Monodromy of the SL_2 Hitchin fibration

We calculate the monodromy action on the mod 2 cohomology for SL(2,C) Hitchin systems and give an application of our results in terms of the moduli space of semistable SL(2,R) Higgs bundles.Comment: 17 pages, 3 figures. v2: Version to appear in IJ

Stability of Affine G-varieties and Irreducibility in Reductive Groups

Let $G$ be a reductive affine algebraic group, and let $X$ be an affine algebraic $G$-variety. We establish a (poly)stability criterion for points $x\in X$ in terms of intrinsically defined closed subgroups $H_{x}$ of $G$, and relate it with the numerical criterion of Mumford, and with Richardson and Bate-Martin-R\"ohrle criteria, in the case $X=G^{N}$. Our criterion builds on a close analogue of a theorem of Mundet and Schmitt on polystability and allows the generalization to the algebraic group setting of results of Johnson-Millson and Sikora about complex representation varieties of finitely presented groups. By well established results, it also provides a restatement of the non-abelian Hodge theorem in terms of stability notions.Comment: 29 pages. To appear in Int. J. Math. Note: this version 4 is identical with version 2 (version 3 is empty

The Scent of Change: A Case Study

Decisions about entering into a new business venture involve a variety of considerations, despite the level of experience an entrepreneur has. This case presents the story of a business owner Bennett Gage and his decisions concerning whether or not he should enter into a business where canines are used to detect bed bugs in hotels. This case study gives the reader an opportunity to wrestle with some of the many questions that are part of entering into the creation of a new service

Removing the Undesirables: A Case Study

The lodging industry has been particularly challenged by the bed bug’s resurgence. Bed bugs are primarily associated with places where people sleep and most of these places are properties associated with the lodging industry such as hotels, motels and bed & breakfasts. In the United States, this industry is massive

The two components of the SO(3)-character space of the fundamental group of a closed surface of genus 2

We use geometric techniques to explicitly find the topological structure of the space of SO(3)-representations of the fundamental group of a closed surface of genus 2 quotient by the conjugation action by SO(3). There are two components of the space. We will describe the topology of both components and describe the corresponding SU(2)-character spaces by parametrizing them by spherical triangles. There is the sixteen to one branch-covering for each component, and the branch locus is a union of 2-spheres or 2-tori. Along the way, we also describe the topology of both spaces. We will later relate this result to future work into higher-genus cases and the SL(3,R)-representations

Cohomological Hall algebras and character varieties

In this paper we investigate the relationship between twisted and untwisted character varieties via a specific instance of the Cohomological Hall algebra for moduli of objects in 3-Calabi-Yau categories introduced by Kontsevich and Soibelman. In terms of Donaldson--Thomas theory, this relationship is completely understood via the calculations of Hausel and Villegas of the E polynomials of twisted character varieties and untwisted character stacks. We present a conjectural lift of this relationship to the cohomological Hall algebra setting.Comment: Slight improvements picked up while editing for publication. To appear in IJM special volume for proceedings of VBAC 201

Superrigid subgroups and syndetic hulls in solvable Lie groups

This is an expository paper. It is not difficult to see that every group homomorphism from the additive group Z of integers to the additive group R of real numbers extends to a homomorphism from R to R. We discuss other examples of discrete subgroups D of connected Lie groups G, such that the homomorphisms defined on D can ("virtually") be extended to homomorphisms defined on all of G. For the case where G is solvable, we give a simple proof that D has this property if it is Zariski dense. The key ingredient is a result on the existence of syndetic hulls.Comment: 17 pages. This is the final version that will appear in the volume "Rigidity in Dynamics and Geometry," edited by M. Burger and A. Iozzi (Springer, 2002

Spectral networks

We introduce new geometric objects called spectral networks. Spectral networks are networks of trajectories on Riemann surfaces obeying certain local rules. Spectral networks arise naturally in four-dimensional N=2 theories coupled to surface defects, particularly the theories of class S. In these theories spectral networks provide a useful tool for the computation of BPS degeneracies: the network directly determines the degeneracies of solitons living on the surface defect, which in turn determine the degeneracies for particles living in the 4d bulk. Spectral networks also lead to a new map between flat GL(K,C) connections on a two-dimensional surface C and flat abelian connections on an appropriate branched cover Sigma of C. This construction produces natural coordinate systems on moduli spaces of flat GL(K,C) connections on C, which we conjecture are cluster coordinate systems.Comment: 87 pages, 48 figures; v2: typos, correction to general rule for signs of BPS count

LpL^p-Spectral theory of locally symmetric spaces with QQ-rank one

We study the $L^p$-spectrum of the Laplace-Beltrami operator on certain complete locally symmetric spaces $M=\Gamma\backslash X$ with finite volume and arithmetic fundamental group $\Gamma$ whose universal covering $X$ is a symmetric space of non-compact type. We also show, how the obtained results for locally symmetric spaces can be generalized to manifolds with cusps of rank one

Rank two quadratic pairs and surface group representations

Let $X$ be a compact Riemann surface. A quadratic pair on $X$ consists of a holomorphic vector bundle with a quadratic form which takes values in fixed line bundle. We show that the moduli spaces of quadratic pairs of rank 2 are connected under some constraints on their topological invariants. As an application of our results we determine the connected components of the $\mathrm{SO}_0(2,3)$-character variety of $X$.Comment: 37 pages, 1 figur