Generalizations of no k-equal spaces

Kosar, Nicholas J

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https://hdl.handle.net/2142/101142 Copy

Description

Title

Generalizations of no k-equal spaces

Author(s)

Kosar, Nicholas J

Issue Date

2018-04-10

Director of Research (if dissertation) or Advisor (if thesis)

Baryshnikov, Yuliy

Doctoral Committee Chair(s)

Hirani, Anil

Committee Member(s)

• Schenck, Hal
• Yong, Alexander

Department of Study

Mathematics

Discipline

Mathematics

Degree Granting Institution

University of Illinois at Urbana-Champaign

Degree Name

Ph.D.

Degree Level

Dissertation

Keyword(s)

• Polychromatic configuration spaces
• k-spanning trees
• discriminantal arrangements

Abstract

We consider generalizations of no $k$-equal spaces as well as their relations to other concepts. For any topological space $X$, the $n^{th}$ no $k$-equal space of $X$ is the space of $n$ points from $X$ such that no $k$ are the same. First, we consider a generalization where each of the points is assigned one of $m$ colors; the interactions between various points are governed by a subset of $\N^m$. We call these spaces polychromatic configuration spaces. We find the homology groups and cohomology rings for two classes of polychromatic configuration spaces of $\R^d$. Next, we consider the relation between no $k$-equal spaces of $\R$ and $k$-trees of simplicial complexes. It was noticed that the first non-trivial homology group of the $n^{th}$ no $k$-equal space of $\R$ has rank equal to the number of facets in a $k$-dimensional spanning tree of the $n$-dimensional hypercube. We give a proof of this that is not reliant on knowledge of these numbers. Furthemore, we prove the analogous fact for a generalization of no $k$-equal spaces: comb no $k$-equal spaces. The $k$-equal arrangements are a generalization of the braid arrangements. In another direction, Manin and Schectman defined discriminantal arrangements as a generalization of braid arrangements. In the final chapter, we combine these two to define codimension-$c$ discriminantal arrangements. These arise geometrically as no $(d+c)$-intersecting translates of hyperplanes. We give results on the first two non-trivial homology groups of no $(d+c)$-intersecting translates of hyperplanes in $\R^d$.

Graduation Semester

2018-05

Type of Resource

text

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http://hdl.handle.net/2142/101142

Copyright and License Information

Copyright 2018 Nicholas Kosar

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