About
I am a mathematician working primarily in algebra and arithmetic / algebraic geometry. My research interests 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.
I am currently an Assistant 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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A large part of my work 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 Lean3, which is built by the leanprover community. I am also a primary member of the Liquid Tensor Experiment, which was completed as of 2022-07-14.
Here is a (incomplete) list of repositories containing formal mathematics where I was/am a contributor:
In Fall 2023, I will be teaching 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.
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Research
My papers are all available on the arXiv. Here is my listing on mathscinet. The PDF links below are always the most up-to-date.
Preprints
- 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. A linear variant of GT (joint with F. Pop). In Overwolfach Reports: Homotopic and Geometric Galois Theory (2021).
- A. Topaz. On the (generic) cohomology of function fields. In Overwolfach 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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