About Me



I am currently a sixth year Math PhD Student at University of Illinois (UIUC), my advisor is Steve Bradlow.

Here is my CV .

Contact

E-mail: collier3 AT illinois DOT edu

Office: 178 Altgeld Hall

Mailing Address:
Department of Mathematics
University of Illinois, Urbana-Champaign
1409 W. Green Street
Urbana, IL 61801

Research


My research is in differential and algebraic geometry. I am particularly interested in Higgs bundles, nonabelian Hodge theory and representation varieties.


Papers



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 (accepted)
            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 $\mathsf{Sp}(4,\mathbb{R})$ 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 $\mathsf{Sp}(4,\mathbb{R})$ 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.



Asymptotics of certain families of Higgs bundles in the Hitchin component
           co-authors: Qiongling Li
            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.



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


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



Past Talks

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


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.


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


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.


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.



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



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



July 13, 2015 Geometry seminar at ICMAT.


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.


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.


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.


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


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.

Speakers were alotted 2-3 hours per talk, my topic is Integrable systems and harmonic maps.


December 1, 2014 Geometry seminar at ICMAT.


Title: Applications of fixed points in the moduli space of Higgs bundles

Abstract: Let $\Sigma$ be a Riemann surface, the nonabelian Hodge correspondence gives a homeomorphism between the moduli space of $G$-Higgs bundle over $\Sigma$ and the moduli space of representations $\pi_1(\Sigma)\rightarrow G.$ In this talk we will focus on Higgs bundles that are fixed points of a roots of unity action. Using special metric splitting properties of this special class of Higgs bundles, we will discuss applications to studying Hitchin representations and certain components of maximal $Sp(4,\mathbb{R})$ representations.



Title: Fixed points in the moduli space of $G$-Higgs bundles

Abstract: For $G$ a real semisimple Lie group, we will study fixed points of a roots of unity action on the moduli space of $G$-Higgs bundles. For certain real groups we will classify these fixed points and give families of examples in the Hitchin component for any split real form and also in certain non-Hitchin maximal components for $\mathsf{Sp}(4,\mathbb{R}).$ If time permits, the relation between fixed points and the harmonic metric solving the Hitchin equations will be discussed.


June 16-20, 2014 Workshop on the Geometry and Physics of Moduli spaces Miraflores de la Sierra (Madrid, Spain).


Title: Fixed points in the Higgs bundle moduli space.

Abstract: Let $(E,\phi)$ be a stable Higgs bundle, the motivating question of this talk is: When is the harmonic metric solving the Higgs bundle equations compatible with a holomorphic splitting of $E$? One well studied example of this occurs when $(E,\phi)$ is a fixed point of the $U(1)$ action. We will show that the harmonic metric also splits when $(E,\phi)$ is only assumed to be fixed by the kth roots of unity $\mathbb{Z}/k\mathbb{Z}⊂U(1)$. Using this metric splitting property, we will to study the asymptotics of the harmonic metrics and flat connections associated to certain families of Higgs bundles in the Hitchin component. The last part is based on recent joint work with Qiongling Li.


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


Title: Fixed points of roots of unity actions on the Higgs bundle moduli space

Abstract: For families of Higgs bundles in the Hitchin component that are fixed points of a roots of unity action, we deduce extra symmetries of the metric solving the Hitchin Equations. For such families, we scale the holomorphic differentials in the Higgs field by a real parameter and analyze the asymptotics of the harmonic metric along with the asymptotics of the corresponding flat connection. This is based on joint work with Qiongling Li.


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


Title: Asymptotics of certain families of Higgs bundles

Abstract: The goal of the talk is to explain the statements of some recent results that are joint with Qiongling Li. Through a guiding example, we will review some Higgs bundle basics and the nonabelian Hodge correspondence. Generalizing this example leads to special metric splitting properties of certain families of Higgs bundles in the Hitchin component. With this setup, we are able to understand the asymptotics of the harmonic metric solving the Higgs bundle equations.



Title: Compatibility of holomorphic and unitary decompositions of Higgs bundles

Abstract: Higgs bundles are holomorphic vector bundles with an auxiliary field, called a Higgs field, over a Riemann surface  $\Sigma.$ Through the nonabelian Hodge theorem the moduli space of polystable Higgs bundles is homeomorphic to the character variety of  $\Sigma.$ One direction of the homeomorphism is given by a Kobayashi-Hitchin correspondence relating polystable Higgs bundles to solutions of certain gauge theoretic equations which yield a special metric and a flat connection. In this talk we will review some Higgs bundle basics and examine when a holomorphic splitting is unitary with respect to the special metric.



Speakers were alotted 2-3 hours per talk, my topic was Semisimple Lie groups. More specifically I talked about Cartan decompositions, parabolic subgroups, associated symmetric spaces and their boundaries. The notes are linked below.



Current Projects



Asymptotics of certain families of maximal $\mathsf{Sp}(4,\mathbb{R})$-Higgs bundles (Joint with Qiongling Li).


For certain families of maximal $\mathsf{Sp}(4,\mathbb{R})$-Higgs bundles, we study the asymptotics of the corresponding family of representations and harmonic metrics.


Maxmial $\mathsf{SO}_0(2,3)$ surface group representations and Labourie's conjecture (Joint with Daniele Alessandrini )


We prove that for all maximal $\mathsf{SO}_0(2,3)$ surface group representations there is a unique conformal structure in which the associated harmonic metric is a minimal immersion.


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.


$\mathsf{Sp}(4,\mathbb{R})$ Workshop


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