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)