# Research

Publications:

1. Hochschild coniveau spectral sequence and the Beilinson residue (with Oliver Braunling). Pacific Journal of Mathematics, to appear.
2. The $A_\infty$-Structure of the Index Map (with Oliver Braunling and Michael Groechenig). Annals of K-Theory, to appear.
3. Etale homological stability and arithmetic statistics (with Benson Farb). Quarterly Journal of Mathematics, vol. 69, no. 3, 2018, pp. 951–974.
4. 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.
5. The Index Map in Algebraic K-Theory (with Oliver Braunling and Michael Groechenig). Selecta Mathematica, vol. 24, no. 2, 2018, pp. 1039-1091.
6. 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.
7. 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.
8. Modular Operads of Embedded Curves (with Satoshi Kondo and Charles Siegel). Geometry & Topology, vol. 21, no. 2, 2017, pp. 903-922.
9. Operator Ideals in Tate Objects (with Oliver Braunling and Michael Groechenig). Mathematical Research Letters, vol. 23, no. 6, 2016, pp. 1565-1631.
10. Coincidences of homological densities, predicted by arithmetic (with Benson Farb and Melanie Wood). “Topologie”, Oberwolfach Research Reports, vol. 16, no. 3, 2016, pp. 2041-2043.
11. 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.
12. Topology and arithmetic of resultants, I (with Benson Farb). New York Journal of Mathematics, vol. 22, 2016, pp. 801-821
13. 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.
14. Descent for $n$-Bundles. Advances in Mathematics, vol. 288, 2016, pp. 527-575.

Pre-prints:

1. Resolvent degree, Hilbert’s 13th Problem and Geometry (with Benson Farb). Submitted for publication.
2. Derived $\ell$-adic zeta functions (with Jonathan Campbell and Inna Zakharevich). Submitted for publication.
3. Coincidences of homological densities, predicted by arithmetic (with Benson Farb and Melanie Matchett Wood). Submitted for publication.
4. A Generalized Contou-Carrere Symbol and its Reciprocity Laws in Higher Dimensions (with Oliver Braunling and Michael Groechenig). Submitted for publication.

In Preparation:

1. Group Extensions and the Tate Tame Symbol on $K_3$ (with Jens Kaad and Ryszard Nest)
2. The Essential Dimension of Congruence Covers (with Benson Farb and Mark Kisin)
3. General upper bounds for the resolvent degree of polynomials (with Benson Farb)
4. Derived motivic integration (with Oliver Braunling, Michael Groechenig and Inna Zakharevich)

Upcoming Talks:

1. USC Colloquium, Sept. 26, 2018

Research Talks:

1. Michigan Representation Stability Conference, Aug. 13-17, 2018
2. UCI Algebra Seminar – The Geometry of Hilbert’s 13th Problem, May 30, 2018
3. Stanford Topology Seminar – The Theory of Resolvent Degree, after Hamilton, Klein, Sylvester, Hilbert and Brauer, May 22, 2018
4. UCLA Algebra Seminar – The Theory of Resolvent Degree, after Hamilton, Klein, Sylvester, Hilbert and Brauer, May 18, 2018
5. Caltech Algebra & Geometry Seminar – The Theory of Resolvent Degree, after Hamilton, Klein, Sylvester, Hilbert and Brauer, May 14, 2018
6. CCR – The Theory of Resolvent Degree, after Hamilton, Klein, Sylvester, Hilbert and Brauer, April 26, 2018
7. USC Tuesday Seminar  – The Geometry of Hilbert’s 13th Problem, April 24, 2018
8. USC Algebra Seminar – The Theory of Resolvent Degree, after Hamilton, Klein, Sylvester, Hilbert and Brauer, April 23, 2018
9. UChicago Colloquium – Braids, Polynomials and Hilbert’s 13th Problem, April 11, 2018
10. Harvard Number Theory Seminar – The Theory of Resolvent Degree, after Hamilton, Klein, Hilbert and Brauer, March 7, 2018
11. Harvard Informal Geometry and Dynamics Seminar – The Geometry of Hilbert’s 13th Problem, March 7, 2018
12. UC Irvine Colloquium – Braids, Polynomials and Hilbert’s 13th Problem, March 1, 2018
13. Kempner Colloquium – Algebraic Topology and Hilbert’s 13th Problem(s), Boulder, CO, Jan. 30, 2018
14. Boulder Topology Seminar – Coincidences of homological densities, predicted by arithmetic, Boulder, CO, Jan. 30, 2018
15. 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
16. UMN Topology Seminar -The theory of resolvent degree, after Hamilton, Sylvester, Hilbert, Segre and Brauer – Twin Cities, MN, Dec. 8, 2017
17. UCSD Number Theory Seminar – The theory of resolvent degree, after Hamilton, Sylvester, Hilbert, Segre and Brauer – San Diego, USA, Nov. 30, 2017
18. UBC Topology Seminar – Coincidences of homological densities, predicted by arithmetic – Vancouver, Canada, Nov. 8, 2017
19. UBC Algebraic Geometry Seminar – The theory of resolvent degree, after Hamilton, Sylvester, Hilbert, Segre and Brauer – Vancouver, Canada, Nov. 6, 2017
20. 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
21. Manifolds and Groups – Coincidences of homological densities, predicted by arithmetic – Regensburg, Germany, Sept. 25-29, 2017
22. UCLA Number Theory Seminar – Coincidences of homological densities, predicted by arithmetic – Jan. 25, 2017
23. UCI Number Theory Seminar – Coincidences of homological densities, predicted by arithmetic – Jan. 23, 2017
24. Notre Dame Topology Seminar – Coincidences of homological densities, predicted by arithmetic – Dec. 8, 2016
25. University of Chicago Geometry/Topology Seminar – Coincidences of homological densities, predicted by arithmetic – Sept. 29, 2016
26. 2016 Oberwolfach Topology Meeting – Coincidences of homological densities, predicted by arithmetic – July 21, 2016
27. 27th Nordic Congress of Mathematicians, Geometry and Topology Session – Counting Problems and Homological Stability – March 17, 2016
28. Purdue Topology Seminar – Counting Problems and Homological Stability – March 2, 2016
29. Wayne State Colloquium – Polynomials, Counting Problems and Algebraic Topology – February 22, 2016
30. IBS-CGP Conference on Homotopical Methods in Quantum Field Theory – Higher Determinants and Double Loop Groups – January 13, 2016
31. Minnesota Topology Seminar – Counting Problems and Homological Stability – October 26, 2015
32. Rice Topology Seminar – Counting Problems and Homological Stability – October 19, 2015
33. Northwestern Topology Seminar – Counting Problems and Homological Stability – October 12, 2015
34. 2015 AMS Summer Institute in Algebraic Geometry – Topology and Arithmetic of Resultants – July 27, 2015
35. Midwest Topology Seminar – Topology and Arithmetic of Spaces of Rational Maps – May 10, 2015
36. Wisconsin Number Theory Seminar – Geometry and Arithmetic of Spaces of Rational Maps – May 7, 2015
37. UIC Homotopy Algebras Seminar – Modular Operads of Embedded Curves – April 17, 2015
38. 2015 Arkansas Spring Lecture Series – Spaces of Rational Functions, Representation Stability, and Statistics over Finite Fields – March 5, 2015
39. Rutgers Algebra Seminar – The Index Map and Reciprocity Laws for Contou-Carrere Symbols – Feb. 4, 2015
40. Yale Geometry, Symmetry and Physics Seminar – The Index Map and Reciprocity Laws for Contou-Carrere Symbols – Feb. 2, 2015
41. MIT – Topology Seminar – The Index Map and Reciprocity Laws – Nov. 17, 2014
42. University of Tokyo – Topology Seminar – The Index Map and Reciprocity Laws for Contou-Carrere Symbols – July 22, 2014
43. IPMU – Get Together Seminar – A Modular Operad of Embedded Curves – July 3, 2014
44. Imperial College London MAGIC Seminar – Index Theory and Reciprocity Laws – February 26, 2014
45. Padua Winter School and Workshop on Higher Structures in Algebraic Analysis – The Index Map in Algebraic K-Theory – February 18, 2014
46. University of Copenhagen Algebra/Topology Seminar – Index Theory and Reciprocity Laws – December 17, 2013
47. Nagoya University K-Theory Seminar – Index Theory in Algebraic $K$-Theory -December 12, 2013
48. Chuo University – K-Theory Seminar – Index Theory in Algebraic $K$-Theory – December 3, 2013
49. IPMU – Get-Together Math Seminar – Index Theory in Algebraic $K$-Theory and Algebraic Geometry – October 10, 2013
50. Northwestern Mini-Conference – Determinants, Delooping and Reciprocity – June 12, 2013
51. Northwestern – Topology Seminar – Descent for $n$-Bundles – June 3, 2013
52. Paris – Institute de Mathematiques de Jussieu – Seminaire d’Analyse Algebrique – Descent for n-Bundles – May 21, 2013
53. Utrecht – Descent for n-Bundles – May 17, 2013
54. Oxford – Algebraic and Symplectic Geometry Seminar – Descent for n-Bundles – May 7, 2013
55. EPFL – Topology Seminar – Determinants and Delooping of Algebraic K-Theory – May 3, 2013
56. EPFL – Geometry Seminar – Determinants, Delooping and Reciprocity – May 1, 2013
57. Max Planck Institute for Mathematics – Higher Differential Geometry Seminar – “Descent for $n$-bundles” – April 24, 2013
58. University of British Columbia – Algebraic Geometry Seminar – “Strictifying Higher Bundles” – Oct. 29, 2012
59. Notre Dame – Topology Seminar – “Extensions of Lie $n$-Groups”  – Sept. 27, 2012
60. 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: