**Exotic slice disks and symplectic surfaces- Kyle Hayden (Columbia University)**

Abstract: One approach to understanding the smooth topology of a 4-dimensional manifold is to study the embedded surfaces it contains. I'll construct new examples of "exotically knotted" surfaces in 4-manifolds, i.e. surfaces that are isotopic through ambient homeomorphisms but not through diffeomorphisms. We'll begin with simple examples of exotic disks in the 4-ball. Then we'll turn to the symplectic setting and exhibit new types of exotic phenomena among symplectic, holomorphic, and Lagrangian surfaces.

**Doubly slice pretzels- Clayton McDonald (Boston College)**

Abstract: A knot K in S^3 is slice if it is the cross section of an embedded sphere in S^4, and it is doubly slice if the sphere is unknotted. Although slice knots are very well studied, doubly slice knots have been given comparatively less attention. We prove that an odd pretzel knot is doubly slice if it has 2n+1 twist parameters consisting of n+1 copies of a and n copies of -a for some odd integer a. Combined with the work of Issa and McCoy, it follows that these are the only doubly slice odd pretzel knots. Time permitting, we might go over some preliminary results involving links as well.

**Topologically embedding spheres in knot traces- Patrick Orson (Boston College)**

Abstract: Knot traces are smooth 4-manifolds with boundary, that are homotopic to the 2-sphere, and obtained by attaching a 2-handle to the 4-ball along a framed knot in the 3-sphere. I will give a complete characterisation for when the generator of the second homotopy group of a knot trace can be represented by a locally flat embedded 2-sphere with abelian exterior fundamental group. The answer is in terms of classical and computable invariants of the knot. This result is directly analogous to the the result of Freedman and Quinn that says a knot with Alexander polynomial 1 is topologically slice, and can be used to exhibit new exotic 4-dimensional phenomena. This is a joint project with Feller, Miller, Nagel, Powell, and Ray.

**Regular homotopy and Gluck twists- Hannah Schwartz (MPIM Bonn)**

Abstract: Any 2-sphere K smoothly embedded in the 4-sphere is related to the unknotted sphere through a finite sequence of locally supported homotopies called finger moves and Whitney moves. We call the minimum number of finger moves (or equivalently Whitney moves) in any such a homotopy the "Casson-Whitney number" of the sphere K. In this talk, I will discuss joint work with Joseph, Klug, and Ruppik showing that if the Casson-Whitney number of K is equal to one, then the unknotted torus can be obtained by attaching a single 1-handle to K. I will also present an application of this result, from joint work with Naylor, that the Gluck twist of any sphere with Casson-Whitney number equal to one is diffeomorphic to the standard 4-sphere, and give some well-known families of 2-spheres for which this is the case.