Abstract: The ultrafilter lemma, saying that every filter can be extended to an ultrafilter, is one of the fundamental consequences of the axiom of choice. By adding closure assumptions, and asking for extension of $\kappa$-complete filters to $\kappa$-complete ultrafilters, we obtain the notion of strongly compact cardinal, which has a very high consistency strength.
Abstract: An observation by Jens Marklof shows that the primitive rational points of a fixed denominator along the periodic unipotent orbit of volume equal to the square of the denominator equidistribute inside a proper submanifold of the modular surface. This concentration as well as the equidistribution are intimately related to classical questions of number theoretic origin. We discuss the distribution of the primitive rational points along periodic orbits of intermediate size. In this case, we can show joint equidistribution with polynomial rate in the modular surface and in the torus.
Title:
Dilations of q-commuting unitaries
Abstract:
Let (u,v) be a pair of unitary operators on a Hilbert space H such that vu=quv for a complex number q of modulus 1. For q' another complex number of modulus 1, we determine the smallest constant c>0 for which there exists a pair of q' commuting unitaries (U,V) on a larger Hilbert space K containing H such that (u,v) is the compression of (cU,cV) to H.
As it was observed a few years ago, there exists a certain signed count of real lines on real projective hypersurfaces of degree 2n+1 and dimension n that, contrary to the honest "cardinal" count, is independent of the choice of a hypersurface, and by this reason provides, as a consequence, a strong lower bound on the honest count. Originally, in this invariant signed count the input of a line was given by its local contribution to the Euler number of an appropriate auxiliary universal vector bundle.
Abstract: We show that under certain conditions, a random walk on the 1-dim torus by affine expanding maps has a unique stationary measure. We then use this result to show that given an IFS of contracting similarity maps of the real line with a uniform contraction ratio 1/D, where D is some integer > 1, under some suitable condition, almost every point in the attractor of the given IFS (w.r.t. a natural measure) is normal to base D.
Abstract:
A collection of polygons with the property that to each side one can find another side parallel to it can be endowed with a translation surface structure by glueing along those edges.
This means that the closed surfaces obtained carries a flat metric outside finitely many conical singularities. Geodesics (which are straight lines) connecting such singularities are called saddle connections.
Title: Projections of Tree-Prikry forcing.
Abstract:
Gitik, Kanovei and Koepke proved that if U is a normal measure over \kappa then the projections of Prikry forcing with U is essentially Prikry forcing with U.
The questions remains regarding to the Tree-Prikry forcing. Gitik and B. showed that without normality, it is possible that a Tree-Prikry generic sequence adds a Add(\kappa,1)
generic function.
In this talk we wish to examine which forcing notions can be projections of Tree-Prikry forcing under different large cardinals assumptions.
Abstract: In their 1985 paper, the above three authors introduced a consistent generalization of Ramsey's theorem to pairs of countable ordinals, which we abbreviate as $OCA_{ARS}$. This axiom asserts that for any continuous coloring (with respect to an appropriate topology) of pairs of countable ordinals, there is a decomposition of $\omega_1$ into countably-many homogeneous sets. The key to their argument is to construct Preassignments of Colors.
Title: Projections of Tree-Prikry forcing.
Abstract:
Gitik, Kanovei and Koepke proved that if U is a normal measure over \kappa then the projections of Prikry forcing with U is essentially Prikry forcing with U.
The questions remains regarding to the Tree-Prikry forcing. Gitik and B. showed that without normality, it is possible that a Tree-Prikry generic sequence adds a Add(\kappa,1)
generic function.
In this talk we wish to examine which forcing notions can be projections of Tree-Prikry forcing under different large cardinals assumptions.
Title: "Paradoxical" sets with no well-ordering of the reals
Abstract: By a Hamel basis we mean a basis for the reals, R, construed as a vecor space over
the field of rationals. In 1905, G. Hamel constructed such a basis from a well-ordering
of R. In 1975, D. Pincus and K. Prikry asked "whether a Hamel basis exists in any
model in which R cannot be well ordered." About two years ago, we answered this positively
in a joint paper with M. Beriashvili, L. Wu, and L. Yu. In more recent joint
A powerful theory due to Ornstein and his collaborators has been successfully applied to many random systems to show that they are isomorphic to independent and identically distributed systems. The isomorphisms provided by Ornstein's theory may not be finitary, that is, effectively realizable in practice. Despite the large number of systems known to be Bernoulli, there are only a handful of cases where explicit finitary isomorphisms have been constructed. In this talk, we will discuss classical and recent constructions, and some long standing open problems.
Title: Cwikel's bound reloaded
Abstract: There are a couple of proofs by now for the famous Cwikel--Lieb--Rozenblum (CLR) bound, which is a semiclassical bound on the number of bound states for a Schr\"odinger operator, proven in the 1970s.
