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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. | ||
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===Fri, Feb 3: Akos Magyar (UBC)=== | ===Fri, Feb 3: Akos Magyar (UBC)=== | ||
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will also discuss some new results on prime solutions to non-diagonal quadratic | 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. | equations of suﬃciently large rank. Most of this is joint work with B. Cook. | ||
+ | |||
+ | ===Fri, Feb 8: Lan-Hsuan Huang (Columbia U)=== | ||
+ | ''Positive mass theorems and scalar curvature problems'' | ||
+ | |||
+ | More than 30 years ago, Schoen-Yau and later Witten made major breakthroughs in | ||
+ | proving the positive mass theorem. It has become one of the most important theorems in | ||
+ | general relativity and differential geometry. In the first part of the talk, I will | ||
+ | introduce the positive mass theorem and present our recent work that extends the | ||
+ | classical three-dimensional results to higher dimensions. In the second part, I will | ||
+ | discuss how the observation from general relativity enables us to solve classical | ||
+ | geometric problems related to the scalar curvature. | ||
===Fri, Feb 10: Melanie Wood (local)=== | ===Fri, Feb 10: Melanie Wood (local)=== |
Revision as of 12:06, 2 February 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 | Akos Magyar (UBC) | On prime solutions to linear and quadratic equations | Street |
Feb 8 | Lan-Hsuan Huang (Columbia U) | Positive mass theorems and scalar curvature problems | Sean |
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 Southern 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: 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 8: Lan-Hsuan Huang (Columbia U)
Positive mass theorems and scalar curvature problems
More than 30 years ago, Schoen-Yau and later Witten made major breakthroughs in proving the positive mass theorem. It has become one of the most important theorems in general relativity and differential geometry. In the first part of the talk, I will introduce the positive mass theorem and present our recent work that extends the classical three-dimensional results to higher dimensions. In the second part, I will discuss how the observation from general relativity enables us to solve classical geometric problems related to the scalar curvature.
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.