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.
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.
Signature of smooth spin (8k+4)-manifold is divisible by 16 (in preparation)
Heights of complex orientable cohomology theories (in preparation)