Automatic sequences are one of the most basic models of computation, with remarkable links to dynamics, algebra and logic (among other fields). In the talk, we will explore a point of view inspired by higher order Fourier analysis. Specifically, we will investigate the behaviour of Gowers norms of some automatic sequences, and (almost) classify all automatic sequences given by generalised polynomial fomulas. The tools used will include some non-trivial results concerning dynamics of nilsystems and their connection
I will discuss joint work with Balazs Barany and Ariel Rapaport on the dimension of self-affine sets and measures. We confirm that under mild irreducibility conditions on the generating maps, the dimension is "as expected", i.e. equal to the affinity or Lyapunov dimension. This completes a program started by Falconer in the 1980s. In the first part of the talk I will explain how the Lyapunov dimension arises from Ledrappier-Young formula for self-affine sets, and then explain how additive combinatorics methods can be used to prove that this is the correct dimension.
In the classical settings of Anosov diffeomorphisms or more general locally maximal hyperbolic sets I describe a new approach for constructing equilibrium measures corresponding to some continuous potentials and for studying some of their ergodic properties. This approach is pure geometrical in its nature and uses no symbolic representations of the system. As a result it can be used to effect thermodynamics formalism for systems for which no symbolic representation is available such as partially hyperbolic systems.
Ergodicity of a nonsingular transformation is equivalent to divergence of ergodic sums of all indicator functions of positive measure. In practice this divergence condition is easy to verify for “nice” sets and hard to verify for general sets. In this talk we will show two different arguments of how to pass from a
In this talk I will introduce the relations of rigidity, equicontinuity and pointwise recurrence between an invertible topological dynamical system (X; T) and the dynamical system (K(X); T_K) induced on the hyperspace K(X) of all compact subsets of X, and show some
Based on joint work with Piotr Oprocha, Xiangdong Ye and Ruifeng Zhang.
Given n uniform points on the surface of a two-dimensional sphere, how can we partition the sphere fairly among them ? "Fairly" means that each region has the same area. It turns out that if the given points apply a two-dimensional gravity force to the rest of the sphere, then the basins of attraction for the resulting gradient flow yield such a partition—with exactly equal areas, no matter how the points are distributed. (See the
In the theory of Diophantine approximations, singular points are ones for which Dirichlet’s theorem can be infinitely improved. It is easy to see that all rational points are singular. In the special case of dimension one, the only singular points are the rational ones. In higher dimensions, points lying on a rational hyperplane are also obviously singular. However, in this case there are additional singular points. In the dynamical setting the singular points are related to divergent trajectories.
A celebrated theorem of Duke states that Picard/Galois orbits of CM points on a complex modular curve, e.g. SL2(Z)\SL2(R)/SO2(R), equidistribute in the limit when the absolute value of the discriminant goes to infinity. Michel and Venkatesh have conjectured that a sequence of some 2-fold self-joinings of CM orbits equidistributes in the product space as long as it escapes any closed orbit of an intermediate subgroup, i.e. Hecke correspondences.
Speaker: Amir Yehudayoff (Technion)
Title: An exposition to topological overlap in the plane
Abstract: We shall discuss Gromov's proof for topological overlap in
the plane. We will also consider a weighted version of Gromov's
theorem and deduce a dual statement.
In this talk we will consider the question of defining descendant invariants in open Gromov-Witten theory. In the closed Gromov-Witten theory, descendant invariants are constructed from Chern classes of certain tautological lines bundles which live on the moduli space of stable curves. The intersection numbers obtained from those classes (and other classes) can be incorporated in a generating function that satisfies various partial differential equations reflecting recurrence relations and which can sometimes be used to calculate the numbers explicitly.
Suppose Z/n acts on a manifold, then if it has a fixed point, the natural homomorphism Z/n --> Out(π) (π = the fundamental group) lifts to Aut(π). If π is centreless, and the aspherical manifold is locally symmetric and the action is isometric, the converse holds. We shall discuss the extent to which this observation is geometric and to what extent it's topological. (It will depend on M and it will depend on n).
לאירוע הזה יש שיחת וידאו.
Let G be a group acting on a projective variety. If G is noncompact, the quotient space X/G is in general "bad". In this talk I will discuss two methods to make this quotient "good", i.e. GIT and symplectic reduction. Both methods include the idea of keeping "good orbits" and throwing away "bad orbits". Hilbert-Mumford criterion provides a way to distinguish good orbits (which are called stable orbits) and the Kempf-Ness theorem tells us two methods produce the same quotient space. I will use several examples to show how Hilbert-Mumford criterion and the Kempf-Ness theorem work.
This talk revolves around the question of how close is one Riemannian manifold to being isometrically immersible in another.
We associate with every mapping $f:(M,g) \to (N,h)$ a measure of distortion - an average distance of $df$ from being an isometry. Reshetnyak's theorem states that a sequence of mappings between Euclidean domains whose distortion tends to zero has a subsequence converging to an isometry.
I will present a generalization of Reshetnyak’s theorem to the general Riemannian setting.
We will start be explaining the difficulties in constructing enumerative open Gromov-Witten theories, and mention cases we can overcome these difficulties and obtain a rich enumerative structure.
We then restrict to one such case, and define the full genus 0 stationary open Gromov-Witten theory of maps to CP^1 with boundary conditions on RP^1, including descendents, together with its equivariant extension. We fully compute the theory.