Doron Puder

In the year 2024-2025 I'm on sabbatical in the School of Mathematics, at the Institute for Advanced Study in Princeton. 

ERC Project: Word Measures in Groups and Random Cayley Graphs

Research Interests: 

I study group theory, and especially combinatorial, geometric and probabilistic group theory. Many of the questions in my research involve word measures on groups: measures induced by word maps. This involves the study of free groups and their automorphism group, as well as certain aspects of the representation theory of the groups on which the word measures are induced. In the background there are problems about expansion and spectral gaps of random objects (random graphs, random surfaces, random Cayley graphs and so on). 

Groups & Dynamics Seminar: Thursdays at 11:00

Preprints:

• The Ring of Stable Characters over GL.(q). With Danielle Ersnt-West and Yotam Shomroni. [arXiv]

• An Extension of the Cogrowth Formula to Arbitrary Subsets of the Tree. [arXiv]

• Stable Invariants and Their Role in Word Measures on Groups. With Yotam Shomroni. [arXiv]

• On the Aldous-Caputo Spectral Gap Conjecture for Hypergraphs. With Gil Alon and Gady Kozma. [arXiv]

Papers:

• Word Measures on GLn(q) and Free Group Algebras. With Danielle Ernst-West and Matan Seidel. [journal, arXiv]
  Algebra & Number Theory, 18(11), 2047-2090, 2024

• Local Statistics of Random Permutations from Free Products. With Tomer Zimhoni. [journal, arXiv]
  International Mathematics Research Notices (IMRN), 2024 (5), 4242-4300, 2024

• The Asymptotic Statistics of Random Covering Surfaces. With Michael Magee. [journal, arXiv, mini-course (part I, part II, part III, part IV)]
  Forum of Mathematics, Pi, 11:e15, 1-51, 2023

• Matrix Group Integrals, Surfaces, and Mapping Class Groups II: O(n) and Sp(n). With  Michael Magee. [journal, arXiv]
  Mathematische Annalen, 388, 1437--1494, 2024

• Core Surfaces. With Michael Magee. [journal, arXiv]
  Geometriae Dedicata, 216:46, 2022

• A Random Cover of a Compact Hyperbolic Surface Has Relative Spectral Gap 3/16-ε. With  Michael Magee and Frédéric Naud. [journal, arXiv]
  Geometric and Functional Analysis (GAFA), 32, 595--661, 2022

• Word Measures on Symmetric Groups. With Liam Hanany. [journal, arXiv]
  International Mathematics Research Notices (IMRN),  2023 (11), 9221--9297, 2023
  Extended abstract in proceedings of FPSAC 2020 [journal, PDF]

• A Note on the Trace Method for Random Regular Graphs. With Joel Friedman. [journal, arXiv]
  Addendum to Expansion of Random Graphs: New Proofs, New Results.
  Israel Journal of Mathematics, 256, 269--282, 2023

• Some Orbits of Free Words that are Determined by Measures on Finite Groups. With Liam Hanany and Chen Meiri. [journal, arXiv]
  Journal of Algebra 555, 305--324, 2020

• Aldous' Spectral Gap Conjecture for Normal Sets. With Ori Parzanchevski. [journal, arXiv]
  Transactions of the American Mathematical Society 373 (10), 7067--7086, 2020

• Surface Words are Determined by Word Measures on Groups. With  Michael Magee. [journal, arXiv]
  Israel Journal of Mathematics 241, 749--774, 2021

• Automorphism-Invariant Positive Definite Functions on Free Groups. With Benoît Collins and Michael Magee. [arXiv]
  Proceedings of the 27th International Conference in Operator Theory (OT27), 115--137, 2020

• Matrix Group Integrals, Surfaces, and Mapping Class Groups I: U(n). With Michael Magee. [journal, arXiv]
  Inventiones Mathematicae 218 (2), 341--411, 2019

• The Markoff Group of Transformations in Prime and Composite Moduli. With Chen Meiri and appendix by Dan Carmon. [journal, arXiv]
  Duke Mathematical Journal 167 (14), 2679--2720, 2018

• Asymptotics for a Class of Meandric Systems, via the Hasse Diagram of NC(n). With Alexandru Nica and Ian P. Goulden. [journal, arXiv]
  International Mathematics Research Notices (IMRN) 2020 (4), 983--1034, 2020

• Ramanujan Coverings of Graphs. With Chris Hall and Will Sawin.  [journal, arXiv, STOC (extended abstract)]
  Advances in Mathematics 323 ,367--410, 2018
  STOC 2016, 48th annual ACM SIGACT Symposium in the Theory of Computing, 533--541, 2016. [Extended abstract]

• Expansion of Random Graphs: New Proofs, New Results. [journal, arXiv] 
  Inventiones Mathematicae 201 (3), 845--908, 2015
  Addendum with Joel Friedman: [arXiv]

• Measure Preserving Words are Primitive. With Ori Parzanchevski.  [journal, arXiv, 50-minute-talk, 4-hours-talk (part I, part II)]
  Journal of the American Mathematical Society 28 (1), 63--97, 2015 

• Growth of Primitive Elements in Free Groups. With Conan Wu. [journal, arXiv, The World of Groups - Question F19 ]
  Journal of the London Mathematical Society 90 (1), 89--104, 2014

• Stalling Graphs, Algebraic Extensions and Primitive Elements in F2. With Ori Parzanchevski. [journal, arXiv]
  Mathematical Proceedings of the Cambridge Philosophical Society 157 (1), 1--11, 2014

• Primitive Words, Free Factors and Measure Preservation. [journal, arXiv]
  Israel Journal of Mathematics 201 (1), 25--73, 2014 

• Word Maps and Spectra of Random Graph Lifts. With Nati Linial. [journal, arXiv]
  Random Structures and Algorithms 37 (1) 100--135, 2010

Not for publication:

• Word Measures on Unitary Groups. With Michael Magee. [arXiv]
  The results of this preprint have been extended and generalized in the paper "Matrix Group Integrals, Surfaces, and Mapping Class Groups I: U(n)"  appearing above. However, this preprint contains useful background, examples and a somewhat different approach in the proofs. 

Book:

• Algebraic Structures: Groups, Rings and Fields (Textbook in Hebrew). With Ehud de Shalit and Alex Lubotzky.
  מבנים אלגבריים: חבורות, חוגים ושדות
  Errata file (please email me about any other mistakes and typos in the book)

Post-docs:

Michal Buran (2020-2021)

Daniele Garzoni (2021-2023)

Noam Kolodner (2022- 2024)

Phd Students:

Matan Seidel

Danielle Ernst-West

Niv Levhari

Master Students:

Danielle Ernst-West  (completed 2019): Word measures on GLn(Fq)

Ziv Greenhut  (completed 2019): A generation criterion for subsets of special linear groups over finite fields

Gal Ordo  (completed 2020): Word measures on the signed symmetric group

Liam Hanany (completed 2021): Word measures on symmetric groups

Yaron Brodsky (completed 2022): Word Measures on Unitary Groups: Improved Bounds for Small Representations

Tomer Zimhoni (completed 2023): Local statistics of random permutations from free products

Yotam Shomroni (completed 2023): Word Measures on Wreath Products I

Dror Frid

Noam Ta-Shma

Mini-courses in conferences:

• In Thematic Program on Randomness and Geometry, Toronto, 29 April - 10 May, 2024
  Title: Random covers of graphs and surfaces

• In Probabilistic Methods in Negative Curvature, Bangalore, February-March 2023
  Title: Random covers of Hyperbolic Surfaces

• In Combinatorics of finite-index subgroups, Paris, November 2022
  Title: The asymptotic statistics of random covering surfaces 

• In Paroles, Paroles: A Summer School on Words, Haifa, July 2022
  Title:  Word measures on symmetric and other groups

• In Group Actions, Geometry and Dynamics, Oholo, July 2022
  Title: Free groups, surfaces and linear programming

• In Laplacians on Random Hyperbolic Surfaces and on Random Graphs, Evanston, May 2022
  Title: The spectrum of random graphs

• In Recent trends in the study of expanders and high dimensional expanders, Rehovot, February 2017
  Title: New trends in expanders based on the method of interlacing polynomials

Advanced Classes Taught:

Random Walks on Groups - Fall 2016
Surfaces: Graphs, Groups and Applications - Spring 2018
Topics in Combinatorial Group Theory - Fall 2018
Representations of Finite Groups - Spring 2020
Introduction to Algebraic Topology - Fall 2020
        Surfaces: Graphs, Groups and Applications - Fall 2021
        Advanced seminar in geometric group theory - Spring 2022

Brief CV:

• 2024-2025: Von-Neumann fellow, School of Mathematics, IAS, Princeton

• 2023 - present: Professor, School of Mathematical Sciences, Tel Aviv University

• 2019-2023: Associate Professor, School of Mathematical Sciences, Tel Aviv University

• 2016-2019: Senior Lecturer, School of Mathematical Sciences, Tel Aviv University

• 2014-2016: Member, School of Mathematics, IAS, Princeton

• 2010-2014: PhD, Einstein Institute of Mathematics, at the Hebrew University of Jerusalem, supervised by Nati Linial.