Colloquium Ical Atom Mailing List

We will discuss the Blaschke branch of integral geometry and its manifestations in pseudo-Riemannian space forms. First we will recall the fundamental notion of intrinsic volumes, known as quermassintegrals in convex geometry. Those notions were extended later to Riemannian manifolds by H. Weyl, who discovered a remarkable fact: given a manifold M embedded in Euclidean space, the volume of the epsilon-tube around it is an invariant of the Riemannian metric on M. We then discuss Alesker’s theory of smooth valuations, which provides a framework and a powerful toolset to study integral geometry, in particular in the presence of various symmetry groups. Finally, we will use those ideas to explain some recent results in the integral geometry of pseudo-Riemannian manifolds, in particular a collection of principal Crofton formulas in all space forms, and a Chern-Gauss-Bonnet formula for metrics of varying signature. Partially based on joint works with S. Alesker, A. Bernig, G. Solanes.

In finite dimensional Riemannian geometry, everything behaves nicely — the Riemannian metric induces a distance function, geodesics exist (at least for some time), and so on. In infinite dimensional Riemannian geometry, however, chaos reigns. In this talk I will focus on diffeomorphism groups, and on a particularly important hierarchy of Riemannian metrics on them: right-invariant Sobolev metrics. These arise in many different contexts, from purely mathematical ones, to applications in hydrodynamics and imaging. I will give a brief introduction to these metrics, why we care about them, and what we know (and don’t know) about their properties. Parts of the talk will be based on joint works with Bob Jerrard and Martin Bauer.

The study of aperiodic order and mathematical models of quasicrystals is concerned with ways in which disordered structures can nevertheless manifest aspects of order. In the talk I will describe examples such as the aperiodic Penrose and pinwheel tilings, together with several geometric, functional, dynamical and spectral properties that enable us to measure how far such constructions are from demonstrating lattice-like behavior. A particular focus will be given to new results on multiscale substitution tilings, a class of tilings that was recently introduced jointly with Yaar Solomon.

In various areas of mathematics there exist “big fiber theorems”, these are theorems of the following type: “For any map in a certain class, there exists a ‘big’ fiber”, where the class of maps and the notion of size changes from case to case.

We will discuss three examples of such theorems, coming from combinatorics, topology and symplectic topology from a unified viewpoint provided by Gromov’s notion of ideal-valued measures.

We adapt the latter notion to the realm of symplectic topology, using an enhancement of a certain cohomology theory on symplectic manifolds introduced by Varolgunes, allowing us to prove symplectic analogues for the first two theorems, yielding new symplectic rigidity results.

Necessary preliminaries will be explained. The talk is based on a joint work with Adi Dickstein, Leonid Polterovich and Frol Zapolsky.

Consider an infinite group G acting by isometries on some metric space X.
How does a “typical” element act? Consider a representation of G into some matrix group. What sort of matrix represents “typical” elements of G?

The answer depends on what we mean by the word “typical,” of which there are at least two reasonable notions. We may take a random walk on G and look where it lands after a large number of steps. We may also fix a generating set for G and look how large balls are distributed.

I will talk about how these two notions of genericity are related and how they differ, focusing on the setting of hyperbolic groups. I will also explain that the following is true with respect to both notions: For a group acting on a Gromov hyperbolic metric space typical elements act loxodromically, i.e. with north-south dynamics.

For a representation of a large class of groups (including hyperbolic groups) into SL_n R, typical elements map to matrices whose eigenvalues are all simple and have distinct moduli.

It is well known that if a finite set of integers A tiles the integers by translations, then the translation set must be periodic, so that the tiling is equivalent to a factorization A+B=Z_M of a finite cyclic group. Coven and Meyerowitz (1998) proved that when the tiling period M has at most two distinct prime factors, each of the sets A and B can be replaced by a highly ordered “standard” tiling complement. It is not known whether this behaviour persists for all tilings with no restrictions on the number of prime factors of M. In joint work with Izabella Laba (UBC), we proved that this is true when M=(pqr)^2. In my talk I will discuss this problem and introduce some ingredients from the proof.

The space Hom(\Gamma,G) of homomorphisms from a finitely-generated group \Gamma to a complex semisimple algebraic group G is known as the G-representation variety of \Gamma. We study this space when G is fixed and \Gamma is a random group in the few-relators model. That is, \Gamma is generated by k elements subject to r random relations of length L, where k and r are fixed and L tends to infinity.

More precisely, we study the subvariety Z of Hom(\Gamma,G), consisting of all homomorphisms whose images are Zariski dense in G. We give an explicit formula for the dimension of Z, valid with probability tending to 1, and study the Galois action on its geometric components. In particular, we show that in the case of deficiency 1 (i.e., k-r=1), the Zariski-dense G-representations of a typical \Gamma enjoy Galois rigidity.

Our methods assume the Generalized Riemann Hypothesis and exploit mixing of random walks and spectral gap estimates on finite groups.

Based on a joint work with E. Breuillard and P. Varju.

In many data-driven applications, the data follows some geometric structure, and the goal is to recover this structure. In many cases, the observed data is noisy and the recovery task is even more challenging. A common assumption is that the data lies on a low dimensional manifold. Estimating a manifold from noisy samples has proven to be a challenging task. Indeed, even after decades of research, there was no (computationally tractable) algorithm that accurately estimates a manifold from noisy samples with a constant level of noise.

In this talk, we will present a method that estimates a manifold and its tangent. Moreover, we establish convergence rates, which are essentially as good as existing convergence rates for function estimation.

Continuous wavelet transforms are mappings that isometrically embed a signal space to a coefficient space over a locally compact group, based on so-called square integrable representations. For example, the 1D wavelet transform maps time signals to functions over the time-scale plane based on the affine group. When using wavelet transforms for signal processing, it is often useful to work interchangeably with the signal and the coefficient spaces. For example, we would like to know what operation in the signal domain is equivalent to multiplication in the coefficient space. While such a point of view is natural in classical Fourier analysis (i.e., “time convolution is equivalent to frequency multiplication”), it is not compatible with wavelet analysis, since wavelet transforms are not surjective. In this talk, I will present the wavelet-Plancherel theory – an extension of classical wavelet theory in which the wavelet transform is canonically extended to an isometric isomorphism. The new theory allows formulating a variety of coefficient domain operations as signal domain operations, with closed form formulas. Using these so-called pull-back formulas, we are able to reduce the computational complexity of some wavelet-based signal processing methods. The theory is also useful for proving theorems in wavelet analysis. I will present an extension of the Heisenberg uncertainty principle to wavelet transforms and prove the existence of uncertainty minimizers using the wavelet-Plancherel theory.

A function at a non-critical point can be converted to a linear form by a local coordinate change. At an isolated critical point one has the weaker statement: higher order perturbations do not change the group orbit. Namely, the function is determined (up to the local coordinate changes) by its (finite) Taylor polynomial.

This finite-determinacy property was one of the starting points of Singularity Theory. Traditionally such statements are proved by vector field integration. In particular, the group of local coordinate changes becomes a ``Lie-type” group.

I will show such determinacy results for maps of germs of (Noetherian) schemes. The essential tool is the “vector field integration” in any characteristic. This equips numerous groups acting on filtered modules with the ``Lie-type” structure. (joint work with G. Belitskii, A.F. Boix, G.M. Greuel.)