# Jiahao Hu

## 胡 家
昊

I am a third-year math PhD student at Stony Brook University. I'm
interested in topology. My advisor is Dennis Sullivan.

**Office**: 5-125B, Math Department, Stony Brook University
**Email**: jiahao.hu@stonybrook.edu

## Research

**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.

## Notes

**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.