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Title:  Generalizations of no kequal spaces 
Author(s):  Kosar, Nicholas J 
Director of Research:  Baryshnikov, Yuliy 
Doctoral Committee Chair(s):  Hirani, Anil 
Doctoral Committee Member(s):  Schenck, Hal; Yong, Alexander 
Department / Program:  Mathematics 
Discipline:  Mathematics 
Degree Granting Institution:  University of Illinois at UrbanaChampaign 
Degree:  Ph.D. 
Genre:  Dissertation 
Subject(s):  Polychromatic configuration spaces
kspanning 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 nontrivial 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 nontrivial homology groups of no $(d+c)$intersecting translates of hyperplanes in $\R^d$. 
Issue Date:  20180410 
Type:  Thesis 
URI:  http://hdl.handle.net/2142/101142 
Rights Information:  Copyright 2018 Nicholas Kosar 
Date Available in IDEALS:  20180904 
Date Deposited:  201805 
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Dissertations and Theses  Mathematics

Graduate Dissertations and Theses at Illinois
Graduate Theses and Dissertations at Illinois