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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.

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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.

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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.

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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.

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(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.

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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.

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The topics to be discussed are: (1) Techniques associated with coset enumeration and subgroup presentations (including Todd-Coxeter and Reidemeister-Schreier). (2) Algorithms associated with abelian, nilpotent and polycyclic groups, and with collection. (3) Techniques associated with rewriting, in particular the Knuth-Bendix process, and computation and use of automatic and coset automatic structures. (4) Testing for hyperbolicity.

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The topics to be discussed are: (1) Techniques associated with coset enumeration and subgroup presentations (including Todd-Coxeter and Reidemeister-Schreier). (2) Algorithms associated with abelian, nilpotent and polycyclic groups, and with collection. (3) Techniques associated with rewriting, in particular the Knuth-Bendix process, and computation and use of automatic and coset automatic structures. (4) Testing for hyperbolicity.

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We will discuss new results and computational illustrations of (i) arithmetic aspects of non-arithmetic triangle groups in SL_2(R) and (ii) Ratner rigidity, and its failure, for planes in hyperbolic 3-manifolds of infinite volume.

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An interactive demonstration of ways to acquire, use and contribute to SnapPy.

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An interactive demonstration of ways to acquire, use and contribute to SnapPy.

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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 3-manifolds. Based on joint projects with M. Brittenham and T. Susse, and with D. Holt, S. Rees, and T. Susse.

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The discrete Heisenberg group can be handled in a very hands-on way, in matrix coordinates (say). An understanding of the large-scale 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.

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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 Harish-Chandra 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

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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 Harish-Chandra 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

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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.

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Vertical arcs and the Markov Unicity Conjecture
Jonah Gaster, University of Wisconsin-Milwaukee
The Markov Unicity Conjecture concerns a correspondence on the modular torus that ties together geometry, topology, and number theory. I will describe a new geometric reformulation of the conjecture

Geometric and arithmetic properties of hyperbolic orbifolds, and the Vinberg algorithm
Nikolay Bogachev, Skoltech
We will discuss recent devolopements and progress in the theory of arithmetic hyperbolic reflection groups, software implementations of the Vinberg algorithm, as well as some other interesting connections between geometric and arithmetic properties of hyperbolic orbifolds. Based on a series of papers, including the recent ones with A. Kolpakov, and with M. Belolipetsky, A. Kolpakov, L. Slavich.

Computing reflection centralisers in hyperbolic reflection groups.
Aleksandr Kolpakov, University of Neuchâtel
In 1996 Brink proved that the non-reflective part of a reflection centraliser in a Coxeter group is a free group. Later on, in 2013, Allcock refined Brink's theorem, and provided a method for computing the Coxeter diagram of the reflective part. We implement Alcock's algorithm and perform some computations with it. This is related to the previous work together with N. Bogachev on (quasi-)arithmetic Coxeter facets of (quasi-)arithmetic hyperbolic Coxeter polytopes.

A quantified local-to-global principle for Anosov representations
Max Riestenberg, University of Texas at Austin
In 2014, Kapovich, Leeb and Porti gave several new characterizations of Anosov representations, including one where geodesics in the word hyperbolic group map to “Morse quasigeodesics” in the associated symmetric space. In analogy with the negative curvature setting, they prove a local-to-global principle for Morse quasigeodesics and describe an algorithm which can verify the Anosov property of a given representation in finite time (unless the representation is not Anosov, in which case the algorithm never terminates). However, some parts of their proof involve non-constructive compactness and limiting arguments, so their theorem does not explicitly quantify the size of the local neighborhoods one needs to examine to guarantee global Morse behavior. In my thesis I obtained explicit criteria for their local-to-global principle by producing new estimates in the symmetric space. This makes their algorithm for verifying the Anosov property effective, however, the balls in the Cayley graph one needs to examine are still prohibitively large. As an alternative application, I produce explicit perturbation neighborhoods of certain Anosov representations

Presentations for cusped arithmetic hyperbolic lattices’
Julien Paupert, Arizona State University
We present a general method to compute a presentation for any cusped arithmetic hyperbolic lattice Gamma, applying a classical result of Macbeath to a suitable Gamma-invariant horoball cover of the corresponding symmetric space. As applications we compute presentations for the Picard modular groups PU(2, 1, O_d) for d = 1, 3, 7 and the quaternion hyperbolic lattice PU(2, 1, H) with entries in the Hurwitz integer ring H. The implementation of the method for these groups is computer-assisted. This is joint work with Alice Mark.

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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).

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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).

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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.

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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 finitely-sided fundamental domain (using either an invariant Riemannian metric or Selberg's 2-point invariant) in the associated symmetric space, (b) the relatively recent notion of {\em Anosov subgroups}. Both definitions allow for geometric local-to-global principles, which, in turn, make computations with such discrete subgroups possible. The lectures will describe theses two concepts, the local-to-global principles and the geometric algorithms.

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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 finitely-sided fundamental domain (using either an invariant Riemannian metric or Selberg's 2-point invariant) in the associated symmetric space, (b) the relatively recent notion of {\em Anosov subgroups}. Both definitions allow for geometric local-to-global principles, which, in turn, make computations with such discrete subgroups possible. The lectures will describe theses two concepts, the local-to-global principles and the geometric algorithms.

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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.

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Can one translate a topological or geometric algorithm into a computable algorithm? We consider the Gilman-Maskit PSL(2,R) two-generator 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 bit-computability. We discuss two models for bit-computation, extended domain bit-computability 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 semi-decidable.

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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.

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We give a new proof of an old result that the congruence subgroup kernel associated to a higher rank non-cocompact arithmetic group is central in the arithmetic completion of the discrete group.

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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.

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A basic consequence of Mostow-Prasad Rigidity is that if M=H^3/G is an orientable hyperbolic 3-manifold of finite volume, then the traces of the elements in $\G$ are algebraic numbers. Say that M has non-integral trace if G contains an element whose trace is an algebraic non-integer. This talk will consider manifolds with non-integral trace and show for example, that there are infinitely many non-homeomorphic hyperbolic knot complements S^3\ K_i with non-integral trace.

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A computer program "SnapPea" and its descendant “SnapPy” compute many invariants of a hyperbolic 3-manifold 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.

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In this talk, I will explore various questions arising from considering the mod-p 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.