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Selected publications and research from Mathematics and Statistics Faculty

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• #### Splice Expression Variation Analysis (SEVA) for inter-tumor heterogeneity of gene isoform usage in cancer ﻿

(Oxford University Press, 2018)
Motivation Current bioinformatics methods to detect changes in gene isoform usage in distinct phenotypes compare the relative expected isoform usage in phenotypes. These statistics model differences in isoform usage in ...
• #### Commutators in Free Products ﻿

(World Scientific Publishing, 1995)
Topological methods are used to show that for certain subgroups S of a free product F, if w∈S is a commutator in F, then w is a commutator in S.
• #### Tesselations of S2 and equations over torsion-free groups ﻿

(Cambridge University Press, 1995)
Let G be a torsion free group, F the free group generated by t. The equation r(t) = 1 is said to have a solution over G if there is a solution in some group that contains G. In this paper we generalize a result due to ...
• #### Compact cores of coverings ﻿

(Elsevier, 1999)
Let H be a subgroup of G. In this paper we discuss the following questions: Is there a two-complex L so that pi(1)(L) = G and so that the cover of L corresponding to H has a compact core? Is there an L so that pi(1)(L) = ...
• #### Equations with Torsion-Free Coefficients ﻿

(Cambridge University Press, 2001)
In this paper we generalize techniques used by Klyachko and the authors to prove some tessellation results about S-2. These results are applied to prove the solvability of certain equations with torsion-free coefficients.
• #### Groups and Several Equations in one Variable ﻿

(World Scientific Publishing, 2001)
This paper addresses the question of simultaneously solving a set of equations in one variable over torsion-free groups.
• #### The Group <G,t|e> When G is Torsion-Free ﻿

(Elsevier, 2001)
In this paper we discuss classes of nontrivial elements in the group with a relative presentation <G, t \e >, where G is torsion free and e is solvable over G.
• #### A Class of Exponent-Sum Two Equations Over Groups ﻿

(Cambridge University Press, 2002)
In this paper, relative pictures are used to analyze a certain family of exponent-sum two equations over groups.
• #### Non-amenable type K equations over groups ﻿

(Cambridge University Press, 2003)
We define a class of equations that are not amenable but are type K and are therefore solvable over torsion-free groups. Moreover, we show that these new equations are solvable over all groups.
• #### Subgroups of free groups and primitive elements ﻿

(De Gruyter, 2010)
An algorithm is developed to determine whether a subgroup of a free group contains a primitive element This answers question 39b in the list of problems on free groups posted on the World of Groups website, www.grouptheory.info.
• #### Two cancellative commutative congruences and group diagrams ﻿

(Springer Verlag, 2011)
Remmers (Adv. Math. 36:283–296, 1980) uses group diagrams in the Euclidean plane to demonstrate how equality in a semigroup S “mirrors” that inside the group G sharing the same presentation with S, when S satisfies Adyan’s ...
• #### Sets of primitive elements in a free group ﻿

(Elsevier, 2012)
A set of elements in a free group F is said to be a primitive set if it is a subset of some basis of F. In this paper several theorems about primitive sets are proven. The results are applications of Whiteheadʼs 3-dimensional ...
• #### Generalized Casson Invariants for SO(3),U(2),Spin(4) , and SO(4) ﻿

(American Mathematical Society, 1994)
We investigate Casson-type invariants corresponding to the low-rank groups $\mathrm{SO}(3), \mathrm{SU}(2) \times S^1, \mathrm{U}(2), \operatorname{Spin}(4)$ and SO(4). The invariants are defined following an approach ...
• #### A decomposition theorem for h-cobordant smooth simply-connected compact 4-manifolds ﻿

(Springer Verlag, 1996)
• #### A Dehn surgery description of regular finite cyclic covering spaces of rational homology spheres ﻿

(World Scientific Publishing, 2001)
We provide related Dehn surgery descriptions for rational homology spheres and a class of their regular finite cyclic covering spaces. As an application, we use the surgery descriptions to relate the Casson invariants of ...
• #### Erratum to “An intersection theory count of the SL(2,C)-representations of the fundamental group of a 3-manifold.” ﻿

(Elsevier, 2003)
• #### An intersection theory count of the -representations of the fundamental group of a 3-manifold ﻿

(Elsevier, 2001)
We define an invariant of closed 3-manifolds counting the signed equivalence classes of representations of the fundamental group in . The invariant is an -analog of the Casson-Walker invariant for SU(2). We reinterpret the ...
• #### A PSL(2,C) Casson Invariant ﻿

(American Mathematical Society, 2005)
We use intersection theory techniques to define an invariant of closed 3-manifolds counting the characters of irreducible representations of the fundamental group in PSL(2,C). We note several properties of the invariant ...
• #### The SL(2,C) Casson invariant for Seifert fibered homology spheres and surgeries on twist knots ﻿

(World Scientific Publishing, 2006)
We derive a simple closed formula for the SL(2,C) Casson invariant for Seifert fibered homology 3-spheres using the correspondence between SL(2,C) character varieties and moduli spaces of parabolic Higgs bundles of rank ...
• #### Splicing and the SL(2,C) invariant ﻿

(American Mathematical Society, 2008)
We establish a formula for the SL2(C) Casson invariant of spliced sums of homology spheres along knots. Along the way, we show that the SL2(C) Casson invariant vanishes for spliced sums along knots in S-3.