EPrint Collection - Mathematics

Permanent URI for this collectionhttps://digital.lib.washington.edu/handle/1773/2129

The EPrint Collection is a permanent and freely available archive of preprints and postprints of work by faculty and students associated with the Department of Mathematics at the University of Washington.

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Now showing 1 - 20 of 112

Affine Partitions and Affine Grassmannians
(2008) Billey, Sara C.; Mitchell, Stephen C.
We give a bijection between certain colored partitions and the elements in the quotient of an aﬃne Weyl group modulo its Weyl group. By Bott’s formula these colored partitions give rise to some partition identities. In certain types, these identities have previously appeared in the work of Bousquet-Melou-Eriksson, Eriksson-Eriksson and Reiner. In other types the identities appear to be new. For type An, the aﬃne colored partitions form another family of combinatorial objects in bijection with n + 1 core partitions and n-bounded partitions. Our main application is to characterize the rationally smooth Schubert varieties in the aﬃne Grassmannians in terms of aﬃne partitions and a generalization of Young’s lattice which reﬁnes weak order and is a subposet of Bruhat order. Several of the proofs are computer assisted.
Fibrations and Sheaves
(2012-12-13) Warner, Garth
The purpose of this book is to give a systematic treatment of fibration theory and sheaf theory, the emphasis being on the foundational essentials.
Homotopical Topos Theory
(2012-05) Warner, Garth
The purpose of this book is two-fold: (1) To give a systematic introduction to topos theory from a purely categorical point of view, thus ignoring all logical and algebraic issues. (2) To give an account of the homotopy theory of the simplicial objects in a Grothendieck topos.
Categorical Homotopy Theory
(2012-01) Warner, Garth
This book is an account of certain developments in categorical homotopy theory that have taken place since the year 2000. Some aspects have been given the complete treatment (i.e., proofs in all detail), while others are merely surveyed. Therefore a lot of ground is covered in a relatively compact manner, thus giving the reader a feel for the "big picture" without getting bogged down in the "nitty-gritty."
Reconstruction Theory
(2011-01) Warner, Garth
Suppose that G is a compact group. Denote by \underline{Rep} G the category whose objects are the continuous finite dimensional unitary representations of G and whose morphisms are the intertwining operators--then \underline{Rep} G is a monoidal *-category with certain properties P_1,P_2, ... . Conversely, if \underline{C} is a monoidal *-category possessing properties P_1,P_2, ..., can one find a compact group G, unique up to isomorphism, such that \underline{Rep} G "is" \underline{C}? The central conclusion of reconstruction theory is that the answer is affirmative.
C*-Algebras
(2010-11-15) Warner, Garth
This book is addressed to those readers who are already familiar with the elements of the theory but wish to go further. While some aspects, e.g. tensor products, are summarized without proof, others are dealt with in all detail. Numerous examples have been included and I have also appended an extensive list of references.
Unsolved Problems in Intuitive Geometry
(1960) Klee, Victor
Lectures on Lost Mathematics
(2010) Grünbaum, Branko
Lectures on Arrangements
(1974) Grünbaum, Branko
The Bilinski dodecahedron, and assorted parallelohedra, zonohedra, monohedra, isozonohedra and otherhedra.
(2010) Grünbaum, Branko
Fifty years ago Stanko Bilinski showed that Fedorov's enumeration of convex polyhedra having congruent rhombi as faces is incomplete, although it had been accepted as valid for the previous 75 years. The dodecahedron he discovered will be used here to document errors by several mathematical luminaries. It also prompted an examination of the largely unexplored topic of analogous non-convex polyhedra, which led to unexpected connections and problems.
Lecture notes on Modern Elementary Geometry
(1997) Grünbaum, Branko
Positivity
(2009-12-23T17:22:45Z) Warner, Garth
These notes provide a systematic account of certain aspects of the statistical structure of quantum theory. Here the all prevailing notion is that of a completely positive map and Stinespring's famous characterization thereof. I have also included a systematic treatment of "quantum dynamical semigroups," culminating in Lindblad's celebrated description of their generators.
Shocks and Business Cycles
(2005) Frankel, David M.; Burdzy, Krzysztof
A popular theory of business cycles is that they are driven by animal spirits: shifts in expectations brought on by sunspots. A prominent example is Howitt and McAfee (AER, 1992). We show that this model has a unique equilibrium if there are payoff shocks of any size. This equilibrium still has the desirable property that recessions and expansions can occur without any large exogenous shocks. We give an algorithm for computing the equilibrium and study its comparative statics properties. This work generalizes Burdzy, Frankel, and Pauzner (2000) to the case of endogenous frictions and seasonal and mean-reverting shocks.
Lagrangian Mechanics
(2009-01-12T18:05:24Z) Warner, Garth
My original set of lectures on Mechanics was divided into three parts: Lagrangian Mechanics, Hamiltonian Mechanics, Equivariant Mechanics. The present text is an order of magnitude expansion of the first part and is differential geometric in character, the arena being the tangent bundle rather than the cotangent bundle. I have covered what I think are the basics. Points of detail are not swept under the rug but I have made an effort not to get bogged down in minutiae. Numerous examples have also been included.
{4,5} cubic lattice sponges
(2009-01-06T18:00:53Z) Gillispie, Steven; Grünbaum, Branko
Isogonal polyhedra are those polyhedra having the property of being vertex-transitive. By this is meant that every vertex can be mapped to any other vertex via a symmetry of the whole polyhedron; in a sense, every vertex looks exactly like any other. The Platonic solids are examples, but these are bounded polyhedra and our focus here is on infinite polyhedra. When the polygons of an infinite isogonal polyhedron are all planar and regular the polyhedra are also known as sponges, pseudopolyhedra, or infinite skew polyhedra. These have been studied over the years but many have been missed by previous researchers. We first introduce a notation for labeling three-dimensional isogonal polyhedra and then show how this notation can be combinatorially used to find all of the isogonal polyhedra that can be created given a specific vertex star configuration. As an example, we apply our methods to the {4,5} vertex star of five squares aligned along the planes of a cubic lattice and prove that there are exactly 15 such unlabeled sponges and 35 labeled ones. Previous efforts had found only 8 of the 15 shapes.
Can Every Face of a Polyhedron Have Many Sides?
(2008-11) Grünbaum, Branko
The simple question of the title has many different answers, depending on the kinds of faces we are willing to consider, on the types of polyhedra we admit, and on the symmetries we require. Known results and open problems about this topic are presented.
An enduring error
(2008-06-05) Grünbaum, Branko
Bosonic Quantum Field Theory
(2008-03-31T20:29:13Z) Warner, Garth
The purpose of these notes is to provide a systematic account of that part of Quantum Field Theory in which symplectic methods play a major role.
Topics in Topology and Homotopy Theory
(Hopf Topology Archive, 2005-01) Warner, Garth
This book is addressed to those readers who have been through Rotman (or its equivalent), possess a wellthumbed copy of Spanier, and have a good background in algebra and general topology. Granted these prerequisites, my intention is to provide at the core a state of the art treatment of the homotopical foundations of algebraic topology. The depth of coverate is substantial and I have made a point to include material which is ordinarily not included, for instance, an account of algebraic K-theory in the sense of Waldhausen. There is also a systematic treatment of ANR theory (but, reluctantly, the connections with modern geometric topology have been omitted). However, truly advanced topics are not considered (e.g., equivalent stable homotopy theory, surgery, infinite dimensional topology, étale K-theory, ...). Still, one should not get the impression that what remains is easy: There are numerous difficult technical results that have to be brought to heel.
Mathematical Aspects of General Relativity
(2006-09-08T18:29:23Z) Warner, Garth
These notes can serve as a mathematical supplement to the standard graduate level texts on general relativity and are suitable for self-study. The exposition is detailed and includes accounts of several topics of current interest, e.g., Lovelock theory and Ashtekar's variables.

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