# TBA

# Mechanical systems exhibiting Arnold diffusion

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# Algebraic Analysis of Dirac Operators

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There are several analogues of the theory of one complex variable, when the values of the functions are taken in the division algebra H of quaternions, or in a suitable Clifford algebra. These theories rely on the construction of operators which somehow imitate the Cauchy-Riemann operator; in the quaternionic case one uses the Cauchy-Fueter operator, and in the Clifford case one uses the Dirac operator. The extension to several variables has remained elusive for a long time, but it can in fact be achieved if one considers these systems from the point of view of their algebraic properties. The analysis of such operators from the point of view of the Palamodov-Ehrenpreis Fundamental Principle allows the construction of a non-trivial theory in several variables. This talk will discuss the strength of this approach, as well as some of the questions which remain open, and will be concluded with a new twist on these theories.

# Indifference pricing with general semimartingales

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We consider a financial market where the discounted prices of the assets available for trading are modeled by semimartingales that are not assumed to be locally bounded. In this case the appropriate class of admissible integrands is defined through a random variable W that controls the losses incurred in trading. Applying the theory of Orlicz spaces, and convex analysis we study the utility maximization problem with an unbounded random endowment.

We then apply the duality relation to compute the indifference price of a claim satisfying weak integrability conditions. The indifference price leads to a convex risk measure defined on the Orlicz space associated to the utility function.

The talk is based on joint works with S. Biagini and with S. Biagini, M. Grasselli.

# Complete conformal metrics with negative Ricci curvature on compact manifolds with boundary.

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# Existence of rough solutions to the Einstein constraint equations without CMC or near-CMC conditions

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There is currently tremendous interest in geometric PDE, due in

part to the geometric flow program used successfully to attack the

Poincare and Geometrization Conjectures. Geometric PDE also play

a primary role in general relativity, where the (constrained) Einstein

evolution equations describe the propagation of gravitational waves

generated by collisions of massive objects such as black holes.

The need to validate this geometric PDE model of gravity has led to

the recent construction of (very expensive) gravitational wave

detectors, such as the NSF-funded LIGO project. In this lecture, we

consider the non-dynamical subset of the Einstein equations called

the Einstein constraints; this coupled nonlinear elliptic system must

be solved numerically to produce initial data for gravitational wave

simulations, and to enforce the constraints during dynamical simulations,

as needed for LIGO and other gravitational wave modeling efforts.

The Einstein constraint equations have been studied intensively for

half a century; our focus in this lecture is on a thirty-year-old open

question involving existence of solutions to the constraint equations

on space-like hyper-surfaces with arbitrarily prescribed mean extrinsic

curvature. All known existence results have involved assuming either

constant (CMC) or nearly-constant (near-CMC) mean extrinsic curvature.

After giving a survey of known CMC and near-CMC results through 2007,

we outline a new topological fixed-point framework that is fundamentally

free of both CMC and near-CMC conditions, resting on the construction of

"global barriers" for the Hamiltonian constraint. We then present

such a barrier construction for case of closed manifolds with positive

Yamabe metrics, giving the first known existence results for arbitrarily

prescribed mean extrinsic curvature. Our results are developed in the

setting of a ``weak'' background metric, which requires building up a

set of preliminary results on general Sobolev classes and elliptic

operators on manifold with weak metrics. However, this allows us

to recover the recent ``rough'' CMC existence results of Choquet-Bruhat

(2004) and of Maxwell (2004-2006) as two distinct limiting cases of our

non-CMC results. Our non-CMC results also extend to other cases such

as compact manifolds with boundary.

Time permitting, we also outline some new abstract approximation theory

results using the weak solution theory framework for the constraints; an

application of which gives a convergence proof for adaptive finite

element methods applied to the Hamiltonian constraint.

This is joint work with Gabriel Nagy and Gantumur Tsogtgerel.

# Complex Finsler Geometry and the Complex Homogeneoous Monge-Ampere Equation."

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The complex analogue of Diecke's Theorem and Brickell's Theorem in

real Finsler geometry. Complex Finsler structures naturally satisfy the complex

homogeneous Monge-Ampere equation and the analogue of Diecke's Theorem and

Brickell's Theorem can be put in the frame work of the classification of

complex manifolds admitting an exhaustion function satisfying the complex

homogeneous Monge-Ampere equation.

# A nonlinear PDE model for lakes and rivers

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I will briefly discuss some simple (and not-so-simple) nonlinear PDE describing growing "sandpiles". I will then introduce a new nonlinear PDE that in an asymptotic limit models the formation of "lakes" and "rivers" resulting from rainfall over a fixed landscape.

# Updating an Abel-Gauss-Riemann Program

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1st year calculus teachers use the equation Tp(cos(&#977;))=cos(p&#977;),

with Tp(w) the pth Chebychev polynomial. It is a map between complex

spheres branched over three points. I will explain why we call Tp a

dihedral function. Functions similar to it form one Mobius class:

equivalent by composing with fractional transformations.

Abel used more general dihedral Mobius classes. These form what we

now call the modular curve Y0(p). In "What Gauss Told Riemann About

Abel's Theorem" a lecture at John Thompson's 70th Birthday,

I cited Otto Neuenschwanden on the 60-year-old Gauss in conversation

with the 20-year-old Rieman. Their goal was to generalize Abel using

Gauss' harmonic functions. Riemann went far, but his early death left

an incomplete program.

To see why the generalization is non-obvious, consider: What is the

alternating (group) version of taking composites of Tp to form Tpk+1,

k 0?

This talk will use (and explain) alternating versions of modular curves

to connect two famous modern problems:

1). The Strong Torsion Conjecture (on Abelian Varieties); and

2). The Regular Inverse Galois Problem.

These spaces have cusps at points on their boundary. A cusp pairing

(the shift-incidence matrix ) helps picture these spaces. They aren't

modular curves. Still, using a result with J.P. Serre, we show how their

cusps resemble those of modular curves. That gives a version of the

renown Merel-Mazur result for these alternating spaces.