VIRTUAL ONLY: Computational Aspects of Discrete Subgroups of Lie Groups
Institute for Computational and Experimental Research in Mathematics (ICERM)
June 14, 2021  June 18, 2021
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Monday, June 14, 2021

9:15  9:30 am EDTWelcomeWelcome  Virtual
 Brendan Hassett, ICERM/Brown University

9:30  10:15 am EDTComputing with hyperbolic structures in dimension 3Seminar  Virtual
 Speaker
 Nathan Dunfield, University of Illinois, UrbanaChampaign
 Session Chair
 Richard Schwartz, Brown University
Abstract
I will discuss the theoretical and practical aspects of working with hyperbolic 3manifolds computationally, illustrating the topic by extensive realtime 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 EDTBreakCoffee Break  Virtual

10:45  11:30 am EDTComputing with hyperbolic structures in dimension 3Seminar  Virtual
 Speaker
 Nathan Dunfield, University of Illinois, UrbanaChampaign
 Session Chair
 Richard Schwartz, Brown University
Abstract
I will discuss the theoretical and practical aspects of working with hyperbolic 3manifolds computationally, illustrating the topic by extensive realtime 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 EDTNonarithmetic 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 nonarithmetic 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 adhoc software). I will sketch an alternative proof via orbifold uniformization, which no longer relies on the computer.

12:30  1:30 pm EDTLunch/Free TimeVirtual

1:30  2:00 pm EDTSupramaximal Representations of Planar Surface GroupsVirtual
 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 4holed 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 EDTBreakCoffee Break  Virtual

2:30  3:00 pm EDTNecklace Theory and Maximal cusps of hyperbolic 3manifoldsVirtual
 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 3manifold with a maximal cusp of volume at most 2.62 then it is obtained by filling one of 16 explicit 2 or 3cusped hyperbolic 3manifolds. As an application, with more rigorous computer assistance, we (with Tom Crawford) show that the figure8 knot complement is the unique 1cusped hyperbolic 3manifold with nine or more non hyperbolic fillings. 
3:15  3:45 pm EDTGraph embeddings in symmetric spacesVirtual
 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 realworld 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 EDTGathertown ReceptionWelcome  Virtual
Tuesday, June 15, 2021

9:15  9:45 am EDTCoffee BreakVirtual

9:45  10:30 am EDTPractical computations with finitely presented groups.Seminar  Virtual
 Speaker
 Sarah Rees, University of Newcastle
 Session Chair
 Olga Kharlampovich, Hunter College, CUNY
Abstract
The topics to be discussed are: (1) Techniques associated with coset enumeration and subgroup presentations (including ToddCoxeter and ReidemeisterSchreier). (2) Algorithms associated with abelian, nilpotent and polycyclic groups, and with collection. (3) Techniques associated with rewriting, in particular the KnuthBendix process, and computation and use of automatic and coset automatic structures. (4) Testing for hyperbolicity.

10:45  11:00 am EDTBreakCoffee Break  Virtual

11:00  11:45 am EDTPractical computations with finitely presented groups.Seminar  Virtual
 Speaker
 Sarah Rees, University of Newcastle
 Session Chair
 Olga Kharlampovich, Hunter College, CUNY
Abstract
The topics to be discussed are: (1) Techniques associated with coset enumeration and subgroup presentations (including ToddCoxeter and ReidemeisterSchreier). (2) Algorithms associated with abelian, nilpotent and polycyclic groups, and with collection. (3) Techniques associated with rewriting, in particular the KnuthBendix process, and computation and use of automatic and coset automatic structures. (4) Testing for hyperbolicity.

12:00  12:30 pm EDTArithmetic and rigidity beyond lattices: Examples from hyperbolic geometryVirtual
 Speaker
 Curtis McMullen, Harvard University
 Session Chair
 Olga Kharlampovich, Hunter College, CUNY
Abstract
We will discuss new results and computational illustrations of (i) arithmetic aspects of nonarithmetic triangle groups in SL_2(R) and (ii) Ratner rigidity, and its failure, for planes in hyperbolic 3manifolds of infinite volume.

12:45  1:45 pm EDTSoftware TutorialTutorial  Virtual
 Speaker
 Marc Culler, University of Illinois at Chicago
 Session Chair
 Michael Kapovich, UC Davis

1:45  2:15 pm EDTSoftware TutorialTutorial  Virtual
 Speaker
 Marc Culler, University of Illinois at Chicago
 Session Chair
 Michael Kapovich, UC Davis

2:30  2:45 pm EDTBreakCoffee Break  Virtual

2:45  3:15 pm EDTWord problems and finite state automataVirtual
 Speaker
 Susan Hermiller, University of Nebraska
 Session Chair
 Michael Kapovich, UC Davis
Abstract
In this talk I will discuss several ways to solve the word problem for groups by finite automata, including automatic and autostackable structures, along with geometric and topological views of these properties. We apply these algorithms to discrete subgroups of Lie groups and fundamental groups of 3manifolds. Based on joint projects with M. Brittenham and T. Susse, and with D. Holt, S. Rees, and T. Susse.

3:30  4:00 pm EDTCalculations in nilpotent groupsVirtual
 Speaker
 Moon Duchin, Tufts University
 Session Chair
 Michael Kapovich, UC Davis
Abstract
The discrete Heisenberg group can be handled in a very handson way, in matrix coordinates (say). An understanding of the largescale geometry can be leveraged to find structure in the numbers. I'll discuss the rationality of growth series for the Heisenberg group and indicate what is known and not known about other nilpotent groups.

4:00  5:00 pm EDTProblem SessionVirtual
 Session Chair
 Alex Kontorovich, Rutgers University
Wednesday, June 16, 2021

9:15  9:45 am EDTCoffee BreakVirtual

9:45  10:30 am EDTAlgorithmic problems for algebraic groupsSeminar  Virtual
 Speaker
 Willem de Graaf, University of Trento
 Session Chair
 Alla Detinko, University of Huddersfield
Abstract
We discuss a number of algorithmic problems, and possible solutions, for algebraic groups in characteristic 0. We will talk about some basic algorithms, that is, how to specify an algebraic group, computing the dimension, the Lie algebra, centralizers and normalizers, the closure of an orbit. Secondly we will look at the problem to compute the Zariski closure of a finitely generated matrix group. This also involves the related question of how to compute the smallest algebraic Lie algebra containing a given Lie algebra. A third topic is the problem how to find generators of arithmetic groups. These arise as the set of integral points of an algebraic group defined over Q. A famous theorem by Borel and HarishChandra asserts that these groups are finitely generated. But it remains a very hard problem to find a finite generating set. Algorithms exist for some classes of algebraic groups

10:45  11:00 am EDTBreakCoffee Break  Virtual

11:00  11:45 am EDTAlgorithmic problems for algebraic groupsSeminar  Virtual
 Speaker
 Willem de Graaf, University of Trento
 Session Chair
 Alla Detinko, University of Huddersfield
Abstract
We discuss a number of algorithmic problems, and possible solutions, for algebraic groups in characteristic 0. We will talk about some basic algorithms, that is, how to specify an algebraic group, computing the dimension, the Lie algebra, centralizers and normalizers, the closure of an orbit. Secondly we will look at the problem to compute the Zariski closure of a finitely generated matrix group. This also involves the related question of how to compute the smallest algebraic Lie algebra containing a given Lie algebra. A third topic is the problem how to find generators of arithmetic groups. These arise as the set of integral points of an algebraic group defined over Q. A famous theorem by Borel and HarishChandra asserts that these groups are finitely generated. But it remains a very hard problem to find a finite generating set. Algorithms exist for some classes of algebraic groups

12:00  12:30 pm EDTPractical computation with infinite linear groupsVirtual
 Speaker
 Dane Flannery, National University of Ireland
 Session Chair
 Alla Detinko, University of Huddersfield
Abstract
We survey some of the progress to date in an ongoing project to enable computation with linear groups defined over infinite domains. This includes computational realization of the finite approximation method, leading up to algorithms for arithmetic groups and beyond. This is joint work with Alla Detinko and Alexander Hulpke.

12:45  1:45 pm EDTLunch/Free TimeVirtual

1:45  2:45 pm EDTLightning TalksVirtual
 Session Chair
 Alex Kontorovich, Rutgers University

3:00  3:15 pm EDTBreakCoffee Break  Virtual

3:15  3:45 pm EDTCalculations in infinite matrix groups using congruence imagesTutorial  Virtual
 Speaker
 Alexander Hulpke, Colorado State University
 Session Chair
 Alla Detinko, University of Huddersfield
Abstract
I will describe, in theory and by demonstrating explicit calculations in the system GAP, algorithms that for investigating infinite matrix groups through suitable congruence images. In an interplay of algorithms for matrix groups and algorithms for finitely presented groups it is possible to prove arithmeticity of certain subgroups (and to prove infinite index if we are lucky).

4:00  4:30 pm EDTCalculations in infinite matrix groups using congruence imagesTutorial  Virtual
 Speaker
 Alexander Hulpke, Colorado State University
 Session Chair
 Alla Detinko, University of Huddersfield
Abstract
I will describe, in theory and by demonstrating explicit calculations in the system GAP, algorithms that for investigating infinite matrix groups through suitable congruence images. In an interplay of algorithms for matrix groups and algorithms for finitely presented groups it is possible to prove arithmeticity of certain subgroups (and to prove infinite index if we are lucky).
Thursday, June 17, 2021

9:15  9:45 am EDTCoffee BreakVirtual

9:45  10:30 am EDTFirst order sentences in random groupsVirtual
 Speaker
 Olga Kharlampovich, Hunter College, CUNY
 Session Chair
 Anna Felikson, Durham University
Abstract
We will use Gromov's density model of randomness and prove, in particular, the following result. Let G be ``the random group" of some fixed density d<1/16. Let f be a universal sentence in the language of groups. Then G almost surely satisfies f if and only if a nonabelian free group F satisfies f. These are joint results with R. Sklinos.

10:45  11:00 am EDTBreakCoffee Break  Virtual

11:00  11:45 am EDTGeometric algorithms for subgroups of Lie groupsSeminar  Virtual
 Speaker
 Michael Kapovich, UC Davis
 Session Chair
 Anna Felikson, Durham University
Abstract
The known geometric algorithms for discrete subgroups of $\mathrm{SL}(n,\R)$ come primarily in two forms, both requiring the subgroup to be ``geometrically nice.'' While the ultimate definition of ``niceness'' is, at this point, very much unclear, the known forms include (a) {\em the traditional geometric finiteness} using a finitelysided fundamental domain (using either an invariant Riemannian metric or Selberg's 2point invariant) in the associated symmetric space, (b) the relatively recent notion of {\em Anosov subgroups}. Both definitions allow for geometric localtoglobal principles, which, in turn, make computations with such discrete subgroups possible. The lectures will describe theses two concepts, the localtoglobal principles and the geometric algorithms.

12:00  12:30 pm EDTGeometric algorithms for subgroups of Lie groups.Seminar  Virtual
 Speaker
 Michael Kapovich, UC Davis
 Session Chair
 Anna Felikson, Durham University
Abstract
The known geometric algorithms for discrete subgroups of $\mathrm{SL}(n,\R)$ come primarily in two forms, both requiring the subgroup to be ``geometrically nice.'' While the ultimate definition of ``niceness'' is, at this point, very much unclear, the known forms include (a) {\em the traditional geometric finiteness} using a finitelysided fundamental domain (using either an invariant Riemannian metric or Selberg's 2point invariant) in the associated symmetric space, (b) the relatively recent notion of {\em Anosov subgroups}. Both definitions allow for geometric localtoglobal principles, which, in turn, make computations with such discrete subgroups possible. The lectures will describe theses two concepts, the localtoglobal principles and the geometric algorithms.

12:45  1:45 pm EDTLunch/Free TimeVirtual

1:45  2:15 pm EDTLie Theory in GAPTutorial  Virtual
 Speaker
 Willem de Graaf, University of Trento
 Session Chair
 William Goldman, University of Maryland
Abstract
We will start with an overview of the functionality for Lie algebras in GAP4 (ways to define a Lie algebra, semisimple Lie algebras and their representations, root systems, nilpotent orbits, real semisimple Lie algebras). Then we will look at some examples of research projects where this functionality has been used.

2:30  2:45 pm EDTBreakCoffee Break  Virtual

2:45  3:15 pm EDTComputability Models: Algebraic, Topological and GeometricVirtual
 Speaker
 Jane Gilman, Rutgers University
 Session Chair
 William Goldman, University of Maryland
Abstract
Can one translate a topological or geometric algorithm into a computable algorithm? We consider the GilmanMaskit PSL(2,R) twogenerator discreteness algorithm under the various models and review complexity bounds in the BSS machine and symbolic computation models. We show that Teichmuller space, T(0,3), and Riemann space, R(0,3) are BSS computable. More generally we discuss the issues for the algorithm with respect to bitcomputability. We discuss two models for bitcomputation, extended domain bitcomputability and two oracle upper and lower computability. These models are currently under development in joint work with Tsvietkova. If we add upper and lower computability oracles, the discreteness problem without parabolics (so that the corresponding quotient has no cusps) is semidecidable.

3:30  4:00 pm EDTPSL(2,C)representations of knot groups by knot diagramsVirtual
 Speaker
 Anastasiia Tsvietkova, RutgersNewark/IAS
 Session Chair
 William Goldman, University of Maryland
Abstract
We will discuss a new method of producing equations for representation and character varieties of the canonical component of a knot group into PSL(2,C). Unlike known methods, it does not involve any decomposition of the knot complement, and uses only a knot diagram. In many cases, it can be applied to an infinite family of knots at once. The idea goes back to computing the complete hyperbolic structure from a link diagram by Thistlethwaite and the speaker, but is generalized to yield the variety. This is joint work with Kate Petersen.
Friday, June 18, 2021

9:15  9:45 am EDTCoffee BreakVirtual

10:00  10:30 am EDTCentrality of the congruence subgroup kernelVirtual
 Speaker
 Tyakal Venkataramana, Tata Institute of Fundamental Research
 Session Chair
 Peter Sarnak, Institute for Advanced Study and Princeton University
Abstract
We give a new proof of an old result that the congruence subgroup kernel associated to a higher rank noncocompact arithmetic group is central in the arithmetic completion of the discrete group.

11:00  11:30 am EDTA cyclotomic family of thin hypergeometric monodromy groups in Sp(4)Virtual
 Speaker
 Simion Filip, University of Chicago
 Session Chair
 Peter Sarnak, Institute for Advanced Study and Princeton University
Abstract
The monodromy of differential equations is a rich source of subgroups of Lie groups. I will describe joint work with Charles Fougeron exhibiting an infinite family of discrete groups in Sp(4), obtained as monodromies of certain hypergeometric differential equations. Besides discreteness, the groups have a number of additional interesting properties. The family was discovered experimentally, but our proof does not rely on computers.

11:45 am  12:15 pm EDTManifolds with nonintegral trace.Virtual
 Speaker
 Alan Reid, Rice University
 Session Chair
 Peter Sarnak, Institute for Advanced Study and Princeton University
Abstract
A basic consequence of MostowPrasad Rigidity is that if M=H^3/G is an orientable hyperbolic 3manifold of finite volume, then the traces of the elements in $\G$ are algebraic numbers. Say that M has nonintegral trace if G contains an element whose trace is an algebraic noninteger. This talk will consider manifolds with nonintegral trace and show for example, that there are infinitely many nonhomeomorphic hyperbolic knot complements S^3\ K_i with nonintegral trace.

12:30  1:45 pm EDTProblem SessionVirtual

1:45  2:15 pm EDTVerified Length SpectrumVirtual
 Speaker
 Maria Trnkova, UC Davis
 Session Chair
 Susan Hermiller, University of Nebraska
Abstract
A computer program "SnapPea" and its descendant “SnapPy” compute many invariants of a hyperbolic 3manifold M. In this talk we will discuss verified computations of geodesics length as a product of matrices and will mention some applications when it is crucial to know the precise length spectrum up to some cut off.

2:30  2:45 pm EDTBreakCoffee Break  Virtual

2:45  3:15 pm EDTMarkoff triples and cryptographyVirtual
 Speaker
 Elena Fuchs, University of California, Davis
 Session Chair
 Susan Hermiller, University of Nebraska
Abstract
In this talk, I will explore various questions arising from considering the modp Markoff graphs as candidates for a hash function. As I discuss several potential path finding algorithms in these graphs, several questions about lifts of mod p solutions to the Markoff equation will come up as well. This is joint work with K. Lauter, M. Litman, and A. Tran.

3:30  4:00 pm EDTBilliards in orthoschemes and pictures of a group cocycle.Virtual
 Speaker
 Richard Schwartz, Brown University
 Session Chair
 Susan Hermiller, University of Nebraska
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