## Hai Long Dao

Let \$R\$ be a standard graded algebra over a field. We investigate how the singularities of \$\Spec R\$ or \$\Proj R\$ affect the \$h\$-vector of \$R\$, which is the coefficients of the numerator of its Hilbert series. The most concrete consequence of our work asserts that if \$R\$ satisfies Serre's condition \$(S_r)\$ and have reasonable singularities (Du Bois on the punctured spectrum or \$F\$-pure), then \$h_0,\dots, h_r\geq 0\$. Furthermore the multiplicity of \$R\$ is at least \$h_0+h_1+\dots +h_{r-1}\$. We also prove that equality in many cases forces \$R\$ to be Cohen-Macaulay. This is joint work with Linquan Ma and Matteo Varbaro.

## Ritvik Ramkumar

The Hilbert scheme of n points in P^2 is smooth of dimension 2n and the tangent space to any (monomial) ideal admits a nice combinatorial description. On the other hand the Hilbert scheme of n points in P^3 is singular and there is a conjecture on what the monomial ideal with the largest tangent space dimension should be. By extending the combinatorial methods used in P^2, we give a proof of a major portion of the conjecture (in a sense we will describe). Along the way we will strengthen previous bounds on the dimension of the tangent space. This is joint (ongoing) work with Alessio Sammartano.