Departmental Colloquium
Spring 2005
Regular Day: Thursday
Regular Time: 4:00PM - 5:00PM
Regular Location: JWB 335
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January 11 Special Colloquium Tuesday |
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The well known transport problem introduced by Monge in 1781 has been studied in many works recently. In these works, the cost of a transport mapping or a transport plan is usually an integral of some function of the distance. However, in many real applications, the actual cost of the transport procedures is not necessarily determined by just knowing some optimal mapping from the starting position to the target position. For example in shipping two items from nearby cities to the same far away city, it may be less expensive to first bring them to a common location and put them on a single truck for most of the transport. In this case, a " Y shaped" path is preferable to a " V shaped" path. In both cases, the transport mapping is trivially the same, but the actual transport path naturally gives the total cost. In general, a ramifying structure is more cost efficient than a "linear" structure. This phenomenon of ramified transportation is very common in nature. Trees, railways, airlines, lightning, electric power supply, the circulatory system, the river channel networks, and cardiovascular systems are some common examples. This subject deserves a more general theoretical treatment and thus I built a model for it in a series of papers. In this talk, I am going to discuss my approach to this interesting problem. We will see how to set up the problem in terms of transport paths. Also, we will discuss the nice properties of optimal transport paths. If time permits, I will also provide the description of the dynamic formation of a tree leaf, as an application of this theory.
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January 12 Special Colloquium Wednesday |
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Biological clocks with a period near one day (circadian) are essential for the survival of most organisms. Circadian clock disorders in man can lead to poor productivity, jet lag, sleep disorders and have been linked to Alzheimer's disease and cancer. The circadian clock within a cell is comprised of a feedback network of genes and proteins. In man, a group of about 20,000 neurons in the brain (the suprachiasmatic nucleus, SCN), many of which have an internal circadian clock, form our central circadian pacemaker and regulate our sleep-wake patterns, core body temperature and the release of most hormones in the body. The first half of the talk will describe a detailed mathematical model of circadian clock in SCN neurons I have developed with Charles Peskin. I will then briefly outline how simulations of this model, and mathematical analysis can used to understand key questions in circadian biology including: 1) How intracellular clocks function accurately despite the inherent stochasticity of the molecular interactions of which they are comprised and 2) How intracellular clocks keep an approximately 24-hour period over a wide range of temperatures. If time permits, I will also discuss: 1) Mathematically predicted biological mechanisms which cause oscillations in genetic feedback loops, and 2) How mathematical models of circadian clocks can help you work productively and avoid jet lag.
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January 13 Special Colloquium Thursday |
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The resonance phenomena and stability of a periodically forced, linear oscillator is well understood. But the problem becomes quite difficult when the mechanical system has more than one degree of freedom and the forcing depends on the state of the system. Multiple scale analysis, Poincare continuation and KAM theory give only partial answers. My talk will focus on recent, rigorous results concerning systems with infinitely many degrees of freedom. I will briefly describe why such systems are ubiquitous in Quantum Mechanics, Statistical Physics and Optics where they are modeled by dispersive partial differential equations. A simplified mechanical example would be a mass-spring system attached to an infinitely long, tense string. The oscillations of the spring excite (resonantly) the string which carries the energy of the excitations to infinity. As a result one sees a decay of the amplitude with which the mass-spring system oscillates. I will present in some detail the mathematical techniques involved in proving that the same phenomenon occurs for the ground state of the cubic nonlinear Schroedinger equation subject to periodic in time perturbation, a result obtained in collaboration with S. Cuccagna and D. Pelinovsky. Then I will connect this result with the ones for random and almost periodic perturbations of linear Hamiltonian partial differential equations obtained in collaboration with M. Weinstein. At the end I will mention some related open problems and argue that the above results and the mathematical techniques developed constitute a solid basis for attacking them.
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January 17 Special Colloquium Tuesday |
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In nature there are millions of distinct networks of biochemical reactions that might present themselves for study at one time or another. Each reaction network gives rise to its own system of differential equations. These are usually high dimensional, nonlinear, and have many unknown parameters. Nevertheless, each reaction network induces its corresponding differential equations (up to parameter values) in a precise way. This raises the possibility that qualitative properties of the induced differential equations might be tied directly to reaction network structure. We will show that reaction diagrams, similar to those that biochemists usually draw, carry subtle information about a reaction network's capacity to exhibit multiple equilibria. Some of these results suggest interesting new problems in real algebraic geometry and graph theory. We will also discuss implications for the interpretation of experiments in cell biology.
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January 18 Special Colloquium Tuesday |
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In nature there are millions of distinct networks of biochemical reactions that might present themselves for study at one time or another. Each reaction network gives rise to its own system of differential equations. These are usually high dimensional, nonlinear, and have many unknown parameters. Nevertheless, each reaction network induces its corresponding differential equations (up to parameter values) in a precise way. This raises the possibility that qualitative properties of the induced differential equations might be tied directly to reaction network structure. We will show that reaction diagrams, similar to those that biochemists usually draw, carry subtle information about a reaction network's capacity to exhibit multiple equilibria. Some of these results suggest interesting new problems in real algebraic geometry and graph theory. We will also discuss implications for the interpretation of experiments in cell biology.
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January 20 Special Colloquium Thursday |
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Random walk in random environment (RWRE) is a basic model of the field of random media. It has been subject to extensive study in the past three decades. Many questions, although seemingly intuitive and simple, present great mathematical challenges. The point of view of the particle is a method that has proved very powerful in the treatment of many (reversible) particle systems. However, in the case of multi-dimensional (usually non-reversible) RWRE this method has been of very limited use until recently. We present new results showing that this method can be successfully extended to answer many important questions about RWRE, such as the law of large numbers, the central limit theorem, and large deviation principles.
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January 24 Special Colloquium Monday |
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I shall discuss two new asymptotic regimes for the nonlinear PDEs governing the tropical atmosphere. Using systematic multiscale asymptotics, we arrive at an asymptotic closure for the ideal fluid equations governing dynamics on large scales in the tropical atmosphere. By selecting a plausible analytic model for smaller scale flows in the tropics, we predict the large scale structure of the Madden-Julian oscillation; this is a planetary scale organization of winds, the understanding of which has been called "the holy grail" of tropical meteorology. In the second problem, we study the same equations, but over longer time and spatial scales. The resultant coupled nonlinear dispersive equations for the amplitudes of interacting wave packets are novel both from the perspective of the atmospheric sciences and from a more general mathematical setting. These equations describe the influence of large scale tropical waves on midlatitude waves and, in particular, are relevant for understanding the effect of the Madden-Julian oscillation on midlatitude weather. Furthermore, the amplitude equations have a Hamiltonian structure and admit analytic solitary wave solutions.
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January 27 Special Colloquium Thursday |
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The theory of local cohomology was developed by Grothendieck, who used it to prove Lefschetz-type theorems. The theory has applications to basic questions such as determining the minimal number of polynomial equations needed to define an algebraic set. Local cohomology modules are typically not finitely generated over the base ring, but still possess useful finiteness properties. We will discuss these finiteness properties, some recent work, and open questions.
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February 1 Special Colloquium Tuesday |
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The property of Rapid Decay is a property of convolution operators that captures certain aspects of the asymptotic geometry of a finitely generated group. In this talk I will give a basic explanation of that property, and describe a theorem joint with Kim Ruane that all lattices in a rank one Lie group have the Rapid Decay property.
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February 3 Special Colloquium Thursday |
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In this talk I will motivate the study of functionals of random trees that satisfy recurrence relations of a simple additive form and describe recent progress in this area. Many important functionals including the space requirement, internal path length, number of leaves, and the so-called "shape functional" fall under this framework. Such functionals also represent the cost of divide-and-conquer algorithms (including QuickSort and Union--Find), where the inherent recursive nature of the algorithms lends itself naturally to such a formulation. In particular, I will describe limit laws of additive functionals on (i) m -ary search tress (natural generalizations of binary search trees) under the random permutation model and the uniform model, (ii) simply generated trees or conditioned Galton--Watson trees (which include ordered trees, d -ary trees, and Cayley trees). Several interesting techniques are employed and extended in this work, including the elementary but powerful "contraction method" and singularity analysis, a complex--analytic technique that relates asymptotics of sequences to singularities of their generating functions.
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February 8 Special Colloquium Tuesday |
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Modular forms play an increasingly central role in modern number theory. One classical example is the famous j -invariant whose values at Heegner points, i.e., at quadratic irrationalities in the upper half plane, are known as singular moduli and are of particular arithmetic interest. Recently, Zagier realized the generating series of the traces of the singular moduli as a meromorphic modular form of weight 3/2. In this talk, we give an introduction to the subject, discussing this work and related results and provide a generalization to modular functions on Riemann surfaces of arbitrary genus. Furthermore, we realize a certain generating series of arithmetic intersection numbers and Faltings heights of Heegner points as the derivative of Zagier's Eisenstein series of weight 3/2.
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February 10 Special Colloquium Thursday |
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A mathematical model of a brain region called the Pre-Botzinger complex consists of a large system of coupled non-linear ODEs. I will present a mathematical analysis of this system that allows to elucidate the multi-dimensional bifurcation structure responsible for transitions between activity modes (quiescence, bursting and spiking). In particular, I will describe a non-standard fast-slow dissection approach, incorporating averaging in the slow subsystem. The results advance the current mathematical understanding of networks of bursting neurons, as well as answer a number of biologically-motivated questions. For example, this analysis clarifies the role of coupling in selecting burst frequency, as well as the mechanisms via which a network can burst synchronously for a much wider range of parameters than in the case of individual cells.
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February 11 Special Colloquium Friday |
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Quadratic differentials on a Riemann surface have played a fundamental role in classical Teichmueller theory. Such a differential determines a special type of Euclidean cone metric on the Riemann surface, a projective line of projective measured foliations, and an isometric embedding of the hyperbolic plane into the Teichmueller space (and more!). Understanding the relationships between these various structures can be incredibly useful. In this talk, I will discuss some of these relationships, how they can be used, and how they produce some very pretty connections with other areas of mathematics.
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February 14 Special Colloquium Monday |
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A quiver variety is a general type of degeneracy locus associated to a quiver of vector bundles and bundle maps over a variety. Examples include Schubert varieties in flag manifolds and determinantal varieties. I have proved a formula for the Grothendieck class of a quiver variety when the underlying quiver is equioriented of type A. This formula is stated in terms of integers called quiver coefficients, which are generalizations of Littlewood-Richardson coefficients. Knutson, Miller, and Shimozono have shown that the lowest degree coefficients, which describe the cohomology class of the quiver variety, are non-negative. I will speak about a proof that general quiver coefficients have signs that alternate with codimension. I will also explain a cohomology formula for non-equioriented quiver varieties, which I have recently proved with R. Rimanyi.
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February 15 Special Colloquium Tuesday |
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Chern classes play a fundamental role in the study of manifolds. In this talk I will address the general problem of finding good notions of Chern classes for singular varieties. I will start by discussing the generalizations proposed by Mather and MacPherson. Then, motivated by the recent discovery (due to Aluffi) that Chern classes of manifolds satisfy a strong birational property, I will propose a different generalization of Chern classes for singular varieties for which such a property is preserved. These new classes are in fact deeply related to certain "stringy invariants" and have a nice orbifold interpretation for quotient varieties. The results are joint work with Lupercio, Nevins and Uribe.
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February 17 Special Colloquium Thursday |
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DNA topology is the study of geometrical (supercoiling) and topological(knotting) properties of DNA loops and circular DNA molecules. Virtually every reaction involving DNA is influenced by DNA topology, or has topological effects. Site-specific recombinases and topoisomerases are enzymes able to change the topology of circular DNA by breaking the DNA and introducing one or more crossing changes. Mathematical analysis of such changes may provide relevant information about the possible enzymatic pathways, and about DNA conformation at the moment of double-stranded break induction. In this talk I will discuss some of the problems that I am currently interested in, and the topological and computational tools used in their analyses.
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February 22 Special Colloquium Tuesday |
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Statistical ensemble simulations play an important role in predicting the behavior of chaotic / noise-driven processes in nature. It is therefore imperative to understand and quantify useful information in a forecast ensemble. Modern methods of estimating predictive skill in an ensemble are often based on its mean state and variance, thus not taking into account the shape of its distribution. Introduced is a novel information theory-based predictability approach via relative entropy which captures extra information in higher order statistical moments of a forecast ensemble.
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February 23 Special Colloquium Wednesday |
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The shape memory effect is an important manifestation of the martensitic phase transformation, a phenomenon observed in various metallic alloys, ceramics, and even biological systems. Because of their technological applications as microactuators, martensitic thin films utilizing the shape memory effect have been intensively studied in recent years. In view of these and other potential applications, it is essential to rigorously understand how effective thin film models can be derived from three-dimensional Nonlinear Elasticity. I will present a new mathematical theory of martensitic thin films and discuss its connections with a number of recent results in this direction.
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February 24 Cancelled; Special Colloquium Thursday |
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Modular forms are certain holomorphic functions on the complex upper half-plane that possess an infinity of symmetry. Number theorists are interested in the systems of eigenvalues obtained from the action of Hecke operators on modular forms. These (a priori complex) numbers are algebraic integers which are often of arithmetic significance. One is interested in studying congruences modulo (powers of) a prime p between these eigenvalues. This is most efficiently done through a systematic study of p-adic analytic variation of modular forms. In my talk I will survey the progress in this area from the conception of the notion of a p -adic analytic family of modular forms to Coleman-Mazur's construction of the eigencurve which is in some sense the "universal" such family.
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March 10 Departmental Colloquium |
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I will present new work with Sylvia Serfaty. The main focus is motion by curvature in two space dimensions. The level-set formulation of this interface evolution law is a degenerate parabolic equation. We show it can be interpreted as the value function of a deterministic two-person game. More precisely, we give a family of discrete-time, two-person games whose value functions converge in the continuous-time limit to the solution of the motion-by-curvature PDE. This result is unexpected, because the value function of a deterministic control problem is normally a first-order Hamilton-Jacobi equation.
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March 17 Spring Break Holiday |
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Archival event entry migrated from the legacy colloquium website.
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March 21 Special Colloquium Monday |
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It is known since 1954 that every 3-manifold bounds a 4-manifold. Thus, for instance, every 3-manifold has a surgery diagram. There are many proofs of this fact, including several constructive ones, but they do not bound the complexity of the 4-manifold. (By "complexity" of a manifold we mean the minimum number of simplices in a triangulation.) Given a 3-manifold M of complexity n, we show how to construct a 4-manifold bounded by M of complexity O(n 2 ). It is an open question whether this quadratic bound can be replaced by a linear bound. The natural setting for this result is shadow surfaces, a representation of 3- and 4-manifolds that generalizes many other representations of these manifolds. One consequence of our results is some intriguing connections between the complexity of a shadow representation and the hyperbolic volume of a 3-manifold. Our results can also be phrased in terms of the singularities of smooth maps. In particular, the minimum number of "crossing singularities" of a map from a hyperbolic 3-manifold to the plane is bounded below and above by the hyperbolic volume. (Joint work with Francesco Costantino.)
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March 24 Departmental Colloquium |
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Several questions in applied analysis motivated by issues in computer vision, physics, materials sciences and other areas of engineering may be treated variationally leading to higher order variational problems and to models involving lower order density measures. Their study often requires state-of-the-art techniques, new ideas, and the introduction of innovative tools in partial differential equations, geometric measure theory, and calculus of variations. In this talk it will be shown how some of these questions may be reduced to well understood first order problems, while in others the higher order plays a fundamental role. Applications to phase transitions, to the equilibrium of foams under the action of surfactants, imaging, micromagnetics and thin films will be addressed.
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April 14 Frontiers of Science Lecture |
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Archival event entry migrated from the legacy colloquium website.
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April 15 Departmental Colloquium Friday |
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The nervous system produces many different rhythms, some simultaneously, in different behavioral situations. This talk focuses on the rhythms of the entorhinal cortex (EC), the gateway of the hippocampus. The EC is believed to have functional modules, each containing at least several different kinds of cells. We model the multiple rhythms that can form within a module, and describe modeling studies about the interaction of these modules. We also discuss potential functional implications of the rhythms.
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