Andrew Kobin’s Lecture Notes

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Research Interests

Click here to read my research statement.

My research lies in three main areas: algebraic geometry, number theory and topology. A unifying theme is the theory of algebraic stacks. For me, these are essential tools for understanding some of the following topics:

  • Arithmetic geometry: covers of curves, wild ramification, étale fundamental groups, formal orbifolds

  • Algebraic geometry: moduli problems, structure of Deligne-Mumford stacks, deformation theory

  • Number theory: modular curves, modular forms, class field theory, zeta and L-functions

  • Homotopy theory: A¹ (or “enriched”) enumerative geometry, homotopy theory of stacks, zeta functions via decomposition spaces/2-Segal spaces, incidence algebras, homotopy linear algebra

One of my main projects is to bring techniques from finite characteristic arithmetic geometry (e.g. Artin-Schreier-Witt theory) into the world of stacks. This has opened a number of exciting doors for future research.

Publications

2. Artin-Schreier root stacks. Journal of Algebra (to appear). Also available at doi.org/10.1016/j.jalgebra.2021.07.023 and arXiv:1910.03146.

1. Crossing number bounds in knot mosaics, with H. Howards. Journal of Knot Theory and its Ramifications, vol. 27, no. 10 (2018). Also available at doi:10.1142/S0218216518500566 and arXiv:1405.7683.

Preprints

2. A primer on zeta functions and decomposition spaces (2020). Also available at arXiv:2011.13903.

1. A¹-local degree via stacks, with L. Taylor (2019). Also available at arXiv:1911.05955.

Other writing

1. “How to Have Lunch in the Time of COVID-19”, with K. DeVleming. Notices of the AMS (January 2021).

Projects

6. Zeta functions and decomposition spaces (in progress). So far has produced the preprint “A primer on zeta functions and decomposition spaces” (see Preprints).

5. A¹-local degree via stacks, with Libby Taylor (in progress). So far has produced the preprint “A¹-local degree via stacks” (see Preprints).

4. Geometric wild ramification invariants, with Vaidehee Thatte (in progress).

3. Wild Ramification and Stacky Curves. PhD dissertation at University of Virginia. Adviser: Andrew Obus. Available at doi:10.18130/v3-y9s6-kt47. So far has produced the article “Artin-Schreier root stacks” (see Publications).

2. Class Field Theory and the Study of Symmetric n-Fermat Primes. Master’s thesis at Wake Forest University. Adviser: Frank Moore.

1. Saturation in Knot Mosaics. Senior thesis at Wake Forest University. Adviser: Hugh Howards. Culminated in the paper “Crossing number bounds in knot mosaics” (see Publications).

Notes

I keep notes on many courses and seminars I have been a part of. There are sure to be some errors, both cosmetic and mathematical, so if you find any, please contact me at akobin (at) ucsc (dot) edu. Also, LaTeX files are available upon request. 

Although my philosophy is that 'all math is connected', I have grouped many of the above notes by area for ease of cross-reference. These larger files are available below. 

Thanks to Matt Feller, George Seelinger, Richard Vradenburgh and many others for noticing errors!