Abstract: A theorem of Beauville implies that the general smooth quartic surface in $P^3$ may be expressed as the zerolocus of a Pfaffian associated to an 8 by 8 skew-symmetric matrix of linear forms. In this talk, I will discuss how the recent work of Aprodu-Farkas on the Green conjecture for curves on K3 surfaces may be used to generalize this statement to all smooth quartic surfaces in $P^3$. This is joint work with Emre Coskun and Rajesh Kulkarni.

Abstract: Let $p$ be a prime. A 1989 theorem of Ossa calculates the connective unitary K-theory of the smash product of two copies of the classifying space for the cyclic group of order $p$ and purports to calculate the corresponding orthogonal connective K-theory when $p=2$. Sadly the latter is wildly wrong!

Using a simple K\"{u}nneth formula short exact sequence I shall derive Ossa's unitary connective K-theory result in an elementary manner. As a corollary, I shall derive the correct version of Ossa's orthogonal theorem.

As an application of this result I shall show that there do not exist stable homotopy classes of $ {\mathbb RP}^{\infty} \wedge {\mathbb RP}^{\infty}$ in dimension $2^{s+1}-2$ with $s \geq 2$ whose composition with the Hopf map to $ {\mathbb RP}^{\infty}$ gives a stable homotopy element having Arf-Kervaire invariant one.