There are many examples in mathematics, both pure and applied, in which
problems with symmetric formulations have non-symmetric solutions.
Sometimes this symmetry breaking is total, as in the example of
turbulence, but often the symmetry breaking is only partial. One technique
that can sometimes be used to constrain the symmetry breaking is
reflection positivity. It is a simple and useful concept that will be
explained in the talk, together with some examples. One of these concerns
the minimum eigenvalues of the Laplace operator on a distorted hexagonal
lattice. Another example that we will discuss is a functional inequality
due to Onofri.
The talk is based on joint work with E. Lieb.

In single-molecule microscopy it is necessary to reconstruct a signal that consists of positive point sources from noisy observations of the spectrum of the signal in the low-frequency band [−fc,fc]. It is shown that the problem can be solved using convex optimization in a stable fashion. The stability of reconstruction depends on Rayleigh-regularity of the support of the signal, i.e., on how many point sources can occur within an interval of length 1.87/fc. The stability estimate is complimented by a converse result: the performance of convex algorithm is nearly optimal. The results are generalized to multi-dimension signals. Applications in microscopy are briefly discussed.

Discussion about the paper "Bivariate Polynomials Modulo Composites and their Applications" by Dan Boneh and Henry Corrigan-Gibbs  

For a closed Riemann surface X and complex reductive Lie
group G, the moduli space of G-Higgs bundles on X
is a hyperkaehler algebraic completely integrable system
that plays an important role in moduli space theory,
representations of surface groups, and supersymmetric gauge
theories.  The uniformization of X and the choice of a principal SL2 in G
give rise to a distinguished point in the moduli space called the
Fuchsian point.  In this talk I will discuss the first order
behavior of certain geometric and dynamical quantities at the
Fuchsian point. These may be regarded as "higher" analogs of
results in Teichmueller theory and for complex projective
structures.  This is joint work with Francois Labourie.

We develop new results on global existence and convergence
of solutions to the gradient flow equation for the Yang-Mills energy
functional on a principal bundle, with compact Lie structure group, over
a closed, four-dimensional, Riemannian, smooth manifold, including the
following. If the initial connection is close enough to a minimum of the
Yang-Mills energy functional, in a norm or energy sense, then the
Yang-Mills gradient flow exists for all time and converges to a
Yang-Mills connection. If the initial connection is allowed to have
arbitrary energy but we restrict to the setting of a Hermitian vector
bundle over a compact, complex, Hermitian (but not necessarily Kaehler)
surface and the initial connection has curvature of type (1,1), then the
Yang-Mills gradient flow exists for all time, though bubble
singularities may (and in certain cases must) occur in the limit as time
tends to infinity. The Lojasiewicz-Simon gradient inequality plays a crucial role in our approach and we develop two versions of that inequality for the
Yang-Mills energy functional.