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

Upcoming Talks:

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

Pre-prints:

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

Publications:

- Tschirnhaus Transformations after Hilbert,
*L’Enseignement Mathematique,*to appear. - Resolvent degree, Hilbert’s 13th Problem and Geometry (with Benson Farb).
*L’Enseignement Mathematique,*vol. 65, 2019, no. 3/4, pp. 303-376. - Derived -adic zeta functions (with Jonathan Campbell and Inna Zakharevich).
*Advances in Mathematics*, vol. 354, 2019, article 106760, 53 pp. - Hochschild coniveau spectral sequence and the Beilinson residue (with Oliver Braunling).
*Pacific Journal of Mathematics*, vol. 300, no. 2, 2019, pp. 257-329. - Coincidences of homological densities, predicted by arithmetic (with Benson Farb and Melanie Matchett Wood).
*Advances in Mathematics*, vol. 352, 2019, pp. 670-716. - The -Structure of the Index Map (with Oliver Braunling and Michael Groechenig).
*Annals of K-Theory,*vol. 3, no. 4, 2018, pp. 581–614. - Etale homological stability and arithmetic statistics (with Benson Farb).
*Quarterly Journal of Mathematics,*vol. 69, no. 3, 2018, pp. 951–974. - 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. - The Index Map in Algebraic K-Theory (with Oliver Braunling and Michael Groechenig).
*Selecta Mathematica,*vol. 24, no. 2, 2018, pp. 1039-1091. - 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.
- 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. - 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. - Modular Operads of Embedded Curves (with Satoshi Kondo and Charles Siegel).
*Geometry & Topology*, vol. 21, no. 2, 2017, pp. 903-922. - Operator Ideals in Tate Objects (with Oliver Braunling and Michael Groechenig).
*Mathematical Research Letters,*vol. 23, no. 6, 2016, pp. 1565-1631. - 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.
- 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*.* - Topology and arithmetic of resultants, I (with Benson Farb).
*New York Journal of Mathematics,*vol. 22, 2016, pp. 801-821 - 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. - Descent for -Bundles.
*Advances in Mathematics,*vol. 288, 2016, pp. 527-575.

In Preparation:

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

Research Talks:

- UNR Topology Seminar – The Geometry of Hilbert’s 13th Problem – Sept. 25
- UNR Colloquium – Braids, Polynomials and Hilbert’s 13th Problem – Sept. 24
- Northwestern Geometry & Physics Seminar – The Geometry of Hilbert’s 13th Problem, May 28, 2020.
- Spring Southeastern AMS Sectional, Special Session on Motivic Aspects of Topology and Geometry – Derived motivic integration, March 13, 2020 (CANCELLED DUE TO COVID-19)
- IAS – Special Seminar on Hilbert’s 13th Problem, Dec. 5 – 6
- UCSB Topology Seminar – Braids, Polynomials and Hilbert’s 13th, April 2, 2019
- IPAM – Braids, Polynomials and Hilbert’s 13th Problem, Feb. 19-21, 2019
- Tufts Colloquium – Braids, Polynomials and Hilbert’s 13th Problem, Feb. 1, 2019.
- Michigan Topology Seminar – Topological Invariants of Algebraic Functions, Jan. 10, 2019.
- UCR Topology Seminar – The Geometry of Hilbert’s 13th Problem, Nov. 8, 2018
- USC Colloquium – Braids, Polynomials and Hilbert’s 13th Problem, Sept. 26, 2018
- Michigan Representation Stability Conference, Aug. 13-17, 2018
- UCI Algebra Seminar – The Geometry of Hilbert’s 13th Problem, May 30, 2018
- Stanford Topology Seminar – The Theory of Resolvent Degree, after Hamilton, Klein, Sylvester, Hilbert and Brauer, May 22, 2018
- UCLA Algebra Seminar – The Theory of Resolvent Degree, after Hamilton, Klein, Sylvester, Hilbert and Brauer, May 18, 2018
- Caltech Algebra & Geometry Seminar – The Theory of Resolvent Degree, after Hamilton, Klein, Sylvester, Hilbert and Brauer, May 14, 2018
- CCR – The Theory of Resolvent Degree, after Hamilton, Klein, Sylvester, Hilbert and Brauer, April 26, 2018
- USC Tuesday Seminar – The Geometry of Hilbert’s 13th Problem, April 24, 2018
- USC Algebra Seminar – The Theory of Resolvent Degree, after Hamilton, Klein, Sylvester, Hilbert and Brauer, April 23, 2018
- UChicago Colloquium – Braids, Polynomials and Hilbert’s 13th Problem, April 11, 2018
- Harvard Number Theory Seminar – The Theory of Resolvent Degree, after Hamilton, Klein, Hilbert and Brauer, March 7, 2018
- Harvard Informal Geometry and Dynamics Seminar – The Geometry of Hilbert’s 13th Problem, March 7, 2018
- UC Irvine Colloquium – Braids, Polynomials and Hilbert’s 13th Problem, March 1, 2018
- Kempner Colloquium – Algebraic Topology and Hilbert’s 13th Problem(s), Boulder, CO, Jan. 30, 2018
- Boulder Topology Seminar – Coincidences of homological densities, predicted by arithmetic, Boulder, CO, Jan. 30, 2018
- 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
- UMN Topology Seminar -The theory of resolvent degree, after Hamilton, Sylvester, Hilbert, Segre and Brauer – Twin Cities, MN, Dec. 8, 2017
- UCSD Number Theory Seminar – The theory of resolvent degree, after Hamilton, Sylvester, Hilbert, Segre and Brauer – San Diego, USA, Nov. 30, 2017
- UBC Topology Seminar – Coincidences of homological densities, predicted by arithmetic – Vancouver, Canada, Nov. 8, 2017
- UBC Algebraic Geometry Seminar – The theory of resolvent degree, after Hamilton, Sylvester, Hilbert, Segre and Brauer – Vancouver, Canada, Nov. 6, 2017
- 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
- Manifolds and Groups – Coincidences of homological densities, predicted by arithmetic – Regensburg, Germany, Sept. 25-29, 2017
- UCLA Number Theory Seminar – Coincidences of homological densities, predicted by arithmetic – Jan. 25, 2017
- UCI Number Theory Seminar – Coincidences of homological densities, predicted by arithmetic – Jan. 23, 2017
- Notre Dame Topology Seminar – Coincidences of homological densities, predicted by arithmetic – Dec. 8, 2016
- University of Chicago Geometry/Topology Seminar – Coincidences of homological densities, predicted by arithmetic – Sept. 29, 2016
- 2016 Oberwolfach Topology Meeting – Coincidences of homological densities, predicted by arithmetic – July 21, 2016
- 27th Nordic Congress of Mathematicians, Geometry and Topology Session – Counting Problems and Homological Stability – March 17, 2016
- Purdue Topology Seminar – Counting Problems and Homological Stability – March 2, 2016
- Wayne State Colloquium – Polynomials, Counting Problems and Algebraic Topology – February 22, 2016
- IBS-CGP Conference on Homotopical Methods in Quantum Field Theory – Higher Determinants and Double Loop Groups – January 13, 2016
- Minnesota Topology Seminar – Counting Problems and Homological Stability – October 26, 2015
- Rice Topology Seminar – Counting Problems and Homological Stability – October 19, 2015
- Northwestern Topology Seminar – Counting Problems and Homological Stability – October 12, 2015
- 2015 AMS Summer Institute in Algebraic Geometry – Topology and Arithmetic of Resultants – July 27, 2015
- Midwest Topology Seminar – Topology and Arithmetic of Spaces of Rational Maps – May 10, 2015
- Wisconsin Number Theory Seminar – Geometry and Arithmetic of Spaces of Rational Maps – May 7, 2015
- UIC Homotopy Algebras Seminar – Modular Operads of Embedded Curves – April 17, 2015
- 2015 Arkansas Spring Lecture Series – Spaces of Rational Functions, Representation Stability, and Statistics over Finite Fields – March 5, 2015
- Rutgers Algebra Seminar – The Index Map and Reciprocity Laws for Contou-Carrere Symbols – Feb. 4, 2015
- Yale Geometry, Symmetry and Physics Seminar – The Index Map and Reciprocity Laws for Contou-Carrere Symbols – Feb. 2, 2015
- MIT – Topology Seminar – The Index Map and Reciprocity Laws – Nov. 17, 2014
- University of Tokyo – Topology Seminar – The Index Map and Reciprocity Laws for Contou-Carrere Symbols – July 22, 2014
- IPMU – Get Together Seminar – A Modular Operad of Embedded Curves – July 3, 2014
- Imperial College London MAGIC Seminar – Index Theory and Reciprocity Laws – February 26, 2014
- Padua Winter School and Workshop on Higher Structures in Algebraic Analysis – The Index Map in Algebraic K-Theory – February 18, 2014
- University of Copenhagen Algebra/Topology Seminar – Index Theory and Reciprocity Laws – December 17, 2013
- Nagoya University K-Theory Seminar – Index Theory in Algebraic -Theory -December 12, 2013
- Chuo University – K-Theory Seminar – Index Theory in Algebraic -Theory – December 3, 2013
- IPMU – Get-Together Math Seminar – Index Theory in Algebraic -Theory and Algebraic Geometry – October 10, 2013
- Northwestern Mini-Conference – Determinants, Delooping and Reciprocity – June 12, 2013
- Northwestern – Topology Seminar – Descent for -Bundles – June 3, 2013
- Paris – Institute de Mathematiques de Jussieu – Seminaire d’Analyse Algebrique – Descent for n-Bundles – May 21, 2013
- Utrecht – Descent for n-Bundles – May 17, 2013
- Oxford – Algebraic and Symplectic Geometry Seminar – Descent for n-Bundles – May 7, 2013
- EPFL – Topology Seminar – Determinants and Delooping of Algebraic K-Theory – May 3, 2013
- EPFL – Geometry Seminar – Determinants, Delooping and Reciprocity – May 1, 2013
- Max Planck Institute for Mathematics – Higher Differential Geometry Seminar – “Descent for -bundles” – April 24, 2013
- University of British Columbia – Algebraic Geometry Seminar – “Strictifying Higher Bundles” – Oct. 29, 2012
- Notre Dame – Topology Seminar – “Extensions of Lie -Groups” – Sept. 27, 2012
- 2012 ESI Program on -Theory and Quantum Fields – “Extensions of Lie -Groupoids by Strict Lie -Groups” – June 14, 2012

Other Talks:

- 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
- IPMU Conformal Blocks Seminar – Loop Groups and Conformal Blocks – Nov. 19, 2013
- MIT Babytop – The Structure of Higher Bundles – Nov. 27, 2012
- 2012 Simons Center Graduate Workshop on SUSY Field Theories – Supermanifolds – March 6, 2012
- 2011 Talbot Workshop – Principal Bundles, Real Structures, Toledo Invariant and Milnor-Wood – May 2, 2011
- Northwestern Graduate Seminar – The Chern-Weil Construction – April 21, 2011
- Northwestern Graduate Seminar – Symmetries of Spaces and Bundles – Jan. 20, 2011
- Northwestern Graduate Seminar – Spin Structures – Oct. 7, 2010
- Discrete and Continuous Geometries, Northwestern Conference for Undergraduates – The Euler Characteristic – June 23, 2010
- 2010 Talbot Workshop – Complex -Theory – May 23, 2010
- Northwestern Graduate Seminar – Introduction to Complex -Theory – May 18, 2010

Posters: