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.
2. A primer on zeta functions and decomposition spaces (2020). Also available at arXiv:2011.13903.
1. “How to Have Lunch in the Time of COVID-19”, with K. DeVleming. Notices of the AMS (January 2021).
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).
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.
Abstract Algebra (Wake Forest, Fall 13 - Spring 14)
Algebraic Geometry of Curves (Virginia, Fall 16)
Algebraic Number Theory (Virginia, Spring 16)
Algebraic Stacks (Virginia, Spring 17 - Spring 20)
Algebraic Topology (Virginia, Spring - Fall 16)
Analysis of Banach Spaces (Wake Forest, Spring 14)
Analytic Number Theory (Wake Forest, Fall 13)
Arithmetic Fundamental Group (Virginia, Spring 17)
Class Field Theory (Wake Forest, Fall 13 - Spring 15)
Commutative Algebra (Virginia, Spring 16/19)
Complex Analysis (Wake Forest, Fall 10)
Complex Surfaces (in progress)
Derived Categories (in progress)
Differential Geometry (Wake Forest, Spring 15)
Differential Topology (Virginia, Fall 15)
Elementary Number Theory (Wake Forest, Spring 13)
Étale Cohomology (in progress)
Étale Theory (Virginia, Fall 18 - Spring 19)
Fibre Bundles (Virginia, Spring 17/18)
Galois Cohomology (in progress)
General Topology (Wake Forest, Fall 13 - Spring 14)
Generalized Jacobians (Virginia, Fall 16 - Spring 17)
Homological Algebra (Wake Forest, Fall 14 / Virginia, Spring 17)
Homotopy Theory (Virginia, Fall 17)
Hyperbolic Geometry (in progress)
L-Functions and Modular Forms (Virginia, Fall 17 - Spring 18)
Lie Groups (Virginia, Fall 15 / Spring 18)
Linear Algebra (Wake Forest, Fall 14)
Measure Theory (Virginia, Spring 16)
Modular Forms (Virginia, Fall 19)
Noncommutative Algebra (Virginia, Fall 15)
Numerical Methods (Wake Forest, Fall 14)
Probabilistic Measure Theory (Wake Forest, Spring 15)
Real Analysis (Wake Forest, Fall 12)
Representation Theory (Wake Forest, Fall 14)
Sheaf Cohomology (Virginia, Fall 18)
Short Course on Schemes (Virginia, Summer 17)
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!