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Organizing Committee
University of Huddersfield
UC Davis
Rutgers University
Institute for Advanced Study and Princeton University
Brown University
Abstract
This workshop is at the interface of algebra, geometry, and computer science. The major theme deals with a novel domain of computational algebra: the design, implementation, and application of algorithms based on matrix representations of groups and their geometric properties. The setting of linear Lie groups is amenable to calculation and modeling transformations, thus providing a bridge between algebra and its applications.
The main goal of the proposed workshop is to synergize and synthesize the independent strands in the area of computational aspects of discrete subgroups of Lie groups. We aim to facilitate solutions of theoretical problems by means of recent advances in computational algebra and additionally stimulate development of computational algebra oriented to other mathematical disciplines and applications.
The main goal of the proposed workshop is to synergize and synthesize the independent strands in the area of computational aspects of discrete subgroups of Lie groups. We aim to facilitate solutions of theoretical problems by means of recent advances in computational algebra and additionally stimulate development of computational algebra oriented to other mathematical disciplines and applications.
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Workshop Schedule
Monday, June 14, 2021
9:15 - 9:30 AM EDT
Welcome
Welcome - Virtual
Brendan Hassett, ICERM/Brown University
9:30 - 10:15 AM EDT
Computing with hyperbolic structures in dimension 3
Seminar - Virtual
Speaker
Nathan Dunfield, University of Illinois, Urbana-Champaign
Session Chair
Richard Schwartz, Brown University
Abstract
I will discuss the theoretical and practical aspects of working with hyperbolic 3-manifolds computationally, illustrating the topic by extensive real-time demonstrations of the program SnapPy. Highlights include rigorous algorithms for determining hyperbolicity, testing for isometry, and solving the word problem.
10:30 - 10:45 AM EDT
Break
Coffee Break - Virtual
10:45 - 11:30 AM EDT
Computing with hyperbolic structures in dimension 3
Seminar - Virtual
Speaker
Nathan Dunfield, University of Illinois, Urbana-Champaign
Session Chair
Richard Schwartz, Brown University
Abstract
I will discuss the theoretical and practical aspects of working with hyperbolic 3-manifolds computationally, illustrating the topic by extensive real-time demonstrations of the program SnapPy. Highlights include rigorous algorithms for determining hyperbolicity, testing for isometry, and solving the word problem.
11:45 AM - 12:15 PM EDT
Non-arithmetic lattices in PU(2,1)
Virtual
Speaker
Martin Deraux, Universite Grenoble Alpes
Session Chair
Richard Schwartz, Brown University
Abstract
In joint work with Parker and Paupert, we constructed several
non-arithmetic lattices in the isometry group of the complex
hyperbolic plane, by describing explicit generating sets and
constructing a fundamental domain (our original argument uses heavy
computation via ad-hoc software). I will sketch an alternative proof
via orbifold uniformization, which no longer relies on the computer.
12:30 - 1:30 PM EDT
Lunch/Free Time
Virtual
1:30 - 2:00 PM EDT
Supramaximal Representations of Planar Surface Groups
Virtual
Speaker
William Goldman, University of Maryland
Session Chair
Jane Gilman, Rutgers University
Abstract
Recently Deroin, Tholozan and Toulisse found connected components of
relative character varieties of surface group representations in a Hermitian
Lie grop G with remarkable properties.
For example, although the Lie groups are noncompact,
these components are compact.
In this way they behave more like relative character varieties for compact Lie
groups. (A relative character variety comprises equivalence classes of
homomorphisms of the fundamental group of a surface S,
where the holonomy around each boundary component of S is constrained
to a fixed conjugacy class in G.)
The first examples were found by Robert Benedetto and myself in an REU in the summer of 1992, and published in Experimental Mathematics in 1999. Here S is the 4-holed sphere and G = SL(2,R). Although computer visualization played an important role in the discovery of these unexpected compact components, computation was invisible in the final proof, and its subsequent extensions.
The first examples were found by Robert Benedetto and myself in an REU in the summer of 1992, and published in Experimental Mathematics in 1999. Here S is the 4-holed sphere and G = SL(2,R). Although computer visualization played an important role in the discovery of these unexpected compact components, computation was invisible in the final proof, and its subsequent extensions.
2:15 - 2:30 PM EDT
Break
Coffee Break - Virtual
2:30 - 3:00 PM EDT
Necklace Theory and Maximal cusps of hyperbolic 3-manifolds
Virtual
Speaker
David Gabai, Princeton University
Session Chair
Jane Gilman, Rutgers University
Abstract
(Joint work with Robert Haraway, Robert Meyerhoff, Nathaniel Thurston and Andrew Yarmola)
With rigorous computer assistance, both discrete and continuous, we show that if N is a complete finite volume hyperbolic 3-manifold with a maximal cusp of volume at most 2.62 then it is obtained by filling one of 16 explicit 2 or 3-cusped hyperbolic 3-manifolds. As an application, with more rigorous computer assistance, we (with Tom Crawford) show that the figure-8 knot complement is the unique 1-cusped hyperbolic 3-manifold with nine or more non hyperbolic fillings.
With rigorous computer assistance, both discrete and continuous, we show that if N is a complete finite volume hyperbolic 3-manifold with a maximal cusp of volume at most 2.62 then it is obtained by filling one of 16 explicit 2 or 3-cusped hyperbolic 3-manifolds. As an application, with more rigorous computer assistance, we (with Tom Crawford) show that the figure-8 knot complement is the unique 1-cusped hyperbolic 3-manifold with nine or more non hyperbolic fillings.
3:15 - 3:45 PM EDT
Graph embeddings in symmetric spaces
Virtual
Speaker
Anna Wienhard, Heidelberg University
Session Chair
Jane Gilman, Rutgers University
Abstract
Learning faithful graph representations has become a fundamental intermediary step in a wide range of machine learning applications. We propose the systematic use of symmetric spaces as embedding targets. We use Finsler metrics integrated in a Riemannian optimization scheme, that better adapt to dissimilar structures in the graph and develop a tool to analyze the embeddings based on the vector valued distance function in a symmetric space. For implementation, we choose Siegel spaces. We show that our approach outperforms competitive baselines for graph reconstruction tasks on various synthetic and real-world datasets and further demonstrate its applicability on two downstream tasks, recommender systems and node classification.
This is joint work with Federico Lopez, Beatrice Pozzetti, Michael Strube and Steve Trettel.
4:00 - 5:00 PM EDT
Gathertown Reception
Welcome - Virtual
Tuesday, June 15, 2021
9:15 - 9:45 AM EDT
Coffee Break
Virtual