Difference between revisions of "Colloquia 2012-2013"
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The swimming kinematics and trajectories of many microorganisms are altered by the presence of nearby boundaries, be they solid or deformable, and often in perplexing fashion. When an organism's swimming dynamics vary near such boundaries a question arises naturally: is the change in behavior fluid mechanical, biological, or perhaps due to other physical laws? We isolate the first possibility by exploring a far-field description of swimming organisms, providing a general framework for studying the fluid-mediated modifications to swimming trajectories. Using the simplified model we consider trapped/escape trajectories and equilibria for model organisms of varying shape and propulsive activity. This framework may help to explain surprising behaviors observed in the swimming of many microorganisms and synthetic micro-swimmers. Along the way, we will discuss the numerical tools constructed to analyze the problem of current interest, but which have considerable potential for more general applicability. | The swimming kinematics and trajectories of many microorganisms are altered by the presence of nearby boundaries, be they solid or deformable, and often in perplexing fashion. When an organism's swimming dynamics vary near such boundaries a question arises naturally: is the change in behavior fluid mechanical, biological, or perhaps due to other physical laws? We isolate the first possibility by exploring a far-field description of swimming organisms, providing a general framework for studying the fluid-mediated modifications to swimming trajectories. Using the simplified model we consider trapped/escape trajectories and equilibria for model organisms of varying shape and propulsive activity. This framework may help to explain surprising behaviors observed in the swimming of many microorganisms and synthetic micro-swimmers. Along the way, we will discuss the numerical tools constructed to analyze the problem of current interest, but which have considerable potential for more general applicability. | ||
+ | |||
+ | ===Fri, Feb 3: Travis Schedler (MIT)=== | ||
+ | ''Symplectic resolutions and Poisson-de Rham homology'' | ||
+ | |||
+ | A symplectic resolution is a resolution of singularities of | ||
+ | a singular variety by a symplectic algebraic variety. Examples | ||
+ | include symmetric powers of Kleinian (or du Val) singularities, | ||
+ | resolved by Hilbert schemes of the minimal resolutions of Kleinian | ||
+ | singularities, and the Springer resolution of the nilpotent cone of | ||
+ | semisimple Lie algebras. Based on joint work with P. Etingof, I define | ||
+ | a new homology theory on the singular variety, called Poisson-de Rham | ||
+ | homology, which conjecturally coincides with the de Rham cohomology of | ||
+ | the symplectic resolution. Its definition is based on "derived | ||
+ | solutions" of Hamiltonian flow, using the algebraic theory of | ||
+ | D-modules. I will give applications to the representation theory of | ||
+ | noncommutative deformations of the algebra of functions of the | ||
+ | singular variety. In the examples above, these are the spherical | ||
+ | symplectic reflection algebras and finite W-algebras (modulo their | ||
+ | center). | ||
===Fri, Feb 3: Akos Magyar (UBC)=== | ===Fri, Feb 3: Akos Magyar (UBC)=== |
Revision as of 10:02, 23 January 2012
Mathematics Colloquium
All colloquia are on Fridays at 4:00 pm in Van Vleck B239, unless otherwise indicated.
Spring 2012
date | speaker | title | host(s) |
---|---|---|---|
Jan 23, 4pm | Saverio Spagnolie (Brown) | Hydrodynamics of Self-Propulsion Near a Boundary: Construction of a Numerical and Asymptotic Toolbox | Jean-Luc |
Jan 27 | Ari Stern (UCSD) | Numerical analysis beyond Flatland: semilinear PDEs and problems on manifolds | Jean-Luc / Julie |
Feb 3, 2:25pm, B215 | Travis Schedler (MIT) | Symplectic resolutions and Poisson-de Rham homology | Andrei |
Feb 3 | Akos Magyar (UBC) | On prime solutions to linear and quadratic equations | Street |
Feb 10 | Melanie Wood (UW Madison) | Counting polynomials and motivic stabilization | local |
Feb 17 | Milena Hering (University of Connecticut) | TBA | Andrei |
Feb 24 | Malabika Pramanik (University of British Columbia) | TBA | Benguria |
March 2 | Guang Gong (University of Waterloo) | TBA | Shamgar |
March 16 | Charles Doran (University of Alberta) | TBA | Matt Ballard |
March 19 | Colin Adams and Thomas Garrity (Williams College) | Which is better, the derivative or the integral? | Maxim |
March 23 | Martin Lorenz (Temple University) | TBA | Don Passman |
March 30 | Wilhelm Schlag (University of Chicago) | TBA | Street |
April 6 | Spring recess | ||
April 13 | Ricardo Cortez (Tulane) | TBA | Mitchell |
April 18 | Benedict H. Gross (Harvard) | TBA | distinguished lecturer |
April 19 | Benedict H. Gross (Harvard) | TBA | distinguished lecturer |
April 20 | Robert Guralnick (University of South California) | TBA | Shamgar |
May 4 | Mark Andrea de Cataldo (Stony Brook) | TBA | Maxim |
May 11 | Tentatively Scheduled | Shamgar |
Abstracts
Mon, Jan 23: Saverio Spagnolie (Brown)
"Hydrodynamics of Self-Propulsion Near a Boundary: Construction of a Numerical and Asymptotic Toolbox"
The swimming kinematics and trajectories of many microorganisms are altered by the presence of nearby boundaries, be they solid or deformable, and often in perplexing fashion. When an organism's swimming dynamics vary near such boundaries a question arises naturally: is the change in behavior fluid mechanical, biological, or perhaps due to other physical laws? We isolate the first possibility by exploring a far-field description of swimming organisms, providing a general framework for studying the fluid-mediated modifications to swimming trajectories. Using the simplified model we consider trapped/escape trajectories and equilibria for model organisms of varying shape and propulsive activity. This framework may help to explain surprising behaviors observed in the swimming of many microorganisms and synthetic micro-swimmers. Along the way, we will discuss the numerical tools constructed to analyze the problem of current interest, but which have considerable potential for more general applicability.
Fri, Feb 3: Travis Schedler (MIT)
Symplectic resolutions and Poisson-de Rham homology
A symplectic resolution is a resolution of singularities of a singular variety by a symplectic algebraic variety. Examples include symmetric powers of Kleinian (or du Val) singularities, resolved by Hilbert schemes of the minimal resolutions of Kleinian singularities, and the Springer resolution of the nilpotent cone of semisimple Lie algebras. Based on joint work with P. Etingof, I define a new homology theory on the singular variety, called Poisson-de Rham homology, which conjecturally coincides with the de Rham cohomology of the symplectic resolution. Its definition is based on "derived solutions" of Hamiltonian flow, using the algebraic theory of D-modules. I will give applications to the representation theory of noncommutative deformations of the algebra of functions of the singular variety. In the examples above, these are the spherical symplectic reflection algebras and finite W-algebras (modulo their center).
Fri, Feb 3: Akos Magyar (UBC)
On prime solutions to linear and quadratic equations
The classical results of Vinogradov and Hua establishes prime solutions of linear and diagonal quadratic equations in suﬃciently many variables. In the linear case there has been a remarkable progress over the past few years by introducing ideas from additive combinatorics. We will discuss some of the key ideas, as well as their use to obtain multidimensional extensions of the theorem of Green and Tao on arithmetic progressions in the primes. We will also discuss some new results on prime solutions to non-diagonal quadratic equations of suﬃciently large rank. Most of this is joint work with B. Cook.
Fri, Feb 10: Melanie Wood (local)
Counting polynomials and motivic stabilization
We will begin with the problem of counting polynomials modulo a prime p with a given pattern of root multiplicity. Here we will discover phenomena that point to vastly more general patterns in configuration spaces of points. To see these patterns, one has to work in the ring of motives--so we will describe this place where a space is equivalent to the sum of its pieces. We will then be able to describe how these patterns in the ring of motives are related to theorems in topology on the homological stability of configuration spaces. This talk is based on joint work with Ravi Vakil.