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

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, surface groups representations, variations of Hodge structure, opers, Anosov representations, higher Teichmüller theory, homogeneous geometries and harmonic maps.

### Papers

Conformal limits and the Białynicki-Birula stratification of the space of λ-connections
coauthors: Richard Wentworth
Submitted for publication
Preprint posted: August, 2018.

Abstract:

The Białynicki-Birula decomposition of the space of λ-connections restricts to the Morse stratification on the moduli space of Higgs bundles and to the partial oper stratification on the de Rham moduli space of holomorphic connections. For both the Morse and partial oper stratifications, every stratum is a holomorphic Lagrangian fibration over a component of the space of complex variations of Hodge structure. In this paper, we generalize known results for the Hitchin section and the space of opers to arbitrary strata. These include the following: an explicit parametrization of the fibers as half-dimensional affine spaces, a biholomorphic identification of the fibers via the "h-bar conformal limit'' of Gaiotto, and a proof that the fibers of the Morse and partial oper stratifications are transverse at complex variations of Hodge structure.

SO(p,q)-Higgs bundles and higher Teichmüller components
coauthors: Marta Aparicio-Arroyo, Steven Bradlow , Oscar Garcia-Prada , Peter Gothen and Andre Oliveira
Submitted for publication
Preprint posted: February, 2018.

Abstract:

Some connected components of a moduli space are mundane in the sense that they are distinguished only by obvious topological invariants or have no special characteristics. Others are more alluring and unusual either because they are not detected by primary invariants, or because they have special geometric significance, or both. In this paper we describe new examples of such `exotic' components in moduli spaces of SO(p,q)-Higgs bundles on closed Riemann surfaces or, equivalently, moduli spaces of surface group representations into the Lie group SO(p,q). Furthermore, we discuss how these exotic components are related to the notion of positive Anosov representations recently developed by Guichard and Wienhard. We also provide a complete count of the connected components of these moduli spaces (except for SO(2,q), with q>3).

Exotic components of SO(p,q) surface group representations, and their Higgs bundle avatars.
coauthors: Marta Aparicio-Arroyo, Steven Bradlow , Oscar Garcia-Prada , Peter Gothen and Andre Oliveira
Comptes Rendus Mathematique Volume 356, Issue 6, June 2018, Pages 666-673.

Abstract:

For semisimple Lie groups, moduli spaces of Higgs bundles on a Riemann surface correspond to representation varieties for the surface fundamental group. In many cases, natural topological invariants label connected components of the moduli spaces. Hitchin representations into split real forms, and maximal representations into Hermitian Lie groups, are the only previously know cases where natural invariants do not fully distinguish connected components. In this note we announce the existence of new such exotic components in the moduli spaces for the groups SO(p,q) with 2< p < q. These groups lie outside formerly know classes of groups associated with exotic components.

SO(n,n+1)-surface group representations and their Higgs bundles
Submitted for publication
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.

The geometry of maximal components of the PSp(4,R) character variety
coauthors: Daniele Alessandrini
Accepted for publication in Geometry and Topology
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.

The geometry of maximal representations of surface groups into SO(2,n)
coauthors: Nicolas Tholozan and Jérémy Toulisse
Submitted for publication
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.

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).

Parallel transport on principal bundles over stacks
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

Maximal Sp(4,R) surface group representations, minimal surfaces and cyclic surfaces
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.

Asymptotics of Higgs bundles in the Hitchin component
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.

A Symplectic Proof of a Theorem of Franks
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

December, 17-21, 2018 Current trends in Hitchin systems at La Plata & Buenos Aires, Argentina.

### Past Talks

November 12-16, 2018 New Trends in Higgs Bundle Theory at ICMAT, Madrid Spain. (here are the slides )

Title: Generalized opers, conformal limits and maximal variations of Hodge structure

Abstract: For a complex semisimple Lie group G, the space of opers is an affine holomorphic Lagrangian submanifold of the moduli space of flat G connections which is biholomorphic to the base of the Hitchin fibration. These objects are related to the interaction of certain holomorphic connections with structure group reductions to a Borel subgroup. In this talk we will introduce a notion of opers for other parabolic subgroups and parameterize these objects. Furthermore, we will see that the so called conformal limit identifies these generalized opers with the stable manifold of certain C^*-fixed points in the Higgs bundle moduli space.

October 22-27, 2018 Harmonic maps and Higgs theory - recent developments at National Research University Higher School of Economics Moscow, Russia.

August 24-26 , 2018 Workshop on Teichmuller theory and related topics at Tsinghua University, Beijing China.

July 1-6, 2018 Workshop on Higgs bundles and harmonic maps of Riemann surfaces at Banff International Research Station for Mathematical Innovation and Discovery at Casa Matemática Oaxaca Mexico. (here is the video of the talk)

Title: Relations between Higgs strata and deRham strata

Abstract: Both the Higgs bundle moduli space and the moduli space of flat connections have a natural stratification induced by a C* action. In both of these stratifications, each strata is a holomorphic fibration over a connected component of complex variations of Hodge structure. While the nonabelian Hodge correspondence provides a homeomorphism between Higgs bundles and flat connections, this homeomorphism does not preserve the respective strata. The closed strata on the Higgs bundle side is the image of the Hitchin section (the Hitchin component) and the closed strata in the space of flat connections is the space of opers. In this talk we will show how many of the relationships between opers and the Hitchin component extend to general strata. This is based on joint work with Richard Wentworth.

June 23-24, 2018 Mini course at RTG Workshop on the Geometry and Physics of Higgs bundles III at University of Illinois at Chicago.

March 14, 2018 Max-Dehn seminar at University of Utah.

Title: Special deformations of Fuchsian representations into SO(p,q)

Abstract: The space of discrete and faithful representations of a closed surface group into Isom(H^2)=SO(1,2) defines a connected component of the character variety. While deformations of such representations into SO(1,3) define interesting hyperbolic 3-manifolds, the space of such geometrically interesting representations is not closed in the SO(1,3) character variety. In fact, such representations can be continuously deformed to have compact Zariski closure. In this talk we will define analogous connected components of the SO(p,p+1) character variety and consider their deformations into the SO(p,p+2) character variety (and more generally SO(p,q)). Unlike the p=1 case, such representations cannot be deformed compact representations. This leads to a dichotomy for the connected components of the SO(p,q) character variety which we will relate to recent notions of positivity introduced by Guichard and Wienhard.

March, 27-31, 2017 Meeting on Complex Analytic Geometry, Tata institute of fundamental research Mumbai, India.

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.

More past talks

March, 20-24, 2017 School and Workshop on Geometry and Physics of Moduli Spaces, Indian Institute of Science in Bangalore, India.

September 5-16, 2016 Nigel Hitchin's 70th birthday conference , at ICMAT in Madrid, Spain.

March 17-20, 2016 Workshop of surface groups , at California Institute of Technology.

January 4-8, 2016 Workshop on holomorophic differentials , at the African Institute for Mathematical Sciences (AIM) in Cape Town, South Africa.

November 2-6, 2015 Higher Teichmüller theory and Higgs bundles: interactions and new trends , at Universität Heidelberg, Germany.

October 5-16, 2015 Workshop and conference on 50 years of the Narasimhan-Seshadri Theorem , at Chennai Mathematical Institute (CMI) in Chennai, India.

April, 18, 2017 Geometry seminar at University of Virginia

October, 28, 2016 Colloquium at Howard University

November 9, 2015 Geometry-Topology seminar , at Unviersity of Maryland.

October 20-24, 2015 Workshop on Geometric Structures, Hitchin Components and Representation Varieties , at the Korean Institute of Advanced Studies (KIAS) in Seoul, South Korea.

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 Teichmüller-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 Teichmüller-Thurston Theory.

## Teaching

#### Current Teaching:

I am not teaching this semester, but next semester I will teach Math 430 (Euclidean and Non-Euclidean geometries)

#### Past Teaching:

• Fall 2017 at University of Maryland:     Math 431 (Geometry for computer graphics applications) Instructor
• Spring 2016 at University of Illinois:     Math 241 (Calc III) TA
• Fall 2013 at University of Illinois:     Math 221 (Calc I) TA Merit section DD2
• Fall 2012 at University of Illinois:     Math 241 (Calc III) TA
• Spring 2012 at University of Illinois:     Math 181 (It's a Mathematical World)
• Fall 2011 at University of Illinois:     Math 231 (Calc II) TA
• Spring 2011 at University of Illinois:     Math 231 (Calc II) TA