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Constant Mean Curvature Surfaces With Three Ends, Karsten Grosse-Brauckmann, Robert Kusner, John M. Sullivan
Constant Mean Curvature Surfaces With Three Ends, Karsten Grosse-Brauckmann, Robert Kusner, John M. Sullivan
Robert Kusner
We announce the classification of complete almost embedded surfaces of constant mean curvature, with three ends and genus zero. They are classified by triples of points on the sphere whose distances are the asymptotic necksizes of the three ends.
An Excursion From Enumerative Geometry To Solving Systems Of Polynomial Equations With Macaulay 2, Frank Sottile
An Excursion From Enumerative Geometry To Solving Systems Of Polynomial Equations With Macaulay 2, Frank Sottile
Mathematics and Statistics Department Faculty Publication Series
Solving a system of polynomial equations is a ubiquitous problem in the applications of mathematics. Until recently, it has been hopeless to find explicit solutions to such systems, and mathematics has instead developed deep and powerful theories about the solutions to polynomial equations. Enumerative Geometry is concerned with counting the number of solutions when the polynomials come from a geometric situation and Intersection Theory gives methods to accomplish the enumeration. We use Macaulay 2 to investigate some problems from enumerative geometry, illustrating some applications of symbolic computation to this important problem of solving systems of polynomial equations. Besides enumerating solutions …
Dynamics Of Lattice Kinks, Panos Kevrekidis, M. I. Weinstein
Dynamics Of Lattice Kinks, Panos Kevrekidis, M. I. Weinstein
Panos Kevrekidis
In this paper we consider two models of soliton dynamics (the sine Gordon and the \phi^4 equations) on a 1-dimensional lattice. We are interested in particular in the behavior of their kink-like solutions inside the Peierls- Nabarro barrier and its variation as a function of the discreteness parameter. We find explicitly the asymptotic states of the system for any value of the discreteness parameter and the rates of decay of the initial data to these asymptotic states. We show that genuinely periodic solutions are possible and we identify the regimes of the discreteness parameter for which they are expected to …
Modular Symbols And Hecke Operators, Paul E. Gunnells
Modular Symbols And Hecke Operators, Paul E. Gunnells
Paul Gunnells
We survey techniques to compute the action of the Hecke operators on the cohomology of arithmetic groups. These techniques can be seen as generalizations in different directions of the classical modular symbol algorithm, due to Manin and Ash-Rudolph. Most of the work is contained in papers of the author and the author with Mark McConnell. Some results are unpublished work of Mark McConnell and Robert MacPherson.
Orbifold Quantum Cohomology, Weimin Chen Chen, Yongbin Ruan
Orbifold Quantum Cohomology, Weimin Chen Chen, Yongbin Ruan
Weimin Chen
This is a research announcement on a theory of Gromov-Witten invariants and quantum cohomology of symplectic or projective orbifolds. Our project started in the summer of 98 where our original motivation was to study the quantum cohomology under singular flops in complex dimension three. In this setting, we allow our three-fold to have terminal singularities which can be deformed into a symplectic orbifold. We spent the second half of 98 and most of spring of 99 to develop the foundation of Gromov-Witten invariants over orbifolds, including the key conceptual ingredient — the notion of good map. In the April of …
Computing Special Values Of Partial Zeta Functions, Gautam Chinta, Paul E. Gunnells, Robert Sczech
Computing Special Values Of Partial Zeta Functions, Gautam Chinta, Paul E. Gunnells, Robert Sczech
Paul Gunnells
We discuss computation of the special values of partial zeta functions associated to totally real number fields. The main tool is the Eisenstein cocycle Ψ, a group cocycle for GL n (ℤ); the special values are computed as periods of Ψ, and are expressed in terms of generalized Dedekind sums. We conclude with some numerical examples for cubic and quartic fields of small discriminant.
Wonderful Blowups Associated To Group Actions, Lev A. Borisov, Paul Gunnells
Wonderful Blowups Associated To Group Actions, Lev A. Borisov, Paul Gunnells
Paul Gunnells
A group action on a smooth variety provides it with the natural stratification by irreducible components of the fixed point sets of arbitrary sub-groups. We show that the corresponding maximal wonderful blowup in the sense of MacPherson-Procesi has only abelian stabilizers. The result is inspired by the abelianization algorithm of Batyrev.
Eisenstein Series Twisted By Modular Symbols For The Group Sln, Dorian Goldfield, Paul Gunnells
Eisenstein Series Twisted By Modular Symbols For The Group Sln, Dorian Goldfield, Paul Gunnells
Paul Gunnells
We define Eisenstein series twisted by modular symbols for the group SLn, generalizing a construction of the first author \cite{goldfeld1, goldfeld2}. We show that, in the case of series attached to the minimal parabolic subgroup, our series converges for all points in a suitable cone. We conclude with examples for SL2 and SL3.
The Large Deviation Principle For Coarse-Grained Processes, Richard S. Ellis, Kyle Haven, Bruce Turkington
The Large Deviation Principle For Coarse-Grained Processes, Richard S. Ellis, Kyle Haven, Bruce Turkington
Richard S. Ellis
The large deviation principle is proved for a class of L2-valued processes that arise from the coarse-graining of a random field. Coarse-grained processes of this kind form the basis of the analysis of local mean-field models in statistical mechanics by exploiting the long-range nature of the interaction function defining such models. In particular, the large deviation principle is used in a companion paper [8] to derive the variational principles that characterize equilibrium macrostates in statistical models of two-dimensional and quasi-geostrophic turbulence. Such macrostates correspond to large-scale, long-lived flow structures, the description of which is the goal of the statistical equilibrium …
Triunduloids: Embedded Constant Mean Curvature Surfaces With Three Ends And Genus Zero, Karsten Grosse-Brauckmann, Robert Kusner, John M. Sullivan
Triunduloids: Embedded Constant Mean Curvature Surfaces With Three Ends And Genus Zero, Karsten Grosse-Brauckmann, Robert Kusner, John M. Sullivan
Robert Kusner
We announce the classification of complete almost embedded surfaces of constant mean curvature, with three ends and genus zero. They are classified by triples of points on the sphere whose distances are the asymptotic necksizes of the three ends.
Boundedness Of Bilinear Operators With Nonsmooth Symbols, John Gilbert, Andrea Nahmod
Boundedness Of Bilinear Operators With Nonsmooth Symbols, John Gilbert, Andrea Nahmod
Mathematics and Statistics Department Faculty Publication Series
We announce the Lp-boundedness of general bilinear operators associated to a symbol or multiplier which need not be smooth. We establish a general result for multipliers that are allowed to have singularities along the edges of a cone as well as possibly at its vertex. It thus unifies ealier results of CoifmanMeyer for smooth multipliers and ones, such the Bilinear Hilbert transform of Lacey-Thiele, where the multiplier is not smooth.