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.

The best way to contact me is through email, using the following address:

`topaz@ualberta.ca`

```
Mathematical and Statistical Sciences
University of Alberta
632 Central Academic Building
Edmonton, Alberta
Canada T6G 2G1
```

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.

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.

- 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

- 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/ℓ*Bull. London Math. Soc. (2014) 46 (1), Pg. 143-154 arXiv Journal^{n}-th Classes of Fields.

- 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