Sam Ballas

About me

I am currently an associate professor in the Department of Mathematics at Florida State University. I also serve as the Director of Pure Mathematics, so feel free to reach out if you have questions about our pure math graduate program.

Previously I was an RTG visiting assistant professor at UC Santa Barbara. Before that I was a graduate student at The University of Texas at Austin under the supervision of Alan Reid.

If you want to learn more about my research check out my short Research Statement or CV. You may also be interested in this artice featuring my research written for school aged children. This article was produced by Futurum, a magazine and online platform aimed at inspiring young people to follow a career in the sciences, research and technology.

Research

My research is primarily concerned with understanding various types of geometric structures on manifolds. Studying these structures typically involves the interplay of techniques from algebra, geometry, and topology.

My research was previously supported by the National Science Foundation (grant DMS-1709097)

Papers

1. On the existence of Parseval frames for vector bundles
   joint with Tom Needham and Clayton Shonkwiler, to appear in Trans. of AMS.
2. Tame and relatively elliptic CP1-structures on the thrice-punctured sphere
   joint with Phil Bowers, Alex Casella and Lorenzo Ruffoni, Alg. Geom. Top. 24-8 (2024), 4589--4650.
3. The moduli space of marked generalized cusps in real projective manifolds
   joint with Daryl Cooper and Arielle Leitner, Conf. Geom. & Dyn. vol 26 (2022) 111-164
4. Gluing equations for real projective structures on 3-manifolds
   joint with Alex Casella, Geom. Dedicata 215 (2021), 69-131.
5. Thin subgroups isomorphic to Gromov--Piatetski-Shapiro lattices
   Pacific J. Math. 309 (2020), no 2, 257--266.
Constructing thin subgroups of SL(n+1,R) via bending
joint with Darren Long, Algebr. Geom. Topol. 20-4 (2020), 2071--2093
Constructing convex projective 3-manifolds with generalized cusps
J. London Math. Soc. 103-4 (2021), 1276--1313
Supplemental Sage notebooks related to this work can be found here
Generalized cusps in real projective manifolds: classification
joint with Daryl Cooper and Arielle Leitner, J. Topol. 13-4 (2020) 1455--1496
Convex projective bendings of hyperbolic manifolds
joint with Ludovic Marquis, Groups, Geom, & Dynam 14 (2020), no. 2, 653-688.
Convex projective structures on non-hyperbolic three-manifolds
joint with J. Danciger and G.-S. Lee, Geom. & Topol. 22 (2018), no. 3, 1593–1646
Constructing thin subgroups commensurable with the figure-eight knot group
joint with Darren Long, Algebr. Geom. Topol. 15 (2015), no. 5, 3011–3024
Finite volume properly convex deformations of the figure-eight knot
Geom. Dedicata 178 (2015), 49–73
Deformations of non-compact, projective manifolds
Algebr. Geom. Topol. 14 (2014), no. 5, 2595–2625
Flexiblity and rigidity of three-dimensional convex projective structures
Ph.D. thesis, University of Texas at Austin, (2013)

Work in progress

Copies may be available upon request

1. Properly convex Dehn filling
   joint with Jeff Danciger, Gye-Seon Lee, and Ludovic Marquis
2. Neumann-Zagier relations for projective gluing equations
   joint with Alex Casella

Notes

1. A Geometric Description of (P)SL(4,R) Hitchin Representations
   notes from workshop on higher teichmuller theory

Teaching

Current Courses

• Spring 2025: MAC 2313 Calculus III

Older Courses

• Fall 2024: MGF 3301 Introduction to Advanced Math
• Spring 2024: MAS 5932 Arithmetic Groups
• Fall 2023: Innovative Teaching
• Summer 2023: MAA 4402 Complex Variables
Spring 2023: MAC2313 Calculus III
Spring 2023: MTG5327 Topology II
Fall 2022: MTG4212 College Geometry
Fall 2022: MTG5326 Topology I
Spring 2021: MTG5932 Differential Geometry of Principal Bundles
Fall 2021: MAC2313 Calculus III
Spring 2021: MTG5932 Geometric Structures on Manifolds
Fall 2020: MAD3105 Discrete Math II
Spring 2020: MAC2313 Calculus II
Fall 2019: MAC2313 Calculus III
Spring 2019: MAC2312 Calculus II
Spring 2019: MTG5327 Topology II
Fall 2018: MAC2312 Calculus III
Fall 2018: MTG5326 Topology I
Spring 2018: MAP2302 Oridinary Differential Equations
Fall 2017: On Leave
Spring 2017: MAC2312 Calculus II
Fall 2016: MAC2313 Calculus III
Spring 2016: Math 260P Projective Structures on Manifolds (at UCSB)

Activities

Upcoming Travel

Invited Talks

1. Frame theory on vector bundles (Slides, Video)
   (CodEx Seminar, Online Seminar, 2/27/2024)
2. A convex projective Dehn filling theorem (Slides)
   (AMS Southwest Sectional Meeting, GA Tech , 3/18/2023)
3. Complex projective structures on surfaces (Slides)
   (University of Florida, Colloquium, 2/11/2022)
4. Properly convex Dehn fillings (Slides)
   (University of Virginia, Topology Seminar, 10/26/2021)
5. Triangulations in low dimensional geometry (Slides)
   (Cal Poly, Colloquium, 2/19/2021)
Gluing equations in low dimensional geometry (Slides)
(Oklahoma State Topology Seminar, 11/4/2020)
Gluing equations for projective structures on 3-manifolds (Slides, Video)
(Trends in Low Dimensional Topology, Online Seminar, 6/23/2020)
Gluing equations for projective structures on 3-manifolds

Cancelled due to COVID-19
(Oklahoma State Topology Seminar, 4/13/2020)
Gluing equations for projective structures on 3-manifolds

Cancelled due to COVID-19
AMS Sectional Meeting, Charlottesville, VA 3/14/2020)
Gluing equations for projective structures on 3-manifolds
(FRG Workshop, University of Michigan, 12/7/2019)
Gluing equations for projective structures on 3-manifolds
(Informal Topology Seminar, Stanford University, 3/19/2019)
Thin subgroups of SL(n+1,R) via bending
(Topology Seminar, Rice University, 2/18/2019)
Thin groups from projective geometry
(Thin groups in number theory, geometry, and topology, Rice University, 5/15/2018)
Convex projective structures on low dimensions
(University of Georgia, Topology Seminar, 4/9/2018)
Generalized Cusps in Convex Projective Manifolds (Slides,Video)
(GEAR Retreat, Stanford Univeristy, 8/7/2017)
Generalized Cusps in Convex Projective Manifolds
(Workshop on Dynamics on Character Varieties, University of Rennes I, 6/29/2017)
Rank 1 Deformations of Non-cocompact Hyperbolic Lattices
(University of Texas at Austin, Topology Seminar, 3/6/2017)
Geometric Structures on Manifolds
(UF/FSU Topology and Dynamics Meeting, University of Florida, 2/25/2017)
Classification of Generalized Cusps (Slides)
(Joint Mathematics Meetings, Special Session on Group Actions and Geometric Structures, Atlanta, GA 1/7/2017 )
Classification of Generalized Cusps
(California Institute of Technology, Geometry and Topology Seminar, 4/22/2016)
Geometric Structures on Manifolds (Slides)
(Florida State University, Mathematics Colloquium, 1/15/2016)
Deforming Properly Convex 3-manifolds (Slides)
(Florida State University, Topology Seminar, 1/14/2016)
Convex Projective Structures on Non-hyperbolic 3-manifolds (Slides)
(Workshop on Interactions in Higher Teichmuller Theory, Heidelberg, Germany)
A Minicourse on Deformations of Convex Projective Structures
(Workshop on Geometric Structures and Representation Varieties, KIAS, Seoul, Korea 11/10-11/14/2014)
The Structure of Properly Convex Manifolds (Slides)
(Geometry, Topology and Physics at Pitt, University of Pittsburgh, 5/14/2014)
The Structure of Properly Convex Manifolds (Slides)
(RTG Workshop on Geometric Structures and Discrete Groups, UT Austin, 5/3/2014)
A Geometric Description of (P)SL(4,R) Hitchin Representations (Notes)
(Workshop on Higher Teichmuller-Thurston Theory, Northport, Maine, 6/29/2013)
Convex Projective Structures on Non-compact 3-manifolds
(California Institute of Technology, Geometry and Topology Seminar, 5/31/2013)
Convex Projective Deformations of the Figure-8 Knot Complement (Slides)
(AMS Special Session on Algebraic and Geometric Structures of 3-manifolds, Boston College, 4/6/2013)
Convex Projective Deformations of Non-Compact Manifolds
(Arizona State University, Differential Geometry Seminar, 1/25/2013)
Convex Projective Deformations of Non-Compact Manifolds
(Purdue University, Special Geometric Analysis Seminar, 1/22/2013)
Convex Projective Structures on 3-manifolds
(University of California at Santa Barbara, Topology Seminar, 10/23/2012)
Twisted Cohomology and Deformation Theory (Slides, Video)
(University of Illinois at Urbana-Champaign, GEAR Junior Retreat, 7/24/2012)

Things I'm Organizing/Have Organized

1. AMS Special Session
   on Recent Advances in Geometry and Topology
   (Florida State University, 3/23-3/24/2024)
2. UF/FSU Topology and Geometry Meeting
   (Florida State University, 11/3-11/4/2023)
3. AMS Special Session
   on Geometric Structures on Manifolds
   (University of Florida, 11/2-11/3/2019)
4. UF/FSU Topology and Geometry Meeting
   (Florida State University, 2/8-2/9/2019)
5. GEAR Junior Retreat 2017
   (Stanford University, 7/31-8/5/2017)

In previous semesters I have organized the topology and geometry seminar.