Event Archive
Seminar given by Hector Sussman
Thursday, March 24, 2016 at 12:00pm - 03:00pm
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MATHEMATICAL PHYSICS SEMINAR
RUTGERS UNIVERSITY
HILL 705
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Hector Sussmann - Rutgers University
Date/Time/Location
Thursday, March 24th, 12:00pm; Hill 705
Title
"Analyticity properties of subriemannian geodesics"
MATHEMATICAL PHYSICS SEMINAR
RUTGERS UNIVERSITY
HILL 705
__________________________________________
Hector Sussmann - Rutgers University
Date/Time/Location
Thursday, March 24th, 12:00pm; Hill 705
Title
"Analyticity properties of subriemannian geodesics"
Abstract
Sub-Riemannian geodesics (i.e., locally length-minimizing paths for s Carnot-Caratheodory metric) are similar to Riemannian geodesics. except that, instead of satisfying a differential equation that implies that they are smooth, they satisfy a differential inclusion that implies smoothness for "most" geodesics, but not for all of them. And for that reason the question whether all sub-Riemannian geodesics are smooth (and analytic for real analytic metrics) has been open for several decades. We prove that that for real analytic sub-Riemannian metrics every geodesic parametrized by arc-length is real analytic on an open dense subset of its interval of definition. We discuss a research program that might lead to stronger regularity results.
Sub-Riemannian geodesics (i.e., locally length-minimizing paths for s Carnot-Caratheodory metric) are similar to Riemannian geodesics. except that, instead of satisfying a differential equation that implies that they are smooth, they satisfy a differential inclusion that implies smoothness for "most" geodesics, but not for all of them. And for that reason the question whether all sub-Riemannian geodesics are smooth (and analytic for real analytic metrics) has been open for several decades. We prove that that for real analytic sub-Riemannian metrics every geodesic parametrized by arc-length is real analytic on an open dense subset of its interval of definition. We discuss a research program that might lead to stronger regularity results.
THERE WILL BE A BROWN BAG LUNCH FROM 1-2PM. PLEASE JOIN US
Eduardo Sontag - Rutgers University
Date/Time/Location
Thursday, March 24th, 2:00pm; Hill 705
Title
"Qualitative features of transient responses A case study: scale-invariance"
Eduardo Sontag - Rutgers University
Date/Time/Location
Thursday, March 24th, 2:00pm; Hill 705
Title
"Qualitative features of transient responses A case study: scale-invariance"
Abstract
An ubiquitous property of sensory systems is "adaptation": a step increase in stimulus triggers an initial change in a biochemical or physiological response, followed by a more gradual relaxation toward a basal, pre-stimulus level. Adaptation helps maintain essential variables within acceptable bounds and allows organisms to readjust themselves to an optimum and non-saturating sensitivity range when faced with a prolonged change in their environment. It has been recently observed that some adapting systems, ranging from bacterial chemotaxis pathways to signal transduction mechanisms in eukaryotes, enjoy a remarkable additional feature: scale invariance or "fold change detection" meaning that the initial, transient behavior remains approximately the same even when the background signal level is scaled.
I will review the biological phenomenon, and formulate a theoretical framework leading to a general theorem characterizing scale invariant behavior by equivariant actions on sets of vector fields that satisfy appropriate Lie-algebraic nondegeneracy conditions. The theorem allows one to make experimentally testable predictions, and I will discuss the validation of these predictions using genetically engineered bacteria and microfluidic devices, as well their use as a "dynamical phenotype" for model invalidation. I will conclude by briefly engaging in some wild and irresponsible speculation about the role of the shape of transient responses in immune system self/other recognition and in evaluating the initial effects of cancer immunotherapy.
An ubiquitous property of sensory systems is "adaptation": a step increase in stimulus triggers an initial change in a biochemical or physiological response, followed by a more gradual relaxation toward a basal, pre-stimulus level. Adaptation helps maintain essential variables within acceptable bounds and allows organisms to readjust themselves to an optimum and non-saturating sensitivity range when faced with a prolonged change in their environment. It has been recently observed that some adapting systems, ranging from bacterial chemotaxis pathways to signal transduction mechanisms in eukaryotes, enjoy a remarkable additional feature: scale invariance or "fold change detection" meaning that the initial, transient behavior remains approximately the same even when the background signal level is scaled.
I will review the biological phenomenon, and formulate a theoretical framework leading to a general theorem characterizing scale invariant behavior by equivariant actions on sets of vector fields that satisfy appropriate Lie-algebraic nondegeneracy conditions. The theorem allows one to make experimentally testable predictions, and I will discuss the validation of these predictions using genetically engineered bacteria and microfluidic devices, as well their use as a "dynamical phenotype" for model invalidation. I will conclude by briefly engaging in some wild and irresponsible speculation about the role of the shape of transient responses in immune system self/other recognition and in evaluating the initial effects of cancer immunotherapy.
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