The number of ends is a numerical invariant of a group. By a theorem of Hopf, the number of ends of a group may be equal to 0,1,2, or infinity. I will define this invariant and give several examples of groups with different numbers of ends. I will also explain an application to geometry, which we obtained jointly with Arapura and Ramachandran. In particular, we proved that the fundamental group of a projective manifold has at most one end.
I will go over some of the very basic concepts in dynamical systems, all within a euclidean space context. If time allows, I will comment on some of the connections of Dynamical Systems theory that connect to my research project.
In analyzing the structure of projective planes, the ability to embed different symmetric configurations plays an important role. Besides the Desarguesian 10-3 and Pappian 9-3 configurations, we will look at the results of a young Stephen Shipman as he attempted, ten years ago, to understand the significance of embedding the 8-3 configuration in a projective plane.
Integrable PDE's may be regarded as infinite dimensional analogues of completely integrable, finite dimensional Hamiltonian systems. As such, they possess infinite sequences of linearly independent constants of motion in involution, for example. In 1971, Hasimoto discovered a transformation relating the Filament Model (FM), which describes the evolution of a non-stretching curve in 3-space, to the cubic non-linear Schrödinger equation (NLS), a well-known integrable PDE. To date, the connection between geometry and integrable PDE's has been made explicit and exploited in a variety of cases and settings. In this first talk on the topic, we will look at some work by Doliwa and Santini and see how the motion of a certain class of spherical, non-stretching curves naturally selects integrable dynamics.
In this talk I will go over some interesting non-orthodox (i.e. mainstream) examples in neuroscience research. I will show some experimental and modeling results that could shed new light on the way how brains function. These results have some philosophical consequences that I will try to explore together with some other difficult questions that still await any reasonable answer.
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We consider a reduced Maxwell-Bloch system and establish integrability, find the soliton solutions and study the geometry of the phase space. Have you ever wondered why you the rules for the multiplication and inverse of elements in GL(n,C), GL(n,R), GL(n,Q) etc. are all the same and independent of what the field (or more generally commutative ring) is? A similar statement can be made pulling out the additive or multiplicative structure for a ring -- e.g. Z has (Z,+) and (Z\0,*). I will give an indication of how these (and other) operations are natural. We will begin by slightly generalizing the notion of a group to a "group object" in a category and see how the notion corresponds to the "Hopf Algebras" in the category of (commutative) rings (with unity).
Have you ever wondered why you the rules for the multiplication and inverse of elements in GL(n,C), GL(n,R), GL(n,Q) etc. are all the same and independent of what the field (or more generally commutative ring) is? A similar statement can be made pulling out the additive or multiplicative structure for a ring -- e.g. Z has (Z,+) and (Z\0,*). I will give an indication of how these (and other) operations are natural. We will begin by slightly generalizing the notion of a group to a "group object" in a category and see how the notion corresponds to the "Hopf Algebras" in the category of (commutative) rings (with unity).
Over F_q(t), Drinfeld modules give an analogue of elliptic curves. We will examine this analogy and see some examples.
In the words of S.S. Chern, it was ``Gauss' fundamental work which elevated differential geometry from a chapter in a Calculus textbook to an independent subject.'' Gauss' basic idea was that a surface has an intrinsic geometry derived from the local notion of arclength alone, and that this surface can be distinguished from another through examination of local invariants, such as curvature. I will discuss the historical derivations of local invariants of plane curves and embedded surfaces, the relationship to global properties, and the impact of these ideas on the development of modern geometry.
Elliptic curves are interesting to study because they are equipped with a great deal of structure. In addition to being varieties, elliptic curves are groups whose group structure respects Galois action on points of the curve. One consequence of this rich structure is the Weil pairing, which sends any pair of torsion points of $E$ to a root of unity. In this talk, I will describe this pairing explicitly, and explain how it applies to my current research. Definitions and examples will be provided.
What do sandpiles, quasars, and James Joyce have in common? We will attempt to answer this and other questions with an introduction to self-organized criticality. This is a science which is attempting to explain not just a particular class of complex systems, but complexity itself. Per Bak started a flurry of activity in this area by introducing "the first theory of complexity with a firm mathematical basis." We will examine his ideas, and focus on applications to studying mass extinctions. We will also review my attempt to add game theory to the soup, and sample the results.
For any p >0, the Bergman space A^p(D) is the set consists of analytic functions f in open unit disk D shich belong to the standard L^p(D) wrt area measure. We will look at some of the natural properties Bergman spaces have. We will focus mainly on function theory in D. If time permits we will examine some operator theory in D and discuss some of the open problems in Bergman spaces.
Probability, linear filters, ODE's, asymptotic and numerical analysis, programming and other tools are used to construct a series of models of the input layer of the mammalian primary visual cortex. Such models increase in complexity upon addition of new assumptions based on neuroanatomy and neurophysiological data obtained experimentally. While the models increase in complexity, the match of computational results with experimental data also increases.
Montel's Theorem, a classical result from complex analysis, has interesting consequences for the Julia set of a rational self-map of the Riemann sphere. In this talk, we shall discuss some of these facts and briefly survey dynamics of self-maps of all Riemann surfaces.