measures. In their fundamental paper Ambrosio, Di Marino and Savare proved that these two approaches are in some sense equivalent within $1<p<\infty$. We consider the limiting case $p=1$ and show that the $AM$-modulus can be obtained alternatively by the plan approach. On the way, we demonstrate unexpected behavior of the $AM$-modulus in comparison with usual capacities.

This is a joint work with Vendula Honzlov\'a Exnerov\'a, Ond\v{r}ej F.K. Kalenda and Olli Martio. Partially supported by the grant GA\,\v{C}R P201/18-07996S of the Czech Science Foundation.

In this talk we discuss new effective methods to test pairwise containment of arbitrary (possibly singular) subvarieties of any smooth projective toric variety and to determine algebraic multiplicity without working in local rings. These methods may be implemented without using Gröbner bases; in particular any algorithm to compute the number of solutions of a zero-dimensional polynomial system may be used. The methods arise from techniques developed to compute the Segre class s(X,Y) of X in Y for X and Y arbitrary subschemes of some smooth projective toric variety T. In particular, this work also gives an explicit method to compute these Segre classes and other associated objects such as the Fulton-MacPherson intersection product of projective varieties.
These algorithms are implemented in Macaulay2 and have been found to be effective on a variety of examples. This is joint work with Corey Harris (University of Oslo).

The moduli space of smooth hypersurfaces in projective space was constructed by Mumford in the 60’s using his newly developed classical (a.k.a. reductive) Geometric Invariant Theory.  I wish to generalise this construction to hypersurfaces in weighted projective space (or more generally orbifold toric varieties). The automorphism group of a toric variety is in general non-reductive and I will use new results in non-reductive GIT, developed by F. Kirwan et al., to construct a moduli space of quasismooth hypersurfaces in certain weighted projective spaces. I will give geometric characterisations of notions of stability arising from non-reductive GIT.

Let $X$ be a K3 surface and let $Z_X(q)$ be the generating series of the topological Euler characteristics of the Hilbert scheme of points on $X$. It is known that $q/Z_X(q)$ equals the discriminant form $\Delta(\tau)$ after the change of variables $q=e^{2 \pi i \tau}$. In this talk we consider the equivariant generalization of this result, when a finite group $G$ acts on $X$ symplectically. Mukai and Xiao has shown that there are exactly 81 possibilities for such an action in terms of types of the fixed points. The analogue of $q/Z_X(q)$ in each of the 81 cases turns out to be a cusp form (after the same change of variables). Knowledge of modular forms is not assumed in the talk; I will introduce all necessary concepts. Joint work with Jim Bryan.

For a finite subgroup $G$ of $SL(2,C)$ and for $n \geq 1$,  the Hilbert scheme $X=Hilb^{[n]}(S)$ of $n$ points on the minimal resolution $S$ of the Kleinian singularity $C^2/G$ provides a crepant resolution of the symplectic quotient $C^{2n}/G_n$, where $G_n$ is the wreath product of $G$ with $S_n$. I'll explain why every projective, crepant resolution of $C^{2n}/G_n$ is a quiver variety, and why the movable cone of $X$ can be described in terms of an extended Catalan hyperplane arrangement of the root system associated to $G$ by John McKay. These results extend the algebro-geometric aspects of Kronheimer's hyperkahler description of $S$ to higher dimensions. This is joint work with Gwyn Bellamy.

I will describe joint work with Bruno Premoselli which gives a new existence theorem for negatively curved Einstein 4-manifolds, which are obtained by smoothing the singularities of hyperbolic cone metrics. Let (M_k) be a sequence of compact 4-manifolds and let g_k be a hyperbolic cone metric on M_k with cone angle \alpha (independent of k) along a smooth surface S_k. We make the following assumptions:

1. The injectivity radius i(k) of M_k tends to infinity (where in defining injectivity radius we ignore those geodesics which hit the cone singularity)

2. The normal injectivity radius of S_k is at least i(k)/2.

3. The area of the singular locii satisfy A(S_k)\leq C \exp(5 i(k)/2) for some C independent of k.

When these assumptions hold, we prove that for all large k, M_k carries a smooth Einstein metric of negative curvature. The proof involves a gluing theorem and a parameter dependent implicit function theorem (where k is the parameter). As I will explain, negative curvature plays an essential role in the proof. (For those who may be aware of our arxiv preprint, https://arxiv.org/abs/1802.00608 [arxiv.org], the work
I will describe has a new feature, namely we now treat all cone angles, and not just those which are greater than 2\pi. This gives lots more examples of Einstein 4-manifolds.)

String compactifications are essential for connecting string theory to low energy particle physics and cosmology. Moduli stabilisation gives rise to effective Lagrangians that capture the low-energy degrees of freedom. Much recent interest has been on swampland consistency conditions on such effective
field theories - which low energy Lagrangians can arise from quantum gravity? Furthermore, given that moduli stabilisation scenarios often exist in AdS space, we can also ask: what do swampland conditions mean in the context of AdS/CFT? I describe work on developing a holographic understanding of moduli stabilisation and swampland consistency conditions. I focus in particular on the Large Volume Scenario, which is especially appealing from a holographic perspective as in the large volume limit all its interactions can be expressed solely in terms of the AdS radius, with no free dimensionless parameters.

Integrable models provide simplified examples of quantum field theories with self-interaction. As often in relativistic quantum theory, their local observables are difficult to control mathematically. One either tries to construct pointlike local quantum fields, leading to possibly divergent series expansions, or one defines the local observables indirectly via wedge-local quantities, losing control over their explicit form.

We propose a new, hybrid approach: We aim to describe local quantum fields; but rather than exhibiting their n-point functions and verifying the Wightman axioms, we establish them as closed operators affiliated with a net of von Neumann algebras. This is shown to work at least in the Ising model.

The renormalized expectation value of the stress energy tensor (RSET) is an object of central importance in quantum field theory in curved space-time, but calculating this on black hole space-times is far from trivial.  The vacuum polarization (VP) of a quantum scalar field is computationally simpler and shares some features with the RSET.  In this talk we consider the properties of the VP for a massless, conformally coupled scalar field on asymptotically anti-de Sitter black holes with spherical, flat and hyperbolic horizons.  We focus on the effect of the different horizon curvature on the VP, and the role played by the boundary conditions far from the black hole.