I am a third-year math PhD student at Stony Brook University. I'm interested in topology. My advisor is Dennis Sullivan.
On the non-existence of almost complex manifolds with sum of Betti number 3 (informal, email me if you spot a mistake) In this note we prove that there does not exist an almost complex manifold whose sum of Betti numbers is 3 in complex dimension greater or equal to 3. Albanese and Milivojevic have already proven that such a manifold does not exist except possibly for dimension being a power of 2. We manage to rule out power of 2 as well. This way, we complete the proof of the assertion that total betti number of a complex manifold of complex dimension grater or equal to 4 is at least 4.
Topological resolution of singularities In this note, we review the topological obstructions to resolving the singularities discovered by Thom, and show these obstructions vanish for complex algebraic varieties of (complex) dimension less or equal to 8 without using Hironaka's theorem.
Adams spectral sequence and applications to cobordism I In topology, there are two sets of invariants that are of the most interest: homotopy and (co)homology. Homology groups are relatively easy to compute but homotopy groups usually tell us more about the space. Adams spectral sequence allows one to extract information from (co)homology to compute homotopy. In this note, we construct Adams spectral sequence and apply it to compute unoriented bordism ring, which is isomorphic to the stable homotopy groups of Thom space of orthogonal group.
Elliptic cohomology and elliptic genera This is a set of notes (1, 2, 3 ,4, 5) for a series of talks I gave at Student Topology Seminar in spring 2020, Stony Brook.