Research

Problems in Arithmetic Topology (with Claudio Gomez-Gonzales) – research problems posed at PIMS 2019 “Arithmetic Topology” workshop.

Upcoming Talks:

  1. UChicago Mini-Course, Braids, Polynomials and Hilbert’s 13th Problem – Winter 2021.
  2. 2021 Spring Western AMS Sectional Special Session on Connections between homotopical algebra and geometry – May 1-2, 2021
  3. UIC Algebraic Geometry Seminar – The Geometry of Hilbert’s 13th Problem (TBD)
  4. UIC Colloquium – Braids, Polynomials and Hilbert’s 13th Problem (TBD)
  5. Mid-Atlantic Topology Conference @UPenn (TBD)

Pre-prints:

  1. Modular functions and resolvent problems, with an appendix by Nate Harman (with Benson Farb and Mark Kisin). Submitted for publication.
  2. The Essential Dimension of Congruence Covers (with Benson Farb and Mark Kisin). Submitted for publication.
  3. A Generalized Contou-Carrere Symbol and its Reciprocity Laws in Higher Dimensions (with Oliver Braunling and Michael Groechenig). Submitted for publication.

Publications:

  1. Tschirnhaus Transformations after Hilbert, L’Enseignement Mathematique,to appear.
  2. Resolvent degree, Hilbert’s 13th Problem and Geometry (with Benson Farb). L’Enseignement Mathematique, vol. 65, 2019, no. 3/4, pp. 303-376.
  3. Derived \ell-adic zeta functions (with Jonathan Campbell and Inna Zakharevich). Advances in Mathematics, vol. 354, 2019, article 106760, 53 pp.
  4. Hochschild coniveau spectral sequence and the Beilinson residue (with Oliver Braunling). Pacific Journal of Mathematics, vol. 300, no. 2, 2019, pp. 257-329.
  5. Coincidences of homological densities, predicted by arithmetic (with Benson Farb and Melanie Matchett Wood). Advances in Mathematics, vol. 352, 2019, pp. 670-716.
  6. The A_\infty-Structure of the Index Map (with Oliver Braunling and Michael Groechenig). Annals of K-Theory, vol. 3, no. 4, 2018, pp. 581–614.
  7. Etale homological stability and arithmetic statistics (with Benson Farb). Quarterly Journal of Mathematics, vol. 69, no. 3, 2018, pp. 951–974.
  8. On the normally ordered tensor product and duality for Tate objects (with Oliver Braunling, Michael Groechenig and Aron Heleodoro). Theory and Applications of Categories, vol. 33, 2018, pp. 296-349.
  9. The Index Map in Algebraic K-Theory (with Oliver Braunling and Michael Groechenig). Selecta Mathematica, vol. 24, no. 2, 2018, pp. 1039-1091.
  10. Resolvent degree, Hilbert’s 13th problem and geometry (with Benson Farb). “Topology of Arrangements and Representation Stability”, Oberwolfach Research Reports, no. 2, 2018, pp. 84–87.
  11. Topology and arithmetic of resultants, II: the resultant = 1 hypersurface (with an appendix by C. Cazanave) (with Benson Farb). Algebraic Geometry, vol. 4, no. 3, 2017, pp. 337-352.
  12. Relative Tate Objects and Boundary Maps in the K-Theory of Coherent Sheaves (with Oliver Braunling and Michael Groechenig). Homology, Homotopy and Applications, vol. 19, no. 1, 2017, pp. 341-369.
  13. Modular Operads of Embedded Curves (with Satoshi Kondo and Charles Siegel). Geometry & Topology, vol. 21, no. 2, 2017, pp. 903-922.
  14. Operator Ideals in Tate Objects (with Oliver Braunling and Michael Groechenig). Mathematical Research Letters, vol. 23, no. 6, 2016, pp. 1565-1631.
  15. Coincidences of homological densities, predicted by arithmetic (with Benson Farb and Melanie Matchett Wood). “Topologie”, Oberwolfach Research Reports, vol. 16, no. 3, 2016, pp. 2041-2043.
  16. Geometric and Analytic Structures on the Higher Adeles (with Oliver Braunling and Michael Groechenig). Research in the Mathematical Sciences, Special Collection in Celebration of the Research of Fedor Bogomolov on the Occasion of his 70th Birthday, vol. 3, no. 1, 2016, article 22.
  17. Topology and arithmetic of resultants, I (with Benson Farb). New York Journal of Mathematics, vol. 22, 2016, pp. 801-821
  18. Tate Objects in Exact Categories (with an appendix by J. Stovicek and J. Trlifaj)  (with Oliver Braunling and Michael Groechenig). Moscow Mathematical Journal, vol. 16, no. 3, 2016, pp. 433-504.
  19. Descent for n-Bundles. Advances in Mathematics, vol. 288, 2016, pp. 527-575.

In Preparation:

  1. A higher Kac-Moody extension for two-dimensional gauge groups (with Jens Kaad and Ryszard Nest)
  2. Derived motivic integration (with Oliver Braunling, Michael Groechenig and Inna Zakharevich)
  3. Lie’s Third Theorem for L_\infty-algebras (with Chris Rogers)

Research Talks:

  1. UNR Topology Seminar – The Geometry of Hilbert’s 13th Problem – Sept. 25
  2. UNR Colloquium – Braids, Polynomials and Hilbert’s 13th Problem – Sept. 24
  3. Northwestern Geometry & Physics Seminar – The Geometry of Hilbert’s 13th Problem, May 28, 2020.
  4. Spring Southeastern AMS Sectional, Special Session on Motivic Aspects of Topology and Geometry – Derived motivic integration, March 13, 2020 (CANCELLED DUE TO COVID-19)
  5. IAS – Special Seminar on Hilbert’s 13th Problem, Dec. 5 – 6
  6. UCSB Topology Seminar – Braids, Polynomials and Hilbert’s 13th, April 2, 2019
  7. IPAM – Braids, Polynomials and Hilbert’s 13th Problem, Feb. 19-21, 2019
    1. After Abel, Feb. 19, 2019
    2. Resolvent Degree and Classical Solutions, Feb. 19, 2019
    3. Resolvent Degree and the Search for Lower Bounds, Feb. 21, 2019
  8. Tufts Colloquium – Braids, Polynomials and Hilbert’s 13th Problem, Feb. 1, 2019.
  9. Michigan Topology Seminar – Topological Invariants of Algebraic Functions, Jan. 10, 2019.
  10. UCR Topology Seminar – The Geometry of Hilbert’s 13th Problem, Nov. 8, 2018
  11. USC Colloquium – Braids, Polynomials and Hilbert’s 13th Problem, Sept. 26, 2018
  12. Michigan Representation Stability Conference, Aug. 13-17, 2018
  13. UCI Algebra Seminar – The Geometry of Hilbert’s 13th Problem, May 30, 2018
  14. Stanford Topology Seminar – The Theory of Resolvent Degree, after Hamilton, Klein, Sylvester, Hilbert and Brauer, May 22, 2018
  15. UCLA Algebra Seminar – The Theory of Resolvent Degree, after Hamilton, Klein, Sylvester, Hilbert and Brauer, May 18, 2018
  16. Caltech Algebra & Geometry Seminar – The Theory of Resolvent Degree, after Hamilton, Klein, Sylvester, Hilbert and Brauer, May 14, 2018
  17. CCR – The Theory of Resolvent Degree, after Hamilton, Klein, Sylvester, Hilbert and Brauer, April 26, 2018
  18. USC Tuesday Seminar  – The Geometry of Hilbert’s 13th Problem, April 24, 2018
  19. USC Algebra Seminar – The Theory of Resolvent Degree, after Hamilton, Klein, Sylvester, Hilbert and Brauer, April 23, 2018
  20. UChicago Colloquium – Braids, Polynomials and Hilbert’s 13th Problem, April 11, 2018
  21. Harvard Number Theory Seminar – The Theory of Resolvent Degree, after Hamilton, Klein, Hilbert and Brauer, March 7, 2018
  22. Harvard Informal Geometry and Dynamics Seminar – The Geometry of Hilbert’s 13th Problem, March 7, 2018
  23. UC Irvine Colloquium – Braids, Polynomials and Hilbert’s 13th Problem, March 1, 2018
  24. Kempner Colloquium – Algebraic Topology and Hilbert’s 13th Problem(s), Boulder, CO, Jan. 30, 2018
  25. Boulder Topology Seminar – Coincidences of homological densities, predicted by arithmetic, Boulder, CO, Jan. 30, 2018
  26. Oberwolfach  Workshop “Topology of Arrangements and Representation Stability” – The theory of resolvent degree, after Hamilton, Sylvester, Hilbert, Segre and Brauer, Oberwolfach, Germany, Jan. 14-20, 2018
  27. UMN Topology Seminar -The theory of resolvent degree, after Hamilton, Sylvester, Hilbert, Segre and Brauer – Twin Cities, MN, Dec. 8, 2017
  28. UCSD Number Theory Seminar – The theory of resolvent degree, after Hamilton, Sylvester, Hilbert, Segre and Brauer – San Diego, USA, Nov. 30, 2017
  29. UBC Topology Seminar – Coincidences of homological densities, predicted by arithmetic – Vancouver, Canada, Nov. 8, 2017
  30. UBC Algebraic Geometry Seminar – The theory of resolvent degree, after Hamilton, Sylvester, Hilbert, Segre and Brauer – Vancouver, Canada, Nov. 6, 2017
  31. No Boundaries – Groups in Algebra, Geometry and Topology – The theory of resolvent degree, after Hamilton, Sylvester, Hilbert, Segre and Brauer – Chicago, USA, Oct. 26-29, 2017
  32. Manifolds and Groups – Coincidences of homological densities, predicted by arithmetic – Regensburg, Germany, Sept. 25-29, 2017
  33. UCLA Number Theory Seminar – Coincidences of homological densities, predicted by arithmetic – Jan. 25, 2017
  34. UCI Number Theory Seminar – Coincidences of homological densities, predicted by arithmetic – Jan. 23, 2017
  35. Notre Dame Topology Seminar – Coincidences of homological densities, predicted by arithmetic – Dec. 8, 2016
  36. University of Chicago Geometry/Topology Seminar – Coincidences of homological densities, predicted by arithmetic – Sept. 29, 2016
  37. 2016 Oberwolfach Topology Meeting – Coincidences of homological densities, predicted by arithmetic – July 21, 2016
  38. 27th Nordic Congress of Mathematicians, Geometry and Topology Session – Counting Problems and Homological Stability – March 17, 2016
  39. Purdue Topology Seminar – Counting Problems and Homological Stability – March 2, 2016
  40. Wayne State Colloquium – Polynomials, Counting Problems and Algebraic Topology – February 22, 2016
  41. IBS-CGP Conference on Homotopical Methods in Quantum Field Theory – Higher Determinants and Double Loop Groups – January 13, 2016
  42. Minnesota Topology Seminar – Counting Problems and Homological Stability – October 26, 2015
  43. Rice Topology Seminar – Counting Problems and Homological Stability – October 19, 2015
  44. Northwestern Topology Seminar – Counting Problems and Homological Stability – October 12, 2015
  45. 2015 AMS Summer Institute in Algebraic Geometry – Topology and Arithmetic of Resultants – July 27, 2015
  46. Midwest Topology Seminar – Topology and Arithmetic of Spaces of Rational Maps – May 10, 2015
  47. Wisconsin Number Theory Seminar – Geometry and Arithmetic of Spaces of Rational Maps – May 7, 2015
  48. UIC Homotopy Algebras Seminar – Modular Operads of Embedded Curves – April 17, 2015
  49. 2015 Arkansas Spring Lecture Series – Spaces of Rational Functions, Representation Stability, and Statistics over Finite Fields – March 5, 2015
  50. Rutgers Algebra Seminar – The Index Map and Reciprocity Laws for Contou-Carrere Symbols – Feb. 4, 2015
  51. Yale Geometry, Symmetry and Physics Seminar – The Index Map and Reciprocity Laws for Contou-Carrere Symbols – Feb. 2, 2015
  52. MIT – Topology Seminar – The Index Map and Reciprocity Laws – Nov. 17, 2014
  53. University of Tokyo – Topology Seminar – The Index Map and Reciprocity Laws for Contou-Carrere Symbols – July 22, 2014
  54. IPMU – Get Together Seminar – A Modular Operad of Embedded Curves – July 3, 2014
  55. Imperial College London MAGIC Seminar – Index Theory and Reciprocity Laws – February 26, 2014
  56. Padua Winter School and Workshop on Higher Structures in Algebraic Analysis – The Index Map in Algebraic K-Theory – February 18, 2014
  57. University of Copenhagen Algebra/Topology Seminar – Index Theory and Reciprocity Laws – December 17, 2013
  58. Nagoya University K-Theory Seminar – Index Theory in Algebraic K-Theory -December 12, 2013
  59. Chuo University – K-Theory Seminar – Index Theory in Algebraic K-Theory – December 3, 2013
  60. IPMU – Get-Together Math Seminar – Index Theory in Algebraic K-Theory and Algebraic Geometry – October 10, 2013
  61. Northwestern Mini-Conference – Determinants, Delooping and Reciprocity – June 12, 2013
  62. Northwestern – Topology Seminar – Descent for n-Bundles – June 3, 2013
  63. Paris – Institute de Mathematiques de Jussieu – Seminaire d’Analyse Algebrique – Descent for n-Bundles – May 21, 2013
  64. Utrecht – Descent for n-Bundles – May 17, 2013
  65. Oxford – Algebraic and Symplectic Geometry Seminar – Descent for n-Bundles – May 7, 2013
  66. EPFL – Topology Seminar – Determinants and Delooping of Algebraic K-Theory – May 3, 2013
  67. EPFL – Geometry Seminar – Determinants, Delooping and Reciprocity – May 1, 2013
  68. Max Planck Institute for Mathematics – Higher Differential Geometry Seminar – “Descent for n-bundles” – April 24, 2013
  69. University of British Columbia – Algebraic Geometry Seminar – “Strictifying Higher Bundles” – Oct. 29, 2012
  70. Notre Dame – Topology Seminar – “Extensions of Lie n-Groups”  – Sept. 27, 2012
  71. 2012 ESI Program on K-Theory and Quantum Fields – “Extensions of Lie n-Groupoids by Strict Lie n-Groups” – June 14, 2012

Other Talks:

  1. IPMU Special Seminar – Nerves and Group Completion for Categories and Operads, A Beginner’s Guide, Parts 1-3 – Aug. 29, Sept. 1 and Sept. 2, 2014
  2. IPMU Conformal Blocks Seminar – Loop Groups and Conformal Blocks – Nov. 19, 2013
  3. MIT Babytop – The Structure of Higher Bundles – Nov. 27, 2012
  4. 2012 Simons Center Graduate Workshop on SUSY Field Theories  – Supermanifolds – March 6, 2012
  5. 2011 Talbot Workshop – Principal Bundles, Real Structures, Toledo Invariant and Milnor-Wood – May 2, 2011
  6. Northwestern Graduate Seminar – The Chern-Weil Construction – April 21, 2011
  7. Northwestern Graduate Seminar – Symmetries of Spaces and Bundles – Jan. 20, 2011
  8. Northwestern Graduate Seminar – Spin Structures – Oct. 7, 2010
  9. Discrete and Continuous Geometries, Northwestern Conference for Undergraduates – The Euler Characteristic – June 23, 2010
  10. 2010 Talbot Workshop – Complex K-Theory – May 23, 2010
  11. Northwestern Graduate Seminar – Introduction to Complex K-Theory – May 18, 2010

Posters:

  1. Newton Institute’s Higher Structure 2013: Operads and Deformation Theory – “Mapping Spaces in Higher Stacks, and Applications” – April 2, 2013