Topology
Seminar
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Cascade Topology Seminar |
Title: The topological structure of the unit octonions and the quantum theory of games
Abstract: We exploit the topological structure of the unit octonions in a quantized version of three player, two strategy games. The structure we exploit is a generalization of the usual “wedge”or “1-point union” of spheres to a construction where our family of spheres all intersect in a common sphere of lower dimension, a construction we call a “posy” of spheres. In the case of the octonions, such arise naturally among the many quaternionic subspaces embedded in the octonions. We find particular use for three embedded copies of the unit quaternions S3 that in the unit octonions S7 meet in a common copy of the unit complexes S1.
Title: Gluing constructions of embedded doubly periodic minimal surfaces
Abstract: In this talk, we will explain how two of the oldest known minimal surfaces, the singly and doubly periodic Scherk surfaces, can be glued together in various ways to construct new examples of embedded minimal surfaces. Background and motivation are included.
Title: Topological index theory for surfaces in 3-manifolds
Abstract: We define an isotopy invariant index of a surface in a 3-manifold, and show that it mimics the index of a (unstable) minimal surface. This allows us to find purely topological analogues to familiar techniques from geometry, such as barrier arguments. Deep results follow concerning Heegaard splittings, bridge positions of knots, and normal surfaces. In addition, this new viewpoint opens up a host of exciting new questions for the field of 3-manifold topology.
Title: Topology and Data
Abstract: As most people are aware, all branches of science and engineering are producing enormous amounts of data, much faster than it can be analyzed. In addition, the data comes in many different forms, such as genomic sequence data network data, binary data, as well as Euclidean data. Often a notion of distance can be constructed, which is very useful in understanding the it, but the notions of distance are not reliable and are not backed up by solid theoretical models. This means that one needs methods of data analysis which are robust to small changes of distance, and which are flexible in the sense that they can be adapted to the various different kinds of data in a straightforward way. It turns out that topology can often supply this kind of method, and in this talk I plan to discuss some of these methods, the underlying topology from which they arise, as well as some examples of applications of the methods.
Title: From fibered symmetric bimonoidal categories to symmetric spectra.
Abstract: In this talk I will introduce the concept of a fibered symmetric bimonoidal categories (FSBC). These are roughly speaking "graded" symmetric bimonoidal categories. I will show, using the machinery of Elemendorf and Mandell on multiplicative infinite loop spaces that there one can correspond an E_{infty}-ring spectrum to a FSBC. FSBC's appear naturally in an attempt to model elliptic cohomology in a joint work with Po Hu and Igor Kriz.
Title: Derived smooth manifolds.
Abstract: A derived manifold is a space with a homotopical sheaf of C^\infty rings, which locally looks like the zero set of smooth functions on a manifold. Non-transverse intersections of submanifolds naturally have the structure of a derived manifold. The category of smooth manifolds embeds as a full subcategory of derived manifolds. The purpose of the larger category is to obtain a functorial intersection theory, in which perturbing submanifolds before intersecting them is unnecessary.
I will describe the construction and some properties of derived manifolds. I will describe the theory of derived cobordism and prove a Thom-Pontrjagin theorem, which endows every compact derived manifold with a fundamental cobordism class. In particular, a cup product formula of the form
[A\cap B] = [A] \cup [B],
for derived submanifolds A and B, holds with full generality.
Title: Homotopy string links over surfaces
Abstract: In his 1947 work "Theory of Braids" Emil Artin asked whether the braid group remained unchanged when one considered classes of braids under link-homotopy, allowing each strand of a braid to pass through itself but not through other strands. The problem remained open for a long time until in her 1974 paper "Homotopy of Braids - in answer to a Question of E. Artin", Deborah Goldsmith described a subgroup of isotopically non-trivial braids that became trivial under the relation of link-homotopy.In a seminal paper "Classification of links up to link- homotopy"(1990) Nathan Habegger and Xiao-Song Lin re-introduced Goldsmith's quotient of the pure braid group as a group of homotopy string links, which they used as a fundamental tool to accomplish classification.
We generalize Artin's question to string links over orientable surface M and show that under link-homotopy surface string links form a group, which is isomorphic to a quotient of the surface pure braid group PBn(M). Our work explores the geometric and visual beauty of the subject as we compute a presentation of the group of homotopy string links in terms of generators and relations.