In Memoriam: Lida Barrett
Exotic continua in modern mathematics – June 9-12, 2022
- Judy Kennedy (Lamar University)
- Olga Lukina (University of Vienna)
- Kim Ruane (Tufts University)
- Nageswari Shanmugalingam (University of Cincinnati)
- Guofang Wei (University of California, Santa Barbara)
- Ryan Alvarado (Amherst College)
- Sylvester Eriksson-Bique (University of Oulu)
- Craig Guilbault (University of Wisconsin, Milwaukee)
- Tamara Kucherenko (City College of New York)
- Constantine Medynets (United States Naval Academy)
- Jiayin Pan (Fields Institute)
- Raquel Perales (National Autonomous University of Mexico)
- Catherine Searle (Wichita State University)
- Sonja Stimac (University of Zagreb)
- Christina Sormani (City University of New York)
- Rodrigo Trevino (University of Maryland)
- Sergio Zamora (Penn State University)
- Scott Zimmerman (Ohio State University)
Function spaces and the geometry of sets.
Abstract: The primary focus of this talk is on how the geometric makeup of a given ambient can directly influence the amount of analysis that the underlying space can support. As an illustration of the interplay between these two branches of mathematics, we will survey some recently obtained results pertaining to the properties of extension and embedding domains for the scale of Besov and Triebel-Lizorkin spaces (𝑁𝑠𝑝,𝑞 and 𝑀𝑠𝑝,𝑞 spaces) in the general setting of quasi-metric spaces. In particular, we will provide examples of several environments, including fractal sets, which highlight how the range of the smoothness parameter, 𝑠, for which these embedding and extension results hold is intimately linked to the geometry of underlying quasi-metric space. This talk is based on joint work with Dachun Yang and Wen Yuan.
Duality techniques in Analysis and Geometry
Abstract: Duality takes various forms in metric spaces and analysis. In this talk we explore some recent geometric and analytic results, which employ duality, and focus on two results. The first of these is a duality of separating sets and connecting curves, which is intimately linked to the max-flow-min-cut theorem. With Pietro Poggi-Corradini, we extended a part of this duality in a sharp way and in the process discovered a new approximation of Lipschitz functions. The second is a duality between moduli of curve families and probability measures on curves (which generate currents). This is also a version of max-flow-min-cut, but with cuts replaced by an admissible function, and the probability measure on curves representing a flow. Such measures can be used to get simple obstructions to embeddings – an idea that was formalized in a “splitting theorem” with Guy C. David.
On the appearance of exotic continua in the study of groups
Abstract: The big idea in geometric group theory is to study a group 𝐺 by finding a 𝐺-action on a space 𝑋, then use geometric and topological properties of 𝑋 to make deductions about 𝐺. One almost always begins with a very nice 𝐺-action (proper, cocompact and often free) on a very nice space 𝑋 (manifolds, CW complexes, and CAT(0) space are especially popular). Since 𝐺 is usually infinite, 𝑋 is normally noncompact. Exotic continua make their appearance indirectly as boundaries of these spaces. By passing to a boundary, much of the information contained in 𝑋 can be compressed into a compact package. Since this package contains a lot of information, boundaries tend to be complicated. Cantor sets, Sierpinski carpets, Menger curves, Pontryagin surfaces, and higer-dimensional analogs are more the rule than the exception. Since the appearance of boundaries is indirect, significant effort is often required to identify or construct them. In this talk, I will begin with a broad picture of the construction and application of group boundaries. From there, I will move to some current work where interesting groups are constructed and-with a blend of modern and classical techniques-the boundaries are identified.
Bizarre topology is natural in dynamical systems
Abstract: A continuum is a compact connected metric space. A continuum is indecomposable if it is not the union of two proper subcontinua (which necessarily intersect). The first question that might arise, if one is not familiar with many continua, is “Do indecomposable continua really exist?” The second thought is probably “Surely such bizarre objects represent extreme pathology.” These objects do exist, and they are not just examples of pathology. They can be expected to arise in many natural situations. We will discuss the history of indecomposable continua and then give examples of how they arise in nonlinear dynamical systems. We will end with an unusual continuum, a Lelek fan, that has arisen recently as the inverse limit of set-valued functions with one bonding map. (A Lelek fan is decomposable.)
Katok’s flexibility paradigm and its realization in symbolic dynamics
Abstract: Katok launched the flexibility program which has been described in a nutshell as follows: “there should be no restrictions on the dynamical characteristics apart from a few obvious ones”. This is a novel direction in dynamics, yet the core problems are clear and accessible to a rather broad community of mathematicians and this has made the program develop at a rapid pace. I will outline the flexibility program and showcase related results obtained within the last few years. Then I will present a striking application of the flexibility paradigm to the pressure function on compact symbolic systems. This is based on joint work with Anthony Quas.
Cantor dynamics of laminations: problems and examples
Abstract: A matchbox manifold, or a lamination in the sense of Sullivan, is a continuum where every point has a product neighborhood with a Euclidean and a Cantor direction. Examples of such laminations are higher-dimensional solenoids and tiling spaces. In this first introductory talk, we discuss basic constructions and give examples of matchbox manifolds. We introduce an important tool in their study, the transverse dynamical system, given by a group action on a Cantor set, and discuss recent classification results.
Weyl groups in Cantor dynamics: applications
Abstract: Maximal tori and Weyl groups for Cantor actions were introduced in my recent joint work with Cortez as a tool to study the properties of such actions. These notions were inspired by analogous notions in the theory of compact Lie groups, where they are fundamental. In the talk, we show how maximal tori and Weyl groups can be used to study properties of representations of absolute Galois groups of number fields as profinite groups acting on trees, and discuss open problems.
Full Groups of Cantor Dynamical Systems: the interplay between group theory and Cantor dynamics
Abstract: Full groups originated from the theory of measurable (and later Cantor) dynamical systems and their von Neumann-algebra (C*-algebra) crossed-products. For a given topological dynamical system (𝑋,𝐺), the full group [𝐺] can be broadly defined as the set of all homeomorphisms of 𝑋 that act within the 𝐺-orbits. Thus, the full groups can be viewed as a generalized symmetric group of the orbit equivalence relation of (𝑋,𝐺). In a series of papers by Giordano-Putnam-Skau, Matui, Medynets, Nekrashevych, and others, it was shown that full groups (as abstract groups) encode complete information about the underlying dynamical systems up to (topological) orbit equivalence. In recent years, the development of the theory of full groups for Cantor minimal systems has been having considerable impact on geometric group theory driven primarily by the fact that by tweaking dynamical properties of the underlying dynamical system (𝑋,𝐺), we can produce a (countable) full group [𝐺] with new and unusual properties, which has been successfully used to solve some open problems in geometric group theory. In this talk, we will discuss the history of the subject and recent developments. We will also touch upon Vershik’s conjecture dealing with the theory of characters for full groups or their finite-type factor representations.
Some examples of open manifolds with positive Ricci curvature
Abstract: We give some examples of open manifolds with positive Ricci curvature. These examples give negative answers to two open questions. One is about the properness of Busemann functions at some point, and the other one regards the singular set of Ricci limit sets. This is joint work with Guofang Wei.
Upper bound on the revised first Betti number and torus stability for RCD spaces
Abstract: It was shown by Gromov and Gallot that for a fixed dimension 𝑛 there exists a positive number 𝜀(𝑛) so that any 𝑛-dimensional closed Riemannian manifold (𝑀,𝑔) satisfying Riccidiam2≥−𝜀(𝑛) must have first Betti number smaller than or equal to 𝑛. Later on, Cheeger and Colding showed that if the first Betti number of 𝑀 equals 𝑛 then (𝑀,𝑔) has to be bi-Hölder homeomorphic to a flat torus. In this talk we will generalize the previous results to the case of RCD(𝐾,𝑁) spaces, which is the synthetic notion of Riemannian manifolds satisfying Ricci≥𝐾 and dim≤𝑁. This class of spaces include Ricci limit spaces and Alexandrov spaces. Joint work with I. Mondello and A. Mondino.
An Introduction to Group Boundaries
Abstract: In this talk, I will give an introduction to the notion of group boundaries via definitions, examples, properties, and some major theorems about these interesting spaces. We will begin with Gromov hyperbolic groups and their boundaries and move to the setting of CAT(0) groups and boundaries. This is meant to be background for my second talk as well as Craig Guilbault’s talk.
Local connectivity, Path Connectivity, and beyond…
Abstract: In this talk I will focus on the connectedness properties of visual boundaries arising from hyperbolic and CAT(0) groups. It is well-known that such a group is one-ended if and only if it’s visual boundarie(s) are connected. It is a major theorem in the theory of hyperbolic groups that one-ended hyperbolic groups have unique locally connected (and thus path-connected) visual boundary. This is the culmination of the work of many. This is not true for CAT(0) groups. In fact, one can have one-ended CAT(0) groups with boundary that is connected, path-connected, but not locally connected and also those that are connected but not path-connected. We will explore these examples and discuss what an open conjecture about what should be true for all of these (connected) boundaries.
Positive Curvature and Discrete Abelian Symmetries
Abstract: Positively curved manifolds are notoriously hard to classify. In the early 90’s, based on the observation that the few known examples are all highly symmetric, Karsten Grove proposed his “Symmetry Program” which suggests trying to classify such manifolds with the additional hypothesis of symmetry. Much work has been focused on the case of continuous symmetries. In this talk, I’ll discuss the case when the isometry group is finite and present some recent results when it is an elementary abelian two-group. This is joint work with Lee Kennard and Elahe Khalili Samani.
First order calculus in non-smooth spaces using upper gradient
Abstract: The goal of this talk is to give an elementary introduction to the theory of Sobolev-type function spaces in the setting of metric measure spaces where the measure is locally finite Borel-regular. This theory is based on the notion of upper gradients, first proposed by Heinonen and Koskela in the late 1990s. We will discuss some of the motivations behind development of this theory as well.
Hyperbolic filling as a tool to turn non-local problems into local problems
Abstract: An aspect of probability, based on the notion of Dirichlet form, generally deals with two types of Dirichlet forms: strongly local, and jump forms. Strongly local Dirichlet forms are generalizations of the idea of derivatives, and are understood more easily by analytic techniques such as those found in classical calculus of variations. Strongly local forms are associated with Sobolev-type function-spaces. Jump Dirichlet forms are non-local in nature, and hence more difficult to study. They are associated with Besov-type function spaces. In this talk we will discuss a way of relating jump forms to strongly local forms.
Open Questions on Scalar Curvature and Convergence
Abstract: We will present a collection of conjectures formulated with Gromov and other members of our IAS Emerging Topics Working Group on the limits of sequences of Riemannian manifolds with uniform lower bounds on their scalar curvature. We will survey results in special cases and present key theorems concerning volume preserving intrinsic flat convergence that have been applied to prove these special cases. For a complete list of papers about intrinsic flat convergence see: https://sites.google.com/site/intrinsicflatconvergence/
Classification of the Lozi maps
Abstract: In this talk, I will show how we classified the Lozi maps (up to conjugacy, for Misiurewicz’s set of parameters) by using two powerful tools, the symbolic dynamics, and the inverse limit spaces. This is joint work with Jan Boronski.
The Lorentz Gas and Quasicrystals
Abstract: The Lorentz gas model dates to the early 20th century, when it was proposed as a model for the free electron in a crystal. The study of the Lorentz gas has driven many of the major developments in ergodic theory, starting with the works of Sinai in the late 60’s and early 70’s, and by now the theory of Lorentz gases for solids with periodic structure is very sophisticated and advanced. In the early 80’s Shechtman discovered quasicrystals, which are solids whose atomic structure is not modeled by any periodic structure such as a lattice, the discovery of which led to a Nobel prize in 2011. Very little is known about the Lorentz gas for solids with aperiodic structure such as quasicrystals. In this talk I will talk about recent developments in the theory of Lorentz gases for solids with aperiodic structure, including joint work with A. Zelerowicz.
Universal Cover of Ricci Limit and RCD Spaces
Abstract: By Gromov’s precompactness theorem, any sequence of n-dim manifolds with uniform Ricci curvature lower bound has a convergent subsequence. The limit spaces are referred to as Ricci limit spaces. Cheeger-Colding-Naber developed great regularity and geometric properties for Ricci limit spaces. However, unlike Alexandrov spaces, these spaces could locally have infinite topological types. Joint with C. Sormani, we gave the first topological result by showing the universal cover of Ricci limit spaces exists. Here the universal cover is in the sense of a universal covering map (need not be simply connected). Joint with A. Mondino, we extended the result to RCD spaces. Recently in a series of work (J. Pan-G.Wei, J.Pan-J.Wang, J.Wang) it is shown that the Ricci limit spaces are semilocally simply connected, therefore the universal covers are simply connected.
Lecture One: Universal Covers of Ricci Limit and RCD Spaces Exist
Lecture Two: Universal Covers of Ricci Limit Spaces are Simply Connected
On collapse of tori under lower sectional curvature bounds
Abstract: Let 𝑋𝑖 be a sequence of closed 𝑛-dimensional Riemannian manifolds of sectional curvature bounded below, and diameter bounded above. It is well known that one can always find a subsequence that converges in the Gromov–Hausdorff sense to a compact geodesic space 𝑋 of dimension ≤𝑛. When 𝑋 has dimension 𝑛, Perelman showed that it is homeomorphic to 𝑋𝑖 for 𝐼 large enough. In other words, the topology stabilizes. On the other hand, the problem of understanding the geometry and topology of 𝑋 when its dimension is strictly less than 𝑛 is hard to attack and not much is known when 𝑋 is highly singular. We will discuss the case when the spaces are homeomorphic to the 𝑛-dimensional torus, where we have shown that the fundamental group can be virtually recovered from the fibration part, generalizing a recent result by Mikhail Katz and showing that in this case, if 𝑋 is a 𝐶1-Riemannian manifold with boundary, then its first Betti is at least its dimension.
Whitney’s extension theorem for curves in the Heisenberg group
Abstract: Given a compact set 𝐾 in ℝ𝑛 and a continuous, real valued function 𝑓 on 𝐾, when is there a 𝐶𝑚,𝜔 function 𝐹 defined on ℝ𝑛 such that 𝐹|𝐾=𝑓? (𝐶𝑚,𝜔 is the space of 𝐶𝑚 functions whose 𝑚th order derivatives are uniformly continuous with modulus of continuity 𝜔.) Whitney famously answered this question in 1934 when extra data is provided on the derivatives of the extension on 𝐾, and less famously answered it in the same year for subsets of ℝ. The question was answered in full by Charles Fefferman in 2009. The Heisenberg group 𝐻 is ℝ3 with a sub-Riemannian and metric structure generated by a class of admissible curves. We will consider Whitney’s original question for curves in 𝐻: given a compact set 𝐾 in ℝ and a continuous, map 𝑓:𝐾→𝐻, when is there a 𝐶𝑚,𝜔 admissible curve 𝐹 such that 𝐹|𝐾=𝑓? I will present a project with Andrea Pinamonti and Gareth Speight in which we answer this question in full.