My work is at the intersection of symplectic geometry and low-dimensional topology. I use methods from bordered Heegaard Floer homology and Fukaya categories to study invariants of such objects as 3-manifolds, knots, mapping classes of surfaces.
In particular, these methods yield geometric invariants of 4-ended tangles, in the form of immersed curves on the boundary 4-punctured sphere. Currently, I think about knot theoretic applications of these immersed curve invariants, in the context of Khovanov, Heegaard Floer and instanton Floer theories. Take a look at my research statement and papers below for more details.
Teachers and collaborators:
- Immersed curves in Khovanov homology ( video | slides )
- Khovanov multicurves are linear
- Cosmetic operations and Khovanov multicurves
- Thin links and Conway spheres
- Khovanov invariants via Fukaya categories: the tangle invariants agree
- The correspondence induced on the pillowcase by the earring tangle ( video | slides ) To appear in Journal of Topology.
- A mnemonic for the Lipshitz-Ozsváth-Thurston correspondence To appear in Algebraic & Geometric Topology.
- Khovanov homology and strong inversions To appear in The Open Book Series.
- Bordered theory for pillowcase homology Mathematical Research Letters 26(5), 1467-1516.
- Comparing homological invariants for mapping classes of surfaces Michigan Mathematical Journal 30(3), 503-560.
- Minimal and Hamiltonian-minimal submanifolds in toric geometry Journal of Symplectic Geometry 14(2), 431-448.
- Blockchains: theory and applications () Stony Brook University, Fall ‘22Blockchain course
This course is an introduction to modern cut-and-paste techniques in knot theory, with a focus on Khovanov homology. Namely, we learn how Khovanov homology changes if one substitutes one 4-ended tangle in a knot with another. Examples of such an operation are Conway mutation and crossing changes.
Here are the main references we use: definition of Khovanov homology, s-invariant, Khovanov homology for tangles, Lagrangian Floer homology for curves, Khovanov homology for tangles via immersed curves.
We cover all this material from a modern viewpoint, using the latest developed algebraic tools. But no special background is needed, all the needed homological algebra is introduced.
- Calculus I ( lecture notes ) Indiana University, Fall ‘20
- Linear Algebra and Applications ( lecture notes ) Indiana University, Spring ‘20
- Linear Algebra with Applications ( lecture notes ) Princeton University, Spring ‘17 + Fall ‘16
“Overall, Dr. Kotelskiy is the best math teacher I have ever had”
“Artem is incredibly knowledgeable and is a great math professor.”
“Artem is enthusiastic about mathematics, and he wants his students to be as well. I believe that this is the best attribute of any instructor.”
“I thought Artem was a great person and cared a lot about the course.”
“I enjoyed his teaching style and how easily accessible Artem was after class. He would respond quickly to emails and would be very helpful during his office hours.”
“I think that the single most valuable thing in this course was Artem's availability and willingness to meet and help with students. At any time, students always felt welcome to engage in a conversation with Artem about anything math, and especially our projects.”
“Artem was an amazing professor who brought an in depth knowledge of blockchain technology to lecture.”
“Artem is incredibly passionate about the subject matter and that was shown throughout his lectures.”
Interestingly, the following evaluation was as enjoyable to read as the ones above. “He doesn't curve the whole class after an exam just so he can get the class average he wants. You're rewarding students who didn't take the time to study as hard as the top students. I have never had a teacher tell students to drop the class as much as Artem. Every other class period after the first exam he kept telling us when the drop date was and when it got closer he just kept saying "You need to strongly think about dropping the class". He is not a motivator at all i.e. you're not a Professor. You're just an asshole who knows a lot about math.”
2013-today: doing math in the US (PhD in Princeton, then postdoc in Indiana University, then in Stony Brook).
2004-2013: studied math in Moscow (high-school #57, then Lomonosov MSU).
1991: born in Yerevan, on the same day when Armenia declared its independence from the USSR.
Apart from math I am also into blockchains, game go (~1dan), chess, volleyball, table tennis. I also have some programming expirience: