Abstract: We first survey a recent progress related to the nonsingular Bernoulli transformations. Then we construct inductively new examples of conservative Bernoulli maps of type III. They appear as a limit of a sequence of Bernoulli maps of type II_1.
Poisson suspensions are random sets of points endowed with a transformation that displaces each point according to a single transformation of the sigma-finite space where the points lie. In this ongoing work, instead of dealing with measure-preserving transformations (which is the classical case), we are going to present our attempt to explore the non-singular case. The difficulties are counterbalanced by new tools that are trivial in the measure-preserving case but highly informative in the non-singular one.
We will present these tools as well as the first basic results we’ve obtained.
Abstract: The Minimal Tower Problem was one of most famous question in Cardinal Invariants. We will present a combinatorial argument of this proof, which without using model theory and forcing, motivated by Malliaris and Shelah's proof.
Repeats every week every Sunday until Sat Jun 29 2019 except Sun Apr 21 2019.
11:00am to 1:00pm
Zlil Sela and Alex Lubotzky "Model theory of groups"
In the first part of the course we will present some of the main results in the theory of free,
hyperbolic and related groups, many of which appear as lattices in rank one simple Lie groups
We will present some of the main objects that are used in studying the theory of these groups,
and at least sketch the proofs of some of the main theorems.
In the second part of the course, we will talk about the model theory of lattices in high rank simple Lie groups.
Repeats every week every Sunday until Sun Jun 23 2019 except Sun Apr 21 2019.
2:00pm to 4:00pm
Yun and Zhang compute the Taylor series expansion of an automorphic L-function over a function field, in terms of intersection pairings of certain algebraic cycles on the so-called moduli stack of shtukas. This generalizes the Waldspurger and Gross-Zagier formulas, which concern the first two coefficients.
The goal of the seminar is to develop the background necessary to state their formula, and then indicate the structure of the proof. If time allows, we may also discuss applications to the Birch and Swinnerton-Dyer conjecture for elliptic curves over function fields.
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.
Dilations of q-commuting unitaries
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.
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.
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.