Applied/ACMS/absS11
Cynthia Vinzant, UC Berkeley
| The central curve in linear programming |
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The central curve of a linear program is an algebraic curve specified by the associated hyperplane arrangement and cost vector. This curve is the union of the various central paths for minimizing or maximizing the cost function over any region in this hyperplane arrangement. Here we will discuss the algebraic properties of this curve and its beautiful global geometry. In the process, we'll need to study the corresponding matroid of the hyperplane arrangement. This will let us give a refined bound on the total curvature of the central curve, a quantity relevant for interior point methods. This is joint work with Jesus De Loera and Bernd Sturmfels appearing in arXiv:1012.3978. |
József Farkas, University of Stirling, Scotland
| Analysis of a size-structured cannibalism model with infinite dimensional environmental feedback
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First I will give a brief introduction to structured population dynamics. Then I will consider a size-structured cannibalism model with the model ingredients depending on size (ranging over an infinite domain) and on a general function of the standing population (environmental feedback). Our focus is on the asymptotic behavior of the system. We show how the point spectrum of the linearised semigroup generator can be characterized in the special case of a separable attack rate and establish a general instability result. Further spectral analysis allows us to give conditions for asynchronous exponential growth of the linear semigroup. |
Tatiana Márquez-Lago, ETH-Zurich
| Stochastic models in systems and synthetic biology |
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Cells prevail as efficient decision makers, despite the intrinsic uncertainty in the occurrence of chemical events, and being embedded within fluctuating environments. The underlying mechanisms of this ability remain widely unknown, but they are critical for the correct understanding of biological systems output and predictability. Some advances have been achieved by considering biological processes as modular units, but the conclusions in many studies vary alongside experimental conditions, or easily break down once the system is no longer isolated. Moreover, sets of seemingly simple biochemical reactions can generate a wide range of highly non-linear complex behaviours, even in the absence of crosstalk.
To illustrate some of these challenges, encountered in everyday biological/pharmaceutical research, I will present three short stories showing how iterations between mathematicians, computer scientists and biologists can generate successful ideas, testable in the laboratory.
The first story revolves around a tunable synthetic mammalian oscillator, from the individual cell perspective and population behavior. The long term importance of this work lies in discerning whether it is possible to influence the underlying genetic clockwork to tune the expression of key genes. Answering this question may prove to be central in the design of future gene therapies, particularly those requiring a periodic input.
In the second story I will show how closures on master equations describing negative self-regulation may yield diametrically opposed noise effects to those expected by exact solutions, discovering how any noise profile (and correlations between mRNA transcription and protein synthesis) can be created by the consideration of specific kinetic rates and network topologies.
Lastly, I will illustrate in a third story how chemical adaptation can many times be considered a purely emergent property of a collective system (even in simple linear settings), how a simple linear adaptation scheme displays fold-change detection properties, and how rupture of biological ergodicity prevails in scenarios where transitions between protein states are mediated by other molecular species in the system. |
Ari Stern, UC San Diego
| Geometric variational crimes: Hilbert complexes, finite element exterior calculus, and problems on hypersurfaces |
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In recent years, the success of "mixed" finite element methods has been shown to have surprising connections with differential geometry and algebraic topology---particularly with the calculus of exterior differential forms, de Rham cohomology, and Hodge theory. In this talk, I will discuss how the notion of "Hilbert complex," rather than "Hilbert space," provides the appropriate functional-analytic setting for the numerical analysis of these methods. Furthermore, I will present some recent results that analyze "variational crimes" (a la Strang) on Hilbert complexes, allowing the numerical analysis to be extended from polyhedral regions in Euclidean space to problems on arbitrary Riemannian manifolds. As a direct consequence, our analysis also generalizes several key results on "surface finite element methods" for the approximation of elliptic PDEs on hypersurfaces (e.g., membranes or level sets undergoing geometric evolution). |
Jian-Guo Liu, Duke University
| Dynamics of orientational alignment and phase transition |
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Phase transition of directional field appears in some physical and biological systems such as ferromagnetism near Currie temperature, flocking dynamics near critical mass of self propelled particles. Dynamics of orientational alignment associated with the phase transition can be effectively described by a mean field kinetic equation. The natural free energy of the kinetic equation is non-convex with a minimum level set consisting of a sphere at super-critical case, a typic spontaneous symmetry breaking behavior in physics. In this talk, I will present some analytical results on this dynamics equation of orientational alignment and exponential convergence rate to the equilibria for both supper and sub critical cases, as well at algebraic convergence rate at the critical case. A new entropy and spontaneous symmetry breaking analysis played an important role in our analysis. |
Tim Reluga, Penn State University
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Alex Kiselev, UW-Madison (Mathematics)
| Biomixing by chemotaxis and enhancement of biological reactions |
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Many processes in biology involve both reactions and chemotaxis. However, to the best of our knowledge, the question of interaction between chemotaxis and reactions has not yet been addressed either analytically or numerically. We consider a model with a single density function involving diffusion, advection, chemotaxis, and absorbing reaction (fertilization). The model is motivated, in particular, by studies of coral broadcast spawning, where experimental observations of the efficiency of fertilization rates significantly exceed the data obtained from numerical models that do not take chemotaxis (attraction of sperm gametes by a chemical secreted by egg gametes) into account. We prove that in the framework of our model, chemotaxis plays a crucial role. There is a rigid limit to how much the fertilization efficiency can be enhanced if there is no chemotaxis but only advection and diffusion. On the other hand, when chemotaxis is present, the fertilization rate can be arbitrarily close to being complete provided that the chemotactic attraction is sufficiently strong. Moreover, an interesting feature of the estimates in chemotactic case is that rates and timescales of the reaction (fertilization) process do not depend on the reaction amplitude coefficient. |
Anne Gelb, Arizona State University
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Vageli Coutsias, University of New Mexico
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Michael Holst, UC San Diego
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Smadar Karni, University of Michigan
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