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# 34th Geometry Festival

## Friday, April 5

### Fluid Mechanics and Geometry

Yann Brenier (ETH, Zurich)
Room 3206 at 3:00 PM
Abstract: Since 1966 and the seminal paper of V.I.Arnold on the geometric interpretation of the Euler equations of incompressible fluids as describing geodesic curves along some suitable groups of volume preserving maps, there have been many further connections between Geometry and Fluid Mechanics. A famous recent example is the work of De Lellis and Szekelyhidi linking the Euler equations to the Nash embedding theorem through "convex integration". I will report on other results, including the existence and uniqueness of a pressure gradient for the minimizing geodesic problem and the fluid mechanics interpretation of various problems related to geometric analysis optimal transportation maps, mean curvature flows, Born-Infeld-Dirac models...). In all these problems, convexity also plays a crucial role.

### Moment maps in symplectic and Kaehler geometry

Dietmar Salamon (CNRS, DMA-Ecole Normale Superieure)
Room 3206 at 4:30 PM
Abstract: In this talk I will explain Hamiltonian group actions on symplectic and Kaehler manifolds, starting with the concept of a moment map and moving on to some of their infinite-dimensional analgues. One such analogue is the Ricci form as a moment map for the action of the group of volume preserving diffeomorphisms on the space of almost complex structures. Another infinite-dimensional analogue leads to the Donaldson geometric flow on the space of symplectic forms on a smooth 4-manifold. This talk is based on joint works with Robin Krom, with Valentina Georgoulas and Joel Robbin, and with Oscar Garcia-Prada and Samuel
Trauwein.

## Saturday, April 6

### Zero sets of Laplace eigenfunctions

Aleksandr Logunov (IAS, Princeton)
Room 3206 at 10:00 AM
Abstract: TBD

### The enumerative geometry and arithmetic of some of the world’s Tiniest Calabi-Yau threefolds

Jim Bryan (University of British Columbia) AG
Room 3206 at 11:30 AM
AbstractWe construct new examples of Calabi-Yau threefolds with some of the smallest known Hodge numbers: ( h^{2,1} , h^{1,1} ) = (0,4) and (0,2) ). We explain how both the enumerative geometry (Gromov-Witten / Donaldson-Thomas theory) and the arithmetic (L-series/Galois representations) lead to interesting modular forms

### Boundary operator associated to sigma_k curvature

Yi Wang (Johns Hopkins University)
Room 3206 at 2:30 PM
Abstract: On a Riemannian manifold (M, g), the \sigma_k curvature is the k-th elementary symmetric function of the eigenvalues of the Schouten tensor A_g. It is known that the prescibing sigma_k curvature equation on a closed manifold without boundary is variational if k=1, 2 or g is locally conformally flat; indeed, this problem can be studied by means of the integral of the sigma_k curvature.  We construct a natural boundary functional which, when added to this energy, yields as its critical points solutions of prescribing sigma_k curvature equations with general non-vanishing boundary data. Moreover, we prove that the new energy satisfies the Dirichlet principle. If time permits, I will also discuss applications of our methods. This is joint work with Jeffrey Case.

### Spectral asymptotics on stationary spacetimes

Steven Zelditch (Northwestern University)
Room 3206 at 4:00 PM
Abstract: Spectral asymptotics on a compact Riemannian manifold (Sigma, h) concerns the igenvalues/eigenfunctions of its Laplacian Delta_Sigma as the eigenvalue tends to infinity. The two cornerstones of spectral asymptotics are the Weyl counting law for eigenvalues and the Gutzwiller-Duistermaat-Guillemin trace formula for the wave group. They are manifestly non-relativistic. My talk will explain how these (and virtually any) theorem of spectral symptotics admits a generalization to globally hyperbolic, stationary spacetimes with compact Cauchy hypersurface. The eigenvalues are `quasi-normal modes' similar in spirit to the well-known ones for black hole spacetimes. Joint work with Alex Strohmaier.

## Sunday, April 7

### Spherical Metrics with Conical Singularities

Xuwen Zhu (University of California, Berkeley)
Room 3206 at 9:30 AM
Abstract: The problem of finding and classifying constant curvature metrics with cone singularities has a long history bringing together several different areas of mathematics. This talk will focus on the particularly difficult spherical case where many new phenomena appear. When some of the cone angles are bigger than $2\pi$, uniqueness fails and existence is not guaranteed; smooth deformation is not always possible and the moduli space is expected to have singular strata. I will give a survey of several recent results regarding this singular uniformization problem, where the compactification obtained by clustering of cone points plays a central role. Based on joint work with Rafe Mazzeo.

### Nearly Fuchsian surface subgroups of finite covolume Kleinian groups

Alex Wright (University of Michigan)
Room 3206 at 11:00 AM
Abstract: We will present joint work with Jeremy Kahn proving that any complete cusped hyperbolic three manifold contains many "nearly isometrically immersed" closed hyperbolic surfaces.