## About Me

I am currently an NSF Postdoctoral Fellow at University of Maryland at College Park. I completed my Ph.D. in 2016 at University of Illinois (UIUC) under the supervision of Steve Bradlow.

Here is my CV .

### Contact

**E-mail:**bcollie2 AT math DOT umd DOT edu

**Office:**4109 William E. Kirwan Hall

**Mailing Address:**

Department of Mathematics

University of Maryland, College Park

4176 Campus Drive - William E. Kirwan Hall

College Park, MD 20742-4015

## Research

My research is in differential and algebraic geometry. I am particularly interested in Higgs bundles, character varieties, Anosov representations, Higher Tiechmüller Theory, and harmonic maps.

### Papers

Preprint posted: October, 2017.

Abstract:

We study the character variety of representations of the fundamental group of a closed surface of genus g>1 into the Lie group SO(n,n+1) using Higgs bundles. For each integer 0< d < n(2g-2)+1, we show there is a smooth connected component of the character variety which is diffeomorphic to the product of a certain vector bundle over a symmetric product of a Riemann surface with the vector space of holomorphic differentials of degree 2,4,...,2n-2.
In particular, when d=n(2g-2), this recovers Hitchin's parameterization of the Hitchin component.
We also exhibit 2^{2g+1}-1 additional connected components of the SO(n,n+1)-character variety and compute their topology.
Moreover, representations in all of these new components cannot be continuously deformed to representations with compact Zariski closure. Using recent work of Guichard-Wienhard on positivity, it is shown that each of the representations which define singularities (i.e. those which are not irreducible) in these 2^{2g+1}-1 connected components are positive Anosov representations.

coauthors: Daniele Alessandrini

Preprint posted: August, 2017.

Abstract:

In this paper we describe the space of maximal components of the character
variety of surface group representations into PSp(4,R) and Sp(4,R). For every
rank 2 real Lie group of Hermitian type, we construct a mapping class group
invariant complex structure on the maximal components. For the groups PSp(4,R)
and Sp(4,R), we give a mapping class group invariant parameterization of each
maximal component as an explicit holomorphic fiber bundle over Teichm\"uller
space. Special attention is put on the connected components which are singular,
we give a precise local description of the singularities and their geometric
interpretation. We also describe the quotient of the maximal components for
PSp(4,R) and Sp(4,R) by the action of the mapping class group as a holomorphic
submersion over the moduli space of curves. These results are proven in two
steps, first we use Higgs bundles to give a non-mapping class group equivariant
parameterization, then we prove an analogue of Labourie's conjecture for
maximal PSp(4,R) representations.

coauthors: Nicolas Tholozan and Jérémy Toulisse

Preprint posted: February, 2017.

Abstract:

In this paper, we study the geometric and dynamical properties of maximal representations of surface groups into Hermitian Lie groups of rank 2. Combining tools from Higgs bundle theory, the theory of Anosov representations, and pseudo-Riemannian geometry, we obtain various results of interest. We prove that these representations are holonomies of certain geometric structures, recovering results of Guichard and Wienhard. We also prove that their length spectrum is uniformly bigger than that of a suitably chosen Fuchsian representation, extending a previous work of the second author. Finally, we show that these representations preserve a unique minimal surface in the symmetric space, extending a theorem of Labourie for Hitchin representations in rank 2.

To appear in the proceedings for Nigel Hitchin's 70th birthday.

Abstract:

Recall that the group PSL(2,R) is isomorphic to PSp(2,R), SO_0(1,2) and PU(1,1). The goal of this paper is to examine the various ways in which Fuchsian representations of the fundamental group of a closed surface of genus g into PSL(2,R) and their associated Higgs bundles generalize to the higher rank groups PSL(n,R), PSp(2n,R), SO_0(2,n), SO_0(n,n+1) and PU(n,n). For the SO_0(n,n+1)-character variety, we parameterize n(2g-2) new connected components as the total space of vector bundles over appropriate symmetric powers of the surface and study how these components deform in the SO_0(n,n+2)-character variety. This generalizes results of Hitchin for PSL(2,R).

My thesis concerns fixed points of the roots of unity action on the moduli space of Higgs bundles (see page ii. of the document for a detailed abstract).

coauthors: Eugene Lerman and Seth Wolbert

Journal of Geometry and Physics Volume 107, September 2016, pp 187-213.

Preprint posted: September, 2015.

Abstract:

In this paper we introduce a notion of parallel transport for principal bundles with connections over differentiable stacks. We show that principal bundles with connections over stacks can be recovered from their parallel transport thereby extending the results of Barrett, Caetano and Picken, and Schreiber and Waldof from manifolds to stacks.
In the process of proving our main result we simplify Schreiber and Waldorf's definition of a transport functor for principal bundles with connections over manifolds and provide a more direct proof of the correspondence between principal bundles with connections and transport functors

Geometriae Dedicata 180 (2015), no. 1, 241–285

Preprint posted: March, 2015.

Abstract:

Let S be a closed surface of genus at least 2. In this paper we prove that for the 2g−3 maximal components of the Sp(4,R) character variety which contain only Zariski dense representations, there is a unique conformal structure on the surface so that the corresponding equivariant harmonic map to the symmetric space Sp(4,R)/U(2) is a minimal immersion. Using a Higgs bundle parameterization of these components, we give a mapping class group invariant parameterization of such components as fiber bundles over Teichmüller space. Unlike Labourie's recent results on the Hitchin components, these bundles are not vector bundles.

co-authors: Qiongling Li

Advances in Mathematics Volume 307, 5 February 2017, Pages 488–558

Preprint posted May 2014

Abstract:

Using Hitchin's parameterization of the Hitchin-Teichm\"uller component of the $SL(n,\mathbb{R})$ representation variety, we study the asymptotics of certain families of representations.
In fact, for certain Higgs bundles in the $SL(n,R)$-Hitchin component, we study the asymptotics of the Hermitian metric solving the Higgs bundle equations. This analysis is used to estimate the asymptotics of the corresponding family of flat connections as we scale the differentials by a real parameter.
We consider Higgs fields that have only one holomorphic differential q_n of degree n or q_{n-1} of degree n-1.
We also study the asymptotics of the associated family of equivariant harmonic maps to the symmetric space SL(n,R)/SO(n) and relate it to recent work of Katzarkov, Noll, Pandit and Simpson.

co-authors: E. Kerman, B. Reiniger, B. Turmunkh, and A. Zimmer

Compositio Mathematica, Volume 148, Issue 06, November 2012, pp 1069--1984.

Preprint posted: July, 2011.

Abstract:

A celebrated theorem in two-dimensional dynamics due to John Franks asserts that every area preserving homeomorphism of the sphere has either two or infinitely many periodic points. In this work we reprove Franks' theorem under the additional assumption that the map is smooth. Our proof uses only tools from symplectic topology and thus differs significantly from all previous proofs. A crucial role is played by the results of Ginzburg and Kerman concerning resonance relations for Hamiltonian diffeomorpisms.

### Upcoming Talks

### Past Talks

Title: Maximal SO(2,n) Higgs bundles and photon structures

Abstract: Like Hitchin representations, the set of maximal representations of a surface group into a Lie group of Hermitian type form connected components of representations with Labourie's Anosov property. For such representations, Guichard and Wienhard have constructed geometric structures modeled on certain flag varieties. In this talk we will use Higgs bundle techniques to construct a unique "maximal" space like surface in the pseudo-Riemannian hyperbolic space H^{2,n-1} associated to any maximal SO(2,n) representation. Using the geometry of this surface, we construct geometric structures on certain homogeneous bundles on the surface and prove that they agree with those of Guichard-Wienhard. As a corollary, we also prove (a generalization of) a conjecture of Labourie for all maximal representations into rank 2 Hermitian Lie groups.This is based joint work with Nicholas Tholozan and Jérémy Toulisse.

Title: Geometric structures and Higgs bundles

Abstract: Let S be a closed oriented surface of genus at least 2. For maximal
representations of the fundamental group of S into SO(2,n), we will use Higgs
bundle techniques to construct geometric structures on certain fiber-bundles
over S. In particular, we will show how these structures recover results of
Guichard and Wienhard and exhibit how our techniques describe the topology
of these geometric manifolds. This is based on a joint work with Nicolas
Tholozan and Jérémy Toulisse.

Title: Parameterizing connected components of SO(p, p+1)-Higgs bundles

Abstract: In this talk I will discuss a parameterization of n(2g-2) connected components of the SO_0(n,n+1)-Higgs bundle moduli space. We will see how this parameterization generalizes both Hitchin's parameterization of the Hitchin component as a vector space of holomorphic differentials of degree 2, 4,\cdots, 2n and Hitchin's parameterization of the nonzero Toledo invariant components of the PSL(2,R)=SO_0(1,2)-Higgs bundle moduli space as vector bundles over certain symmetric products of the Riemann surface. Time permitting, we will give a connected count of the SO_0(p,p+1)-Higgs bundles moduli space and discuss the Zariski closure of these the representations in each connected component.

Title: Holomorphic differentials and the group SO_0(n,n+1)

Abstract: In this talk I will discuss a parameterization of n(2g-2) connected components of the SO_0(n,n+1)
Higgs bundle moduli space. We will see how this parameterization generalizes both Hitchin's
parameterization of the Hitchin component as a vector space of holomorphic differentials of
degree 2,4,...,2n and Hitchin's parameterization of the nonzero Toledo invariant components of
the PSL(2,R)=SO_0(1,2) Higgs bundle moduli space by holomorphic quadratic differentials twisted
by an effective divisor.

Title: Maximal SO_0(2,3) representations and beyond

Abstract:
For a closed surface S of genus g > 1, the space of maximal PSp(4,R) representations is
especially diverse. For example, there are 2(2
2g −1) +4g − 3 connected components,
and for each integer 0 < d < 4g − 3 there is a particularly interesting smooth connected
component of the character variety which we call a Gothen component. When d = 4g − 4
the Gothen component is the Hitchin component and when d < 4g − 4 the Gothen
components are noncontractible and contain only Zariski dense representations.
Generalizing Labourie's results for Hitchin representations, we will give a mapping class
group invariant parameterization of the Gothen components as fiber bundles over
Teichmüller space. For n > 2 there is no component of the maximal PSp(2n,R)
representations which generalize the Gothen representations. However, motivated by the
isomorphism PSp(4, R)=SO_0(2,3), we will use a Higgs bundle description of the Gothen
components to show that the Gothen representations are an SO_0(n,n+1) phenomenon.

September 14, 2015
Geometry, Toplology and Dynamics seminar
, at University of Illinois Chicago.

July 13, 2015 Geometry seminar at ICMAT.

June 9-13, 2015 Workshop on Higgs Bundles and Character
Varieties, at the joint meeting of the American, European and Portuguese Mathematical Societies, in Oporto, Portugal.

February 17, 2015 Geometry, Groups, and Dynamics/GEAR seminar at University of Illinois (here is a video)

January 3-11, 2015 Workshop on Higgs Bundles and Harmonic maps in North Carolina.

December 1, 2014 Geometry seminar at ICMAT.

July 7-18, 2014 Summer school on the Geometry, Topology and Physics of Moduli spaces of Higgs bundles, National University of Singapore.

Workshop on the Geometry and Physics of Moduli spaces Miraflores de la Sierra (Madrid, Spain).

May 23-June 1, 2014, Gear Junior Retreat at University of Michigan Ann Arbor.

April 4-5, 2014, Graduate Student Toplogy and Geometry Conference at University of Texas Austin.

December 9th, 2013, Geometry Analysis seminar at Rice University.

June 23-30, 2013 Workshop on Higher Teichmuller-Thurston Theory.

## Notes

Slides from my talk in Porto Portugal on June 13, 2015 during the Workshop on Higgs Bundles and Character Varieties at the joint meeting of the American, European and Portuguese Mathematical Societies, in Oporto, Portugal.

Video from my talk in Geometry, Groups, and Dynamics/GEAR seminar February 17, 2015

Slides from my talk in Singapore on July 14, 2014 at Summer school on the Geometry, Topology and Physics of Moduli spaces of Higgs bundles, National University of Singapore. (The talk was part chalk talk part slides.)

Slides from my talk on Fixed points in the Higgs bundle moduli space and asymptotics of certain families of Higgs bundles in the Hitchin component on June 17, 2014 at the Workshop on the Geometry and Physics of Moduli Spaces in Miraflores de la Sierra.

Slides from my talk on asymptotics of certain families of Higgs bundles on April 06, 2014 in Austin.

Harmonic Reductions of Structure . (Incomplete, updated 2-9-14)

This is a document I wrote to help me learn about harmonic maps and harmonic reductions of structure; it is the result of a reading course I did with Pierre Albin in the Fall of 2013 . The goal was to understand how the different notions of harmonicity of a metric on a flat bundle are equivalent. One notion arises from thinking of the metric as an equivariant harmonic map from the universal cover to the symmetric space; the other comes from thinking of the metric as a reduction of structure group satisfying the harmonic bundle equations.

Semisimple Lie Groups .

These are notes from a talk I gave on semisimple Lie groups and Lie algebras at a Workshop on Higher Teichmuller-Thurston Theory.

#### Workshops

Here is the information on the Compactifying moduli spaces of representations workshop June 10-18, 2017.

Here is the information on Steve Bradlow's 60th Birthday conference Geometry and Physics of Augmented bundles May 5-7, 2017.

Here is the information on the Sp(4,R) Workshop January 10-18, 2016.

Here is the information on the Higgs bundles and harmonic map workshop January 3-11, 2015.

## Teaching

#### Current Teaching:

Math 431, Geometry for computer applications

#### Past Teaching:

- Spring 2016 (@UIUC): Math 241 (Calc III) TA
- Fall 2013 (@UIUC): Math 221 (Calc I) TA Merit section DD2
- Fall 2012 (@UIUC): Math 241 (Calc III) TA
- Spring 2012 (@UIUC): Math 181 (It's Mathematical World)
- Fall 2011 (@UIUC): Math 231 (Calc II) TA
- Spring 2011 (@UIUC): Math 231 (Calc II) TA