**Note: This talk begins with a pre-seminar (aimed at graduate students) at 3:30–4:00. The main talk starts at 4:10.**

**Join Zoom Meeting: https://washington.zoom.us/j/ 91547335974Meeting ID: 915 4733 5974**

A simplicial complex on $n$ vertices is \$s\$-neighborly if it has the same \$(s − 1)\$-skeleton as the \$(n − 1)\$-simplex on the same vertex set. While a \$(d-1)\$-dimensional sphere with \$n>d+1\$ vertices cannot be more than \$\lfloor d/2\rfloor\$-neighborly, \$\lfloor d/2\rfloor\$-neighborly \$(d-1)\$-dimensional spheres with \$n\$ vertices do exist. How many such neighborly spheres are there? We will present a recent construction showing that for \$d\geq 5\$, there are at least \$2^{\Omega(n^{\lfloor (d-1)/2\rfloor})}\$ distinct combinatorial types of neighborly spheres.

Joint work with Hailun Zheng.