The geometry of the moduli space of 4d  N=2  moduli spaces, and in particular of their Coulomb branches (CBs), is very constrained. In this talk I will show that through its careful study, we can learn general and somewhat surprising lessons about the properties of N=2  super conformal field theories (SCFTs). Specifically I will show that we can prove that the scaling dimension of CB coordinates, and thus of the corresponding operator at the SCFT fixed point, has to be rational and it has a rank-dependent maximum value and that in general the moduli spaces of N=2 SCFTs can have metric singularities as well as complex structure singularities. 

Finally I will outline how we can explicitly perform a classification of geometries of N>=3 SCFTs and carry out the program up to rank-2. The results are surprising and exciting in many ways.

Recovery of reaction pathways under the DCSC framework.

I will discuss joint work with Gismatullin, Halupczok, and Simonetta on the following problem: given a henselian valued field of characteristic 0, possibly equipped with analytic structure (in the sense stemming originally from Denef and van den Dries), describe the possibilities for a definable group G in the valued field sort which is definably almost simple, that is, has no proper infinite definable normal subgroups. We also have results for an algebraically closed valued field K in characteristic p, but assuming also that the group is a definable subgroup of GL(n, K).

I will explain how complex varieties which have asymptotically large intersections with finite grids can be seen to correspond to projective geometries, exploiting ideas of Hrushovski. I will describe how this leads to a precise characterisation of such varieties. Time permitting, I will discuss consequences for generalised sum-product estimates and connections to diophantine problems. This is joint work with Emmanuel Breuillard.

abstract:

In my talk I shall try to explain the following speculation and present some
evidence in the form of "correlations" between categoricity conjectures in
model theory and motivic conjectures in algebraic geometry.

Transfinite induction constructions developed in model theory are by now
sufficiently developed to be used to build analogues of objects in algebraic
geometry constructed with a choice of topology, such as a singular cohomology theory,
the Hodge decomposition, and fundamental groups of complex algebraic varieties.
Moreover, these algebraic geometric objects are often conjectured to satisfy
homogeneity or freeness properties which are true for objects constructed by
transfinite induction.


An example of this is Hrushovski fusion used to build Zilber pseudoexponentiation,
i.e. a group homomorphism  $ex:C^+ \to C^*$ which satisfies Schanuel conjecture,
a transcendence property analogous to Grothendieck conjecture on periods.


I shall also present a precise conjecture on "uniqueness" of Q-forms (comparison isomorphisms)
of complex etale cohomology, and will try to explain its relation to conjectures on l-adic
Galois representations coming from the theory of motivic Galois group.
 

In 1985, Babai and Sós asked whether there exists a constant c>0 such that every finite group of order n has a product-free set of size at least cn, where a product-free set of a group is a subset that does not contain three elements x,y and z  satisfying xy=z. Gowers showed that the answer is no in the early 2000s, by linking the existence of product-free sets of large density to the existence of low dimensional unitary representations.

In this talk, I will provide an answer to the aforementioned question by model theoretic means. Furthermore, I will relate some of Gowers' results to the existence of nontrivial definable compactifications of nonstandard finite groups.
 

Authors:

Anne Balter and Antoon Pelsser

Models can be wrong and recognising their limitations is important in financial and economic decision making under uncertainty. Robust strategies, which are least sensitive to perturbations of the underlying model, take uncertainty into account. Interpreting

the explicit set of alternative models surrounding the baseline model has been difficult so far. We specify alternative models by a stochastic change of probability measure and derive a quantitative bound on the uncertainty set. We find an explicit ex ante relation

between the choice parameter k, which is the radius of the uncertainty set, and the Type I and II error probabilities on the statistical test that is hypothetically performed to investigate whether the model specification could be rejected at the future test horizon.

The hypothetical test is constructed to obtain all alternative models that cannot be distinguished from the baseline model with sufficient power. Moreover, we also link the ambiguity bound, which is now a function of interpretable variables, to numerical

values on several divergence measures. Finally, we illustrate the methodology on a robust investment problem and identify how the robustness multiplier can be numerically interpreted by ascribing meaning to the amount of ambiguity.

In this talk, we will present results regarding the regularity and

rigidity of solutions to the Monge-Ampere equation, inspired by the role

played by this equation in the context of prestrained elasticity. We will

show how the Nash-Kuiper convex integration can be applied here to achieve

flexibility of Holder solutions, and how other techniques from fluid

dynamics (the commutator estimate, yielding the degree formula in the

present context) find their parallels in proving the rigidity. We will indicate

possible avenues for the future related research.