Space fullerenes
Space fullerenes are tilings of Euclidean space by fullerenes, occurring in metallurgy, crystallography, and optimization. I performed the enumeration of small space fullerenes and new Frank-Kasper structures.
Scientific and Technical Work
The Hermite constant in dimension 9
We solved the lattice packing problem in dimension 9 using a highly optimized implementation of Voronoi's algorithm. This work determined the exact value of the Hermite constant γ9 and classified all extreme lattices in this dimension.
Geometry of Numbers
We designed specific algorithms for enumerating Delaunay polytopes in lattices under symmetry. These polytopes form tessellations dual to Voronoi partitions, which are essential in computational geometry and physics.
Combinatorial types
Building on Fedorov's 1885 classification of 3D lattice Delaunay polytopes, I enumerated them in dimension 6. This research identified over 6,000 distinct combinatorial types.
Perfect Delaunay polytopes
Perfect Delaunay polytopes correspond to the extreme rays of the Erdahl polyhedral cone. I introduced combinatorial optimization techniques to find extreme Delaunay polytopes in dimensions 7 and 8.
L-type domains and the lattice covering problem
Using semidefinite programming, we optimized covering densities over L-type domains. This allowed us to find record lattice coverings in dimensions 9 through 15 and classify thin algebraic number fields.
Polyhedral computation with symmetry
I developed highly efficient algorithms for computing the dual description of polytopes with symmetry. This led to the classification of 10,916 perfect forms in dimension 8, a feat previously deemed impossible.
Elementary polycycles and Face regular maps
We classified elementary elliptic (R,q)-polycycles to create an efficient enumeration technique. This method was successfully applied to the classification and study of face-regular spheres and tori.
Zigzags and central circuits
We studied the structure of zigzags and central circuits in 3- and 4-valent plane graphs. This led to classification results for 2-connected graphs and new insights into the Goldberg-Coxeter construction.
Simplicial complexes
All simplicial n-complexes with specific face containment properties have been classified. This work provides a classification based on the partitions of set elements related to the dimension of the complex.
Wythoff polytopes
The Wythoff construction was used to determine L1-embeddable polytopes derived from regular polytopes. It was also applied to compute the third homology group of the Mathieu group M24.
Cube packings and tilings
We investigated periodic cube packings and tilings in R^n, proving extensibility theorems for dense packings. Our research provides lower bounds on non-extendible packings and explores their continuous parametrization.
Ginzburg Landau model
I established the phase diagram for the bulk Ginzburg-Landau model of superconductivity using complex hermitian geometry. This included establishing the Abrikosov asymptotic expansion for minimal energy states.
EVM Support in Linera
I implemented Ethereum Virtual Machine (EVM) support within the Linera protocol, allowing Ethereum-based applications to run on microchains. The implementation supports complex protocols like Morpho using specialized precompiles.
Bathymetry smoothing
We proposed a new linear programming method for smoothing sea depths in terrain-following oceanic models. This approach performs significantly better than existing methods by maintaining stability while staying near the real bathymetry.
Coupling of circulation and wave models
I coupled the circulation model ROMS with the wave model WWM to study their interactions. This coupling revealed significant effects on wave height in regions like the Bura jet and the Istrian coast.