The study of properties that makes a combinatorial object (such as a permutation) to be quasirandom, i.e., to resemble a truly random object of the same kind, is an important and active line of research in combinatorics, which has various applications in computer science and statistics. The standard combinatorial way of comprehending quasirandomness evolved from nowadays classical results by Rödl, Thomason, and Chung, Graham and Wilson from the 1980s and appears in fundamental concepts in modern combinatorics such as the Regularity Method of Szemerédi.
The talk will start with discussing how the above mentioned classical results on quasirandom objects can be viewed through the lenses of the theory of combinatorial limits. We then employ analytic tools provided by the theory of combinatorial limits to solve several open problems concerning quasirandom objects of various kinds (in particular, directed graphs, permutations and Latin squares). At the end of the talk, we briefly explore the relation of presented results to the hypergraph regularity, which was developed independently by Gowers, and by Nagle, Rödl, Schacht and Skokan about two decades ago, and to the stochastic block model in statistics and network science.
We begin by introducing an action principle defined on a finite set of points. This action principle is causal in the sense that it generates a relation on pairs or points which distinguishes between spacelike and timelike separation. In this way, minimizing the action gives rise to a "discrete causal structure". We generalize our action principle to include continuum space-times and review existence results. We outline how the same action principle can be formulated in Minkowski space to obtain a formulation of quantum field theory.
In the second part of the talk, we consider as a special case a variational principle for Borel measures on the two-sphere. We prove that the support of every minimizing measure has no interior. This can be understood that when minimizing the action, a spontaneous symmetry breaking effect leads to the formation of a discrete structure.
We discuss several theorems relating the connection between the local CR-embeddability of 3-dimensional CR manifolds, and the existence of algebraically special Maxwell and gravitational fields. We reduce the Einstein equations for spacetimes associated with such fields to a system of CR invariant equations on a 3-dimensional CR manifold that is defined by the fields. Using the reduced Einstein equations, we construct two independent CR functions, which give the embedding. We also point out that the Einstein equations imply that the spacetime metric, after rescaling, become well defined on a circle bundle over the CR manifold.
Optimization problems governed by partial differential equations arise in a lot of technical applications. Newly, such problems occur also in medicin and finance.
Practial applications are characterized by several challenges: coupled systems of different types of equations, nonlinearities in the equations, nontrivial geometries (for instance reentrant corners) and pointwise inequality constraints for control and state.
In the talk new approaches and results are presented. Strategic goal is to solve such type of problems with given accuracy at low costs.
Low temperature Potts model is the simplest statistical mechanical model of q co-existing phases. In this talk we shall explain how to prove that in two dimensions Potts equilibrium crystal shapes are always smooth and strictly convex. In other words, in two dimensions Potts models do not undergo roughening transition. Since the models in question (except for the Ising q=2 case) are not exactly soluble, the proof relies on an intrinsic probabilistic analysis of random phase separation lines. The main step of the latter is to develop finite scale renormalization procedures which enable a coding of the interface distribution in terms of Ruelle operator for full shifts over countable alphabets.
Joint work with Massimo Camapanino and Yvan Velenik
The theory of gravity is non-renormalizable. Thus we either have to modify the theory or to face the problem of dealing with a non-renormalizable theory if we insist that gravity should be quantized. Here we follow the latter path and define quantum gravity as a non-perturbative sum over geometries in an exceedingly simple way. The approach is background independent but neverless we "observe" in computer simulations the emergence of a four-dimensional universe which can be viewed as a classical universe with superimposed quantum fluctuations. The results might be related to the ideas of Hartle, Hawking and Vikenkin about the creation of our universe from nothing.