## About Me

I am currently an NSF Postdoctoral Fellow at University of Maryland at College Park. I will be at the MSRI for the Fall 2019 and Spring 2020 semseters and will be an Assistant Professor at University of California Riverside starting Fall 2020. 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, surface groups representations, variations of Hodge structure, opers, Anosov representations, higher Teichmüller theory, homogeneous geometries and harmonic maps.

### Papers

Survey article roughly based on Summer 2018 mini-course given at UIC.

SIGMA 15 (2019), 010, 32 pages (Special Issue on Geometry and Physics of Hitchin Systems).

Abstract:

This is a survey article whose main goal is to explain how many components of the character variety of a closed surface are either deformation spaces of representations into the maximal compact subgroup or deformation spaces of certain Fuchsian representations. This latter family is of particular interest and is related to the field of higher Teichmüller theory. Our main tool is the theory of Higgs bundles. We try to develop the general theory of Higgs bundles for real groups and indicate where subtleties arise. However, the main emphasis is placed on concrete examples which are our motivating objects. In particular, we do not prove any of the foundational theorems, rather we state them and show how they can be used to prove interesting statements about components of the character variety. We have also not spent any time developing the tools (harmonic maps) which define the bridge between Higgs bundles and the character variety. For this side of the story we refer the reader to the survey article of Q. Li [arXiv:1809.05747]

coauthors: Richard Wentworth

Accepted for publication in Advances in Mathematics.

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.

coauthors: Marta Aparicio-Arroyo, Steven Bradlow, Oscar Garcia-Prada, Peter Gothen and Andre Oliveira

Accepted for publication in Inventiones mathematicae.

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).

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.

Accepted for publication in Annales scientifiques de l'ENS.

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

Accepted for publication in Geometry and Topology.

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

Accepted for publication in Duke Mathematical Journal.

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.

Geometry and Physics: a Festschrift in honour of Nigel Hitchin, Oxford University Press.

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.

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.

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.

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.

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

January 28, 2019
Colloquium at Rutgers University Newark.

January 23, 2019
Colloquium at North Carolina State University.

January, 19, 2019
Joint Mathematics Meetings AMS Special Session on Geometry of Representation Spaces in Baltimore, Maryland.

January 14, 2019
Colloquium at Iowa State University.

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

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

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)

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

February 24- March 3, 2018
Workshop on relative character varieties and parabolic Higgs bundles.

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

August, 6-12, 2017
GEAR Retreat University of Stanford.
(here is the video of the talk)

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

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.

## Videos and Notes

Video from my talk at the Simons Center February 5, 2019.

Video from my talk at Banff International Research Station at Casa Matemática Oaxaca Mexico, July 5, 2018.

Video from my talk at the GEAR Retreat at University of Stanford, August 12, 2017.

Video from my talk in Geometry, Groups, and Dynamics/GEAR seminar, May 4, 2016

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

Slides from my talk at New Trends in Higgs Bundle Theory at the ICMAT, Madrid Spain.

Slides from my talk at the Workshop on Higgs Bundles and Character Varieties.

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 (The talk was part chalk talk part slides.)

Slides from my talk at the Workshop on the Geometry and Physics of Moduli Spaces in Miraflores de la Sierra, Spain.

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

Notes on 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.

Notes on 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.

#### 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 430 Euclidean and Non-Euclidean geometries (all class information is on ELMS)

#### Past Teaching:

- Fall 2017 at University of Maryland: Math 431 (Geometry for computer graphics applications)
- 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