Departmental Colloquium
Spring 2000
Regular Day: Thursday
Regular Time: 4:00PM - 5:00PM
Regular Location: JWB 335
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February 10 |
Duane Nykamp
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The neural networks of even small functional units in the brain are enormously complex. Conventional simulation methods, where one models thousands of individual neurons, can take large amounts of computer time even for models of small cortical areas. The population density approach can be used to speed up large-scale neural network simulations. In this method, one groups neurons into large populations of similar neurons. By calculating the evolution of a probability density function for each population, one obtains population firing rates and the distribution of neurons over state space. I demonstrate a population density method for simulating networks of integrate-and-fire neurons with instantaneous synapses or with slow inhibitory synapses. Through comparisons with conventional Monte-Carlo simulations for a model of a hypercolumn in cat visual cortex, I demonstrate the speed and accuracy of the population density method.
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February 11 |
Brendan Hassett
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Let X be a smooth cubic hypersurface of dimension d in complex projective space. By definition, X is rational if its field of algebraic functions is purely transcendental over the complex numbers. Classically, it was known that X is irrational when d=1 and rational when d=2. Clemens and Griffiths proved that X is irrational when d=3, but no cubic of dimension d>3 is known to be irrational. My talk will focus on the case of cubic fourfolds. First, I will review the known examples of rational cubic fourfolds, which form a countably infinite union of subvarieties in the moduli space of cubic fourfolds. Then I will discuss how these subvarieties are `special' in the moduli space. For instance, the known examples of rational cubic fourfolds possess `associated K3 surfaces,' isomorphic to surfaces blown up in a birational map P^4 ---> X. The definition of an associated K3 surface is intrinsic to (the Hodge structure of) the cubic fourfold. Unfortunately, experimental evidence suggests the presence of such a surface does not guarantee rationality. This led to recent joint work with Tschinkel. If X contains an algebraic surface with certain invariants then X is necessarily rational. We study the problem of representing homology classes on X by such algebraic surfaces. This leads to general conjectures describing the effective 1-cycles on symplectic varieties naturally arising from X.
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February 11 |
Igor Pak
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I will give a somewhat biased review of recent results on theoretical and practical methods for generating random group elements. Basically we will start by introducing the algorithms and discuss problems from various fields as they arise. No previous knowledge of the subject is assumed.
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February 16 |
Angela Stevens
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Selforganization in microbiology is not only interesting in itself but also serves as a model problem for questions in morphogenesis, e.g."how do cells manage to cooperate and build higher organized structures." Aggregation or selforganization of a species frequently involves movement towards or away from an external stimulus which sometimes is also modified. This behavior will be modeled by a self-attracting reinforced random walk. The random walk for a single particle can be formally approximated by a so-called chemotaxis equation which describes the dynamics of the particles probability distribution. To approximate the dynamics of many particles, their interaction has to be taken into account. Starting from a model where the dynamics of each particle is described by a stochastic differential equation which includes moderate interaction with the other particles, the chemotaxis equation can be rigorously derived as a model for the dynamics of the total population. For suitable parameters the solutions of the chemotaxis equation blowup in finite time which here reflects possible selforganization.
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February 17 |
Stephan Luckhaus
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In continuum physics phase transitions are modelled by a change in the constitutive laws coupling the thermodynamic quantities. In systems with heat and mass diffusion this law may be taken to be the entropy as a function of energy and mass density (for isothermic systems entropy is replaced by free energy). The overall entropy is then the convex hull of the entropies of all the phases. The equations of evolution are degenerate parabolic systems of Stefan type and exhibit regions of phase mixtures, so called mushy regions. Following Gibbs, mesoscopic models include terms proportional to the area of the phase interface in the entropy, thus avoiding mushy regions. The evolution equations for these models are parabolic in the bulk coupled to either mean curvature or mean curvature flow equations on the phase interface. A notion of solution is discussed which is closed under weak convergence. This is a slight variation of the one introduced by Brakke for mcf and extended by Soner to the Mullins Sekerka problem. In this context the phase interface becomes an integer varifold.
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February 18 |
Cristian D. Popescu
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In the 1970s and 1980s Stark developed a remarkable conjecture aimed at interpreting the first non-vanishing derivative of an Artin L -function $L_{K/k, S}(s,\chi)$ at s=0 in terms of the arithmetic properties of the Galois extension of global fields K/k. Work of Stark, Tate, and Chinburg has revealed far reaching applications of Stark's Conjecture to Hilbert's 12-th Problem and the theory of Galois module structure of groups of units and ideal-class groups. In his search for new examples of Euler systems, Rubin has formulated in 1994 a strong version ("over Z ", in Tate's terminology) of Stark's Conjecture for abelian L -functions of arbitrary order of vanishing at s=0}. Our recent study of the functorial base-change behavior of Rubin's Conjecture led us to formulating a seemingly more natural Stark-type conjecture "over Z ". We will discuss and provide evidence for this new statement, as well as briefly describe the main goals of the conjectural program initiated by Stark.
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February 22 |
Anette Hosoi
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We consider the flow in a thin film generated by partially submerging a rigid plate into a reservoir of ethanol- or methanol-water solutions and wetting its surface. Evaporation leads to concentration and surface tension gradients that drive flow up the plate. An experimental study indicates that the thin film is subject to two distinct instabilities. The first is a convective instability characterized by flattened (wide) rolls aligned in the direction of flow and accompanied by free surface deformations; in the meniscus region, this instability gives rise to pronounced ridge structures. The second instability was evident only when the plate was nearly vertical, and was characterized by transverse waves propagating up the plate. We demonstrate that the observed longitudinal rolls are driven by the combined influence of surface deformations and alcohol concentration gradients. Guided by the observation that the rolls are flat, we develop a quasi-two-dimensional theoretical model for the instability of the film based on lubrication theory which includes the effects of gravity, capillarity and Marangoni stresses at the surface. We develop stability criteria for the film which are in qualitative agreement with our experimental observations. Our analysis yields an equation for the shape of the interface which is solved numerically and reproduces the salient features of the experimental flows, including the slow lateral drift and merger of the ridges.
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February 24 |
David Vogan
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The simplest noncommutative compact Lie group is SU(2), the group of unit quaternions. If G* is any compact Lie group, a natural problem is to understand the set D(G*) of conjugacy classes of homomorphisms of SU(2) into G*. Dynkin showed in the 1950s that D(G*) is a finite set, and calculated it in all cases. Suppose now that k is a local field. The structure theory of reductive groups attaches to G* and k a split reductive group G. The compact group G* is a "Langlands dual" of G. A fundamental unsolved problem in abstract harmonic analysis is to parametrize the "purely real" unramified unitary representations G. It's known that such representations are parametrized by a compact polytope P(G,k). It turns out that the polytope depends very little on the field k ; for example, it's the same for all p -adic fields k (for a fixed G* ). A conjecture of Arthur realizes D(G*) as a subset of P(G,k) ; these finitely many special points seem to control most of the geometry of the polytope P(G,k). I'll discuss how the two problems illuminate each other.
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February 29 |
Clifford J. Nolan
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Intensive research is currently taking place in the geophysics community to model wave propagation in anisotropic models of the earth's subsurface. We investigate the associated inverse problem of estimating elastic and electrical properties of the earth from measurements of scattered elastic and electro-magnetic waves. We adopt the high-frequency linearized inversion approach to see what model parameters of the earth can be recovered and how to recover them. The results of this research are useful in determining locations of oil and mineral deposits. Other applications include ultra-sound (medical) imaging, mine detection, location of cracks in materials, etc.
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March 2 |
Leonid Ryzhik
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The enhancement effect of wind or fluid motion on the rate of chemical reactions, and front propagation in general, is a well known phenomenon, that occurs in many applications ranging from combustion to biology. We will review some previous mathematical results regarding front propagation in the presence of advection based on homogenization theory and existence of travelling waves for some classes of flows. However, these methods do not provide explicit bounds on the propagation speed in terms of the advecting flow, and are applicable for a limited range of flows. We will introduce an unambiguous way to measure the reaction rate in the situations when these approaches may not work. We will describe bounds on the reaction rate in terms of the magnitude, geometry and scale of oscillations of the advecting flow. In particular this will allow us to estimate the speed of the travelling fronts in shear and vortical flows.
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March 9 |
Alex Mogilner
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Motility of animal cells is fundamentally important and is the most striking process underlying the phenomena of wound healing, morphogenesis and cancerogenesis. Despite recent radical advances in cell biology and the biophysics of the motile cell, we still do not have a complete picture of how animal cells move across surfaces. One reason for this is that a huge variety of molecular mechanisms are involved in locomotion, which leads to a multiplicity and redundancy in force generation machineries and regulatory pathways. Theoretical modeling helps to search for truth in this situation. Amoeboid motility, in all its forms save one, is associated with the actin cytoskeleton. The crawling sperm of nematodes are the exception. An intriguing aspect of nematode sperm motility is that these cells discard their actin-based cytoskeleton and deploy an entirely new motility machinery based on a major sperm protein. Nematode sperm offer at least one advantage for investigating principles of cell crawling: these cells are remarkably simple and dedicated entirely to locomotion, yet their migrating behavior is essentially indistinguishable from that of actin-based cells. I will present some preliminary results of quantitative modeling of the sperm cells. Two approaches will be discussed: one is based on Monte Carlo simulations of ensembles of cytoskeletal polymers. Another one incorporates what is known about relevant molecular mechanisms into partial integro-differential equations for the essential spatio-angular cytoskeletal densities. These equations are solved on a domain with a moving boundary of the cell. I will demonstrate how these models produce sperm-like shapes, movement and forces and advance our understanding of the dynamic principles of cell locomotion.
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March 30 |
Peter E. Trapa
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Suppose G is a real Lie group. A classical and still open problem is to describe all continuous, irreducible, length-preserving actions of G on a Hilbert space. An impressive array of ideas has been applied to this problem with varying degrees of partial success. In many cases, these ideas have taken on a life of their own and have revealed deep and interesting insights into nonunitary representations. The purpose of this talk is to survey a handful of those ideas --- some algebraic, some geometric, and some arithmetic --- with a view toward optimistic applications of them.
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April 10 |
Angelo Vistoli
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Vector bundles are among the basic objects of study in all of geometry. Out of vector bundles on an algebraic variety or a topological space X one forms a ring K_0(X), the K -theory ring of X, introduced by Grothendieck. If furthermore there is a group G operating on X one forms the equivariant K -theory ring K_0(X,G), using equivariant vector bundles on X. For example if X is a point then K_0(X,G) is the representation ring of G, already a very interesting object. The ring K_0(X,G) has some of the same formal properties of the nonequivariant ring K_0(X), but its theory is richer, because of the interaction of the algebra coming from G with the geometry coming from X. In particular K_0(X,G) tends to decompose in interesting ways, reflecting the properties of the fixed point subsets. I will start by motivating the introduction of the K -theory rings, and end by illustrating some decomposition results in their simplest context.
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April 18 |
Paul Bresloff
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We show that the circuitry of the primary visual cortex has approximate Euclidean symmetry with a novel group action. This action gives rise to the spontaneous formation of cortical patterns that generate images in the visual field (via a conformal mapping) consistent with some of the common forms of geometric visual hallucinations. Various techniques from group theory and nonlinear analysis will be applied to the problem of computing the types of patterns and their stability.
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May 4 |
John Warnock
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Archival event entry migrated from the legacy colloquium website.
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