| UCSC Mathematics Colloquia Fall 1997 |
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| September 30, 1997 | |
| Pascal Chossat
chossat@math.ucsc.edu |
Institut Nonlineaire de Nice at Sophia-Antipolis, France |
| Title: An overview of equivariant bifurcation theory and its applications | |
| ABSTRACT. Equivariant bifurcation theory has proven to be more than just an ad'hoc tool to compute branches of solutions in dynamical systems with symmetry. It can help understanding complicated behaviour and patterns which are found in physical models, even when symmetry occurs as an idealisation of the real physical problem. I shall introduce some basic concepts and describe some examples to illustrate these statements. | |
| October 7, 1997 | |
| James Milgram
milgram@gauss.stanford.edu |
Stanford University |
| Title: The Geometry of Mapping Spaces and Moduli Spaces | |
| ABSTRACT.
Moduli spaces are topological spaces where the individual
points represent equivalence classes of objects of one kind
or another. For example the complex projective space, CP^n
is the moduli space of complex lines through the origin
in the complex vector space C^{n+1}. Similarly, the moduli
spaces of instantons are the equivalence classes of solutions
of the Yang-Mills equations under the action of the Gauge
group. There are also the classical moduli spaces of Riemann
surfaces.
Closely related to these spaces are spaces of holomorphic maps from Riemann surfaces to other complex manifolds and moduli spaces of of complex vector bundles over Riemann surfaces and other complex manifolds. These spaces also appear in applications of mathematics. For example the moduli space of linear control systems with $n$ inputs and $m$ outputs is identified with the mapping space Hol(P^1, G_{n,m}), where P^1 is the Riemann sphere, and G_{n,m} is the complex Grassmannian of complex n-planes in C^{n+m}. In recent work in topology the geometry of many of these spaces has begun to be clarified. Many years ago topologists learned how to understand the structure of spaces such as the mapping spaces consisting of ALL continuous maps Map(P^1, X), and more recently a number of theorems of the following type have been proved: Theorem: Let X be a complex manifold with "enough complex lines". Let A \in H_2(X, Z) be a torsion free class represented by a complex line. Then in the limit over n, Hol_{na}(P^1, X) has the homotopy type of a component in the full mapping space Map(P^1, X). (Here Hol_{na} means homolorphic maps which represent the class na \in H_2(X, Z).) These results have been used to determine, for example, the homology and stable homotoy types of the space Hol(P^1, F) where F is CP^n, the complex Grassmannian, G_{n,m}, or more generally, any complex flag manifold. Extensions of thest techniques to holomorphic maps of Riemann surfaces into X have also been successful in many cases, in particular in the cases of the elliptic and hyperelliptic curves. As applications one has, for example, the proof of the Atiyah-Jones conjecture, which shows that large energy instantons on the 4-sphere approximate through higher and higher dimensions the full mapping space of maps from the 4-sphere to the infinite quaternionic projective space HP^{\infty}. One also has similar approximation results for other real four dimensional manifolds, such as the ruled complex surfaces. The main problems in the area, currently, are to determine the generality of these approximation results. One reason for interest in this kind of question is the possibility of a generalized Morse theory for understanding the structure of mapping spaces of the form Map(V, M^n) where V and M are manifolds with V of dimension at least two. |
|
| October 14, 1997 | |
| Jochen Denzler
denzler@rz.mathematik.uni-muenchen.de |
Ludwig--Maximilians--Universität München |
| Title: Existence of windows with minimal heat leakage | |
| ABSTRACT.
For a given bounded Lipschitz domain $\Omega\subset\Bbb{R}^n$ and a
measurable subset $D\subset\partial\Omega$ of prescribed area
($n-1$-dimensional measure), but otherwise arbitrary, the lowest
eigenvalue of the Laplace operator, with Dirichlet boundary conditions
on $D$ and Neumann conditions elsewhere, will be called heat leakage
rate for the window $D$. The author can prove the existence of a window of prescribed area with minimal heat leakage rate and show that in the case of $\Omega$ a ball, this optimal window will be a spherical cap (uniquely). The talk will exhibit the basic ideas and ingredients of the proof and also give some complementary results. The subject touches the area of direct methods in the calculus of variations, Sobolev space theory, classical methods for elliptic PDEs and in part relies on recent work by C.~Kenig on harmonic analysis. Moreover, symmetrization arguments are used in the result for the ball. |
|
| October 21, 1997 | |
| John Baez
baez@math.ucr.edu |
UC Riverside |
| Title: Higher-dimensional algebra and quantum gravity | |
| ABSTRACT. Recently there has been a flourishing of topological techniques in physics, and also interesting applications of physical ideas to topology. A simple and beautiful example of this is the Turaev- Viro model, which on the one hand gives an invariant of 3-dimensional manifolds, and on the other hand can be thought of as a quantum theory of gravity in 3-dimensional spacetime. We give an elementary introduction to some of the ideas underlying the Turaev-Viro model and related (still very tentative) theories of 4-dimensional quantum gravity. | |
| October 28, 1997 | |
| Felipe Linares
linares@msri.org linares@impa.br |
Institute for Pure and Applied Mathematics (Rio de Janeiro, Brazil) |
| Title: On Dispersive Blow-up | |
| ABSTRACT. In this talk we will describe the so-called "Dispersive Blow-up". Some smoothness properties presented in solutions of dispersive equations will be discussed and how these properties are used to establish this kind of blow-up. | |
| November 4, 1997 | |
| David Ebin ebin@math.sunysb.edu |
SUNY at Stony Brook |
| Title: Motion of Incompressible Inviscid Fluids with Free Boundary and Surface Tension | |
| ABSTRACT.
I will explain what one might call the basic theorem for
incompressible fluids with surface tension; that is, I will show that
the initial value problem is well-posed. (Existence, Uniqueness, and
Continuous dependence on initial data). Basically the situation is the following: One starts with a domain \Omega in R^n and a divergence-free vector field v defined on \Omega. One can describe the fluid motion as a curve of maps \eta(t):\Omega --> R^n. A particle initally at x goes to \eta(t)(x) at time t. Because of incompressibility \eta(t) has Jabobian 1 for all t and x . Usual physical principles (I use Hamilton's principle) result in an equation for \eta(t). I show that there is a time interval (-a,b) on which there exists a unique \eta(t) which satisfies the equation and the conditions: \eta(0)=identity, d/dt(\eta)(0) = v. Furthermore \eta depends continuously on v. To prove the theorem, I decompose \eta(t) into a diffeomorphism \beta(t) of \Omega, with Jacobian 1, and a gradient of a function f(t) defined on \Omega. This is done by the identity: (*) \eta(t)(x) = \beta(t)(x) + grad f(t)(\beta(t)(x)) The equation satified by \eta(t) yields equations for \beta(t) and f(t), with the equation for f being particularly interesting. It looks like: (**) d^2/dt^2 f = Af + lower order terms depending on \beta where A is a third order elliptic pseudo-differential operator. Since \eta and \beta have Jacobian 1, f satisfies: det(Id + D^2 f) = 1 where Id means the identity matrix and D^2 f is the matrix of second partials of f in the x variables. This is an elliptic equation for f and it implies that f is determined by its values on the boundary of \Omega. Thus we can say that the decomposition (*) decomposes the motion of the fluid into a motion of the interior namely \beta and a displacement of the boundary given by grad f. If the surface tension is large compared to other parameters of the problem, (this would usually be true in the case of a liquid drop) then one get get a good approximation for \eta in the following way: Let \beta(t) be the motion of the fluid which one would get if the boundary were fixed and then let f(t) be a solution of (**). Then construct \eta(t) from (*). Such an \eta will be an approximate solution with an error of the order of \kappa^{-2} where \kappa is the coefficient of surface tension. Thus the proof of the theorem can be used to get approximate solutions which are easier to compute then the exact solution. Such approximations should be useful in applied problems involving motion of drops such as combustion studies involving fuel droplets. In such studies it is necessary to find the temperature distribution on a drop and this is affected both by internal motion in the drop and the shape of the boundary. |
|
| November 11, 1997 | |
| Peter Bouwknegt
pbouwkne@physics.adelaide.edu.au |
University of Adelaide |
| Title: Do particles know about combinatorics? | |
| ABSTRACT. In this talk we discuss how particles with 'funny statistics' can be used to derive elegant results in combinatorics in terms of q-series identities (such as the Rogers-Ramanujan identities). We illustrate this procedure in a few elementary examples before we proceed to discuss how to construct (quasi-) particle type bases for modules of affine Lie algebras. | |
| November 18, 1997 | |
| Note: There are two talks
on this date in different rooms. The Tea will be between talks. |
|
| Jack Morava
jack@chow.mat.jhu.edu |
The John Hopkins University |
| Time: 2:00-3:00pm | Place: 358 Applied Sciences |
| Title: Topological 2D gravity and vertex operator algebras | |
| ABSTRACT. The rational cohomology of the moduli space of Riemann surfaces in the large genus limit is not completely understood. However, it contains a large subalgebra with remarkable formal properties which closely resemble those of vertex operator algebras (VOAs). This talk will be an informal introduction to topological gravity in two dimensions, focussing on its VOA-like features. | |
| November 18, 1997 | |
| Lionel Mason
lmason@msri.org |
Oxford University and MSRI |
| Time: 4:00 - 5:00pm | Place: 372 Applied Sciences |
| Title: Spinors and Twistors in Einstein's Theory of Gravity | |
| ABSTRACT. Penrose's twistor theory attempts to replace space-time as the primary arena for physics with a complex manifold, the twistor space. The framework has yet to be extended to curved space-times as required by Einstein's theory of gravity, general relativity. This lecture will explain, without assuming prior knowledge of spinors or twistors, some of the underlying ideas and progress on the programme, in particular focussing on recent developments based on helicity 3/2 fields, Rarita-Schwinger fields. I will then go on to discuss applications of the methods and ideas to problems in classical general relativity such as providing a deeper understanding of gravitational energy and giving rise to a framework for proving existence theorems for the field equations. | |
| November 25, 1997 | |
| PLEASE NOTE: Tea starts at 4:00PM, talk at 4:30PM (half an hour delay). | |
| Ana Cannas da Silva
acannas@math.berkeley.edu |
UC Berkeley and Instituto Superior Técnico |
| Folding symplectic manifolds | |
| ABSTRACT. Symplectic manifolds are manifolds equipped with a closed non-degenerate 2-form. Many cut-and-paste constructions force a symplectic geometer to leave the symplectic category. However, surgery often produces a closed 2-form with mild degeneracies, which is enough to apply some symplectic techniques on these more general (non-symplectic) manifolds. Folding-type degeneracies occur in important examples, including the 4-sphere, and not just by surgery from symplectic manifolds. Folded symplectic manifolds are manifolds equipped with a closed 2-form having only folding-type degeneracies; they will be explained and illustrated. | |
| December 2, 1997 | |
| Gunnar Carlsson
gunnar@gauss.stanford.edu |
Stanford University |
| Title: | |
| Speakers for winter and spring quarters include: | ||
|---|---|---|
| Tetsuji Miwa | . | RIMS, Kyoto |
| Estelle Basor | . | Cal. Poly, San Luis Obispo |
| Craig Evans | . | UC Berkeley |
| Sasha Givental | . | UC Berkeley |
| Vaughn Jones | vfr@math.berkeley.edu | UC Berkeley |