About
I am a mathematician working primarily in formalized mathematics, algebra, and arithmetic / algebraic geometry. Here is a link to my CV. I am currently an Associate Professor in the department of Mathematical and Statistical Sciences at the University of Alberta. I previously held postdoctoral positions at Oxford and Berkeley. I completed my Ph.D. in 2013 at the University of Pennsylvania.
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Email
The best way to contact me is through email, using the following address:
topaz@ualberta.ca
Mailing Address
Mathematical and Statistical Sciences
University of Alberta
632 Central Academic Building
Edmonton, Alberta
Canada T6G 2G1
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Research
A large part of my research since 2020 has been related to the formalization of pure mathematics, primarily using the Lean interactive theorem prover. I am one of the maintainers of mathlib, the mathematics library of Lean, which is built by the leanprover community. I am also a primary member of the Liquid Tensor Experiment, which was completed on 2022-07-14.
Here is a (incomplete) list of repositories containing formal mathematics where I was/am a contributor:
In Fall 2023, I taught a PIMS network-wide graduate course on the formalization of mathematics. Information about this course can be found on the PIMS webpage, and some additional comments can be found in this PDF.
Pure mathematics
My research interests in pure mathematics mostly revolve around Galois theory, with an emphasis on anabelian geometry and other related topics in arithmetic/algebraic geometry. I am also interested in various other topics, such as algebraic cycles, Hodge theory (Archimedean and p-adic), valuation theory, differential Galois theory, model theory of fields, etc.
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Papers
My papers are mostly all available on the arXiv. Here is my listing on mathscinet. The PDF links below are always the most up-to-date.
Preprints
- D. Asgeirsson, R. Brasca, N. Kuhn, F. A. E. Nuccio and A. Topaz. Categorical Foundations of Formalized Condensed Mathematics. arXiv
- A. Topaz. Algebraic dependence and Milnor K-theory. PDF
- A. Topaz. Alternating pairs with coefficients. PDF
- F. Pop and A. Topaz. A linear variant of GT. arXiv
- A. Topaz. Recovering function fields from their integral ℓ-adic cohomology with the Galois action. arXiv
Accepted/Published
- J. Commelin and A. Topaz. Abstraction boundaries and spec driven development in pure mathematics. To appear in Bull. Amer. Math. Soc. arXiv
- A. Topaz. A Torelli-like theorem for higher-dimensional function fields. To appear in JEMS (2022). arXiv
- J. Bell, R. Moosa and A. Topaz. Invariant Hypersurfaces. J. Inst. Math. Jussieu 21 (2022), no. 2, 713–739. arXiv DOI
- P. Guillot, J. Mináč and A. Topaz. Appendix by O. Wittenberg. Four-fold Massey products in Galois cohomology. Compositio Mathematica (2018) 154 (9), 1921-1959. arXiv DOI
- A. Topaz. The Galois action on geometric lattices and the mod-ℓ I/OM. Invent. Math. (2018) 213 (2), 371-459. Journal (open)
- A. Topaz. Abelian-by-Central Galois Groups I: A Formal Description. Trans. Amer. Math. Soc. (2017) 368, pg. 2721-2745. arXiv Journal
- A. Topaz. Commuting-Liftable Subgroups of Galois Groups II. J. reine angew. Math. (2017) 730, pg. 65-133. arXiv Journal
- A. Topaz. Reconstructing Function Fields from Rational Quotients of Mod-ℓ Galois Groups. Math. Annalen (2016) 366 (1), Pg. 337-385. arXiv Journal
- A. Topaz. Abelian-by-Central Galois Groups II: Definability of Inertia / Decomposition Groups. Israel J. Math. (2016) 215 (2), Pg. 713-748. arXiv
- A. Topaz. On the Nature of Base Fields. Appendix in ‘On The Minimized Decomposition Theory of Valuations’ by F. Pop, Bull. Math. Soc. Sci. Math. Roumanie. Tome 58(106) No. 3. Journal
- J. Mináč, J. Swallow and A. Topaz. Galois Module Structure of ℤ/ℓn-th Classes of Fields. Bull. London Math. Soc. (2014) 46 (1), Pg. 143-154 arXiv Journal
Other
- A. Topaz. Algebraic dependence and Milnor K theory. In Oberwolfach Reports: Arithmetic Homotopy and Galois Theory (2023).
- A. Topaz. A linear variant of GT (joint with F. Pop). In Oberwolfach Reports: Homotopic and Geometric Galois Theory (2021).
- A. Topaz. On the (generic) cohomology of function fields. In Oberwolfach Reports: Field Arithmetic (2018).
- A. Topaz. On Milnor K-groups of Function Fields. In Oberwolfach Reports: Valuation Theory and its Applications (2014).
- A. Topaz. Detecting Valuations Using Small Galois Groups. Valuation Theory in Interaction (Proceedings of the 2nd International Conference on Valuation Theory). Link
- A. Topaz. Pro-ℓ Galois groups and valuations. In Oberwolfach Reports: Arithmetic of Fields (2013).
- A. Topaz. Commuting-liftable subgroups of Galois groups. Ph.D. Thesis at the University of Pennsylvania (2013). Link
- A. Topaz. Almost-commuting-liftable subgroups of Galois groups. Manuscript (2012). Will not be submitted for publication. arXiv
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