My academic research topic is Presburger Arithmetic and its connections to various problems in Discrete Geometry. This is motivated by Integer Points in Polyhedra and their connections to Integer Programming and Optimization. The questions that I investigate mostly carry some algorithmic/complexity flavors.
I was greatly honored to receive the 2018 Sacks Prize by the Association of Symbolic Logic for the most outstanding PhD dissertations in logic. Here is a version of my dissertation.
My articles/preprints:
12. On the number of integer points in translated and expanded polyhedra,
with Igor Pak,
Discrete Comput. Geom. (2020), journal link; arxiv.
11. Presburger Arithmetic with algebraic scalar multiplications,
with Philipp Hieronymi and Igor Pak,
Log. Methods Comput. Sci. (2021), journal link; arxiv.
10. Parametric Presburger Arithmetic: Complexity of Counting and Quantifier Elimination,
with Tristram Bogart, John Goodrick and Kevin Woods,
Math. Log. Q. 65 (2019), journal link; arxiv.
9. VC-dimension of short Presburger formulas,
with Igor Pak,
Combinatorica 39 (2019), journal link; arxiv.
8. Short Presburger Arithmetic is hard,
with Igor Pak,
SIAM J. Comput. (2022), journal link; conference version in FOCS 2017; arxiv.
7. The computational complexity of integer programming with alternations,
with Igor Pak,
Math. Oper. Res. 45 (2019), journal link; conference version in CCC 2017; arxiv.
6. Complexity of short generating functions,
with Igor Pak,
Forum of Mathematics, Sigma 6 (2018) E1., journal link; arxiv.
5. Complexity of short Presburger arithmetic,
with Igor Pak,
4. Enumerating projections of integer points in unbounded polyhedra,
with Igor Pak,
SIAM J. Discrete Math. 32 (2018), journal link; conference version in IPCO 2017; arxiv.
3. Tiling the integer lattice with translated sublattices,
with Maciej Borodzik an Sinai Robins,
Moscow Journal of Combinatorial Number Theory 6 (2016), 407–430, journal link; arxiv.
2. Solid angles associated to Minkowski reduced bases,
from my undergraduate thesis,
preprint; arxiv.
1. The inverse moment problem for convex polytopes: implementation aspects,
with Nick Gravin, Dmitrii Pasechnik and Sinai Robins,
technical report; arxiv.