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Talk 01 — Tomoharu Suda (Tokyo University of Science, Japan) — [computing…]Local time: click here to view in your time zoneTitle: Dynamical systems as path spaces
Abstract
While dynamical systems are usually interpreted as maps or flows on a state space, this framework is insufficient to consider systems lacking uniqueness of the time evolution, which often arise in practical applications. A useful approach here is to treat a dynamical system as the collection of all its possible time evolution, namely, a path space. In this talk, I will motivate the study of dynamical systems without uniqueness using a few examples and introduce the Yorke formulation of the axiomatic theory of ordinary differential equations as a tool for studying such systems. We will review the fundamental results of this theory, in particular, the asymptotic properties of path-based systems. If time permits, I will also discuss the categorical aspects of the theory that enable generalizations.
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Talk 02 — Shubhajit Goswami (Tata Institute of Fundamental Research, India) — [computing…]Local time: click here to view in your time zoneTitle: Roughness of geodesics in Liouville quantum gravity
Abstract
Liouville quantum gravity (LQG) is a one-parameter family of random fractal surfaces which was first studied by physicists in the 1980's. The corresponding metric has been recently constructed for all parameter values as a culmination of several works. It was a well-known conjecture in the community that the associated geodesics should be "rough" in the sense that their Euclidean Hausdorff dimension should be larger than 1. In this talk, we discuss a proof of this conjecture. Based on a joint work with Zherui Fan.
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Talk 03 — Shilpak Banerjee (Indian Institute of Technology Tirupati. India) — [computing…]Local time: click here to view in your time zoneTitle: Weak mixing behaviour for the projectivized derivative extension
Abstract
We start with a discussion of some old results on IM diffeomorphisms by Gunesch, Katok and Kunde. We will next describe the approximation by conjugation method developed by Anosov and Katok in 1970 and mention how it has been useful in construction of such diffeomorphisms. Finally we describe some recent results showing the existence of weak mixing diffeomorphisms in both smooth analytic categories, whose projectivized derivative extension also exhibit certain weak mixing properties. (This is a joint work with D. Khurana and P. Kunde.)
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Talk 04 — Dillon Cislo (Rockefeller University, USA) — [computing…]Local time: click here to view in your time zoneTitle: Shaping Fate: Geometric Methods for Cell Fate Transitions and Tissue Patterning
Abstract
Biology currently possesses unprecedented capabilities to generate genome-wide readouts of cells as they develop in embryos. It remains a challenge to parse this data into comprehensible models that can reveal the underlying principles of development and guide methods to engineer target cell and tissue fates. It has recently been demonstrated that Waddington's metaphor of development as downhill flow in an "epigenetic landscape" can be made mathematically rigorous using techniques from differential geometry and dynamical systems. I will discuss applications of these methods to lineage bifurcations in the early mouse embryo and in vitro recapitulations of floorplate induced dorsoventral patterning of the neural tube. In both cases, geometric reasoning simplifies complicated dynamic processes to enable testable predictions in terms of small sets of effective variables. I will show results, from both single-cell assays and whole-tissue live imaging, illustrating how cell signaling coordinates with biochemical and mechanical regulation at the scale of the embryo/organoid to dynamically organize developing tissues.
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Talk 05 — Mahesh Nerurkar (Rutgers University – Camden, USA) — [computing…]Local time: click here to view in your time zoneTitle: Forced oscillators, classical and quantum
Abstract
Using methods of ergodic theory and tools from geometric control theory we establish that dynamical/spectral behavior of a typical classical/quantum forced oscillator is unstable/chaotic.
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Talk 06 — Christian Wolf (Mississippi State University, USA) — [computing…]Local time: click here to view in your time zoneTitle: Computability of Measures of Maximal Entropy on Coded Shift Spaces
Abstract
In this talk, we present recent results about the computability of measures of maximal entropy (MMEs) on coded shift spaces. Coded shifts are natural generalizations of sofic shifts and are defined as the closure of all bi-infinite concatenations of words (generators) from a countable generating set G. Our first main result is that if an MME puts full measure on the set of concatenations, then the measure is unique, G-Bernoulli and computable as long as the Vere-Jones parameter ofG is computable. This establishes the computability of the MME for several well-known classes of shift spaces including S-gap shifts, multiple gap shifts, and beta-shifts. Our result even implies the computability of two ergodic MMEs of the Dyck shift. We also consider the case when an MME puts zero measure on the concatenation set, in which case it is known that the MME is, generally, not unique. We demonstrate that, even if the MME is unique and the Vere-Jones parameter is not computable, the MME is, in general, not computable. The results presented in this talk are joint work with Tamara Kucherenko and Marco Lopez.