Welcome

In spring 2020 I am teaching a graduate course: Geometry in Group Theory. Here is my CV and below are my papers. At the bottom of this page there are also some expository notes. Please email me at allcock at math.utexas.edu if you have trouble downloading, printing or understanding my papers. As part of a grant application I had to summarize five years of research in two pages. I'm posting the summary here in case it is interesting to someone. (I'm afraid it will only be readable to mathematicians.)

  Moduli spaces of tropical curves are simply connected
      with Daniel Corey and Sam Payne; submitted 2019
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  Best play in Dots and Boxes endgames
      submitted 2019
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  The tetrahedron and automorphisms of Enriques and Coble surfaces of Hessian type
      with Igor Dolgachev, submitted 2018
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  Generators for a complex hyperbolic braid group
      with Tathagata Basak, Geometry & Topology 20 (2018) 747-778.
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  Congruence subgroups and Enriques surface automorphisms
      J. Lond. Math. Soc. 98 (2018) 1-11.
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  The Conway-Sloane calculus for 2-adic lattices
      with Itamar Gal and Alice Mark; to appear in L'Enseignement mathematique (arxiv 1511.04614)
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  Spherical space forms revisited
      Transactions of the A.M.S. 370 (2018) 5561-5582.
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  A fake projective plane via 2-adic uniformization with torsion
      with Fumiharu Kato, Tohoku Math. J. 69 (2017) 221-237
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  Prenilpotent pairs in the E10 root system
      Math. Proc. Camb. Phil. Soc. 164 (2018) 473-483.
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  Presentation of hyperbolic Kac-Moody groups over rings
      with Lisa Carbone, J. Algebra 445 (2016) 232-243
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  Presentation of affine Kac-Moody groups over rings
      Algebra and Number Theory 10 (2016) 533-556
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  Geometric generators for braid-like groups
      with Tathagata Basak
      Geometry and Topology 20 (2016) 747-778
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  Steinberg groups as amalgams
      Algebra & Number Theory 10 (2016) 1791-1843
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  Binary Relations from Tournament Solutions, and Back Again
      with Scott Moser; submitted 2013
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  Root systems for Lorentzian Kac-Moody algebras in rank 3
      Bull. Lond. Math. Soc. 47 (2015) 325-342
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  Reflection centralizers in Coxeter groups
      Transformation Groups 18 (2013) 599-613
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  Completions, branched covers, Artin groups and Singularity Theory
      Duke Math. J. 162 (2013) 2645-2689.
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  The densest lattices in PGL₃(Q₂)
      with Fumiharu Kato, Advances in Math. 242 (2013) 102-115
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  Normalizers of parabolic subgroups of Coxeter groups
      Algebraic & Geometric Topology 12 (2012) 1137-1143
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  The reflective Lorentzian lattices of rank 3
      Memoirs of the A.M.S. 220 (number 1033) (2012). Erratum.
      Download PDF, tex/perl or the (huge) unabridged table PDF or tex/perl
  Complex hyperbolic geometry and the monster simple group (conjectural)
      Oberwolfach Reports 5 (2008) 2586-2588
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  Triangles of Baumslag-Solitar groups
      Canadian J. of Math. 64 (2012) 241-253
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  A Banach space determined by the Weil height
      with Jeffrey Vaaler, Acta Arithmetica 136 (2009), 279-298
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  Monodromy groups of Hurwitz-type problems
      with Chris Hall; Advances in Mathematics 225 (2010) 69-80.
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  On the Y₅₅₅ complex reflection group
      J. Alg. 322 (2009) 1454-1465
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  A new approach to rank one linear algebraic groups
      J. Algebra 321 (2009) 2540-2544.
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  Hyperbolic geometry and moduli of real cubic surfaces
      with J. Carlson and D. Toledo, Ann. Sci. Ecole Norm. Sup. 43 (2010) 69-115
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  The moduli space of cubic threefolds as a ball quotient
      with J. Carlson and D. Toledo, Memoirs of the A.M.S. 209 (2011) no. 985
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  A monstrous proposal
      in Groups and symmetries, CRM Proc. Lecture Notes 47 (2009) 17-24
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  Non-arithmetic uniformization of some real moduli spaces
      with J. Carlson and D. Toledo; Geometriae Dedicata 122 (2006) 159-169. Erratum 122 (2006) 171.
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  Hyperbolic geometry and the moduli space of real binary sextics
      with J. Carlson and D. Toledo; in Algebra and Geometry around Hypergeometric Functions, R.P. Holzapfel, A.M. Uludag, M. Yoshida (editors), Progress in Mathematics series v. 260, Birkauser, 1-22.
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  Infinitely many hyperbolic Coxeter groups through dimension 19
      Geometry and Topology 122 (2006), 159-169.
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  Hyperbolic surfaces with prescribed infinite symmetry groups
      Proc. Amer. Math. Soc. 134 (2006) 3057-3059.
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  Orbits in the Leech Lattice
      Experimental Math. 14 (2005) 491-509
      Download PDF or check the errata
  Real cubic surfaces and real hyperbolic geometry
      with J. Carlson and D. Toledo; CRAS 337 (2003) 185-188
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  Orthogonal Complex Hyperbolic Arrangements
      with J. Carlson and D. Toledo; Symposium in Honor of C. H. Clemens, Contemp. Math. 312 (2002) 1-8
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  The moduli space of cubic threefolds
      J. Alg. Geom. 12 (2003) 201-223
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  The complex hyperbolic geometry of the moduli space of cubic surfaces
      with J. Carlson and D. Toledo; J. Alg. Geom. 11 (2002) 659-724
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  Cubic surfaces and Borcherds products
      with E. Freitag; Comm. Math. Helv. 77 (2002) 270-296
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  The period lattice for Enriques surfaces
      Math. Ann. 317 (2000) 483-488
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  Asphericity of moduli spaces via curvature
      J. Diff. Geom. 55 (2000) 441-451
      previously entitled Metric curvature of infinite branched covers
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  Braid pictures for Artin groups
      Trans. A.M.S. 354 (2002) 3455-3474
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  Ideals in the integral octaves
      J. Alg. 220 (1999) 396-400
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  A complex hyperbolic structure for moduli of cubic surfaces
      with J. Carlson and D. Toledo, CRAS 326 (1998) 49-54
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  The Leech lattice and complex hyperbolic reflections
      Invent. Math. 140 (2000) 283-301
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  A homological characterization of hyperbolic groups
      with S. Gersten, Invent. Math. 135 (1999) 723-742
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  Reflection groups on the octave hyperbolic plane
      J. Alg. 213 (1999) 467-498
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  New complex- and quaternion-hyperbolic reflection groups
      Duke Math. J. 103 (2000) 303-333
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  An isoperimetric inequality for the Heisenberg groups
      GAFA 8 (1998) 219-233
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  Identifying models of the octave projective plane
      Geom. Dedicata 65 (1997) 215-217
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  Spotting infinite groups
      Math. Proc. Camb. Phil. Soc. 125 (1999) 39-42
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Expository notes:

  Hilbert's Nullstellensatz
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  The finite reflection groups
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My current research is supported by a Simons Foundation Collaboration Grant
Here is a Dots & Boxes board.
You can play Euclid: The Game! to challenge yourself with compass and ruler constructions.
Daniel Allcock
University of Texas at Austin
Department of Mathematics, RLM 8.100
2515 Speedway Stop C1200
Austin, Texas 78712-1202

phone (512) 471-1120
fax (512) 471-9038
email: allcock at math.utexas.edu