## 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

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 $2$.

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 $\mathsf{Sp}(4,\mathbb{R})$ 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 $\mathsf{Sp}(4,\mathbb{R})/\mathsf{𝖴}(2)$ 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 $\mathsf{SL}(n,\mathbb{R})$-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 $\mathsf{SL}(n,\mathbb{R})/\mathsf{SO}(n,\mathbb{R})$ 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: Parameterizing connected components of $\mathsf{SO}(p, p+1)$-Higgs bundles

Abstract: In this talk I will discuss a parameterization of $n(2g-2)$ connected components of the $\mathsf{SO}_0(n,n+1)$-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 $\mathsf{PSL}(2,\mathbb{R})=\mathsf{SO}_0(1,2)$-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 $\mathsf{SO}_0(p,p+1)$-Higgs bundles moduli space and discuss the Zariski closure of these the representations in each connected component.

Title: Holomorphic differentials and the group $\mathsf{SO}_0(n,n+1)$

Abstract: In this talk I will discuss a parameterization of n(2g-2) connected components of the \mathsf{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 $\mathsf{PSL}(2,R)=\mathsf{SO}_0(1,2)$ Higgs bundle moduli space by holomorphic quadratic differentials twisted
by an effective divisor.

Title: Maximal $\mathsf{SO}_0(2,3)$ representations and beyond

Abstract:
For a closed surface S of genus g > 1, the space of maximal $\mathsf{PSp}(4,\mathbb{R})$ 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 $\mathsf{PSp}(2n, \mathbb{R} )$
representations which generalize the Gothen representations. However, motivated by the
isomorphism $PSp(4, \mathbb{R})=\mathsf{SO}_0(2,3)$, we will use a Higgs bundle description of the Gothen
components to show that the Gothen representations are an $\mathsf{SO}_0(n,n+1)$ phenomenon.

Title: Maximal $\mathsf{SO}_0(2,3)$ surface group representations and Labourie's conjecture

Abstract: The nonabelian Hodge correspondence proveides a homeomorphism between the character variety of surface group representations into a real Lie group $G$ and the moduli space of $G$-Higgs bundles. This homoeomorphism however breaks the natural mapping class group action on the character variety. Generalizing techniques and a conjecture of Labourie, we restore the mapping class group symmetry for all maximal $\mathsf{SO}_0(2,3)=\mathsf{PSp}(4,\mathbb{R})$ surface group representations. More precisely we prove that for each maximal $\mathsf{SO}_0(2,3)$ representation, there is a unique conformal structure in which the corresponding equivariant harmonic map to the symmetric space is a conformal immersion, or, equivalently, a minimal immersion. This is done by exploiting finite order fixed point properties of the associated maximal Higgs bundles.

Title: A mapping class group invariant parameterization of maximal $\mathsf{Sp}(4,\mathbb{R})$ surface group representations

Abstract: Let $S$ be a closed surface of genus $g\geq 2$, and consider the moduli space of representations $\rho:\pi_1(S)\rightarrow Sp(4,\mathbb{R}).$ There is an invariant $\tau\in\mathbb{Z},$ called the Toledo invariant, which satisfies a Milnor-Wood inequality $|\tau|\leq 2g-2,$ and helps to distinguish connected components. Representations with maximal Toledo invariant have many geometrically interesting properties, for instance, they are all discrete and faithful. In this talk, we will give a mapping class group invariant parameterization of the $2g-3$ special connected components of the maximal $Sp(4,\mathbb{R})$ representations. Our main tool is Higgs bundles. However, to utilize Higgs bundle techniques, one has to fix a conformal structure of the surface $S,$ hence breaking the mapping class group symmetry. To restore the symmetry, we associate a unique `preferred' conformal structure to each such representation. This is done by exploiting the relationship between the associated Higgs bundles and minimal surfaces.

Title: A mapping class group invariant parameterization of maximal $\mathsf{Sp}(4,\mathbb{R})$ surface group representations

Abstract: Let $S$ be a closed surface of genus at least $2$, and consider the moduli space of representations $\rho:\pi_1(S)\rightarrow\mathsf{Sp}(4,\mathbb{R})$. There is an invariant $\tau\in\mathbb{Z}$, called the Toledo invariant, which helps to distinguish connected components. The Toledo invariant satisfies a Milnor-Wood inequality $|\tau|\leq 2g-2$. Representations with maximal Toledo invariant have many geometrically interesting properties, for instance, they are all discrete and faithful. In this talk, we will give a mapping class group invariant parameterization of all smooth connected components of the maximal $\mathsf{Sp}(4,\mathbb{R})$ representations. Our main tool is Higgs bundles. However, to utilize Higgs bundle techniques, one has to fix a conformal structure of the surface $S$, hence breaking the mapping class group symmetry. To restore the symmetry, we associate a unique `preferred' conformal structure to each such representation. This is done by exploiting the relationship between the associated Higgs bundles and minimal surfaces.

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

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 $\mathsf{Sp}(4,\mathbb{R})$ Workshop January 10-18, 2016.

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

## Teaching

#### Current Teaching:

I am not teaching this semester

#### Past Teaching:

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