Maria Chudnovsky, Columbia University

Perfect graphs are a class of graphs that behave particularly well with respect to coloring. In the 1960’s Claude Berge made two conjectures about this class of graphs, that motivated a great deal of research in the field in the second half of the 20th century. One of the conjectures, The Weak Perfect Graph Conjecture, was solved by Laszlo Lovasz in 1972, the other, The Strong Perfect Graph Conjecture, by Robertson, Seymour, Thomas and the speaker in the early 2000’s.

The following remained open however: design a combinatorial algorithm that produces an optimal coloring of a perfect graph. Recently, in joint work with Lo, Maffray, Trotignon and Vuskovic we were able to construct such an algorithm under the additional assumption that the input graph is square-free (contains no induced four-cycle). Historically, the class of square-free perfect graphs seems to be a good special case to look at; in particular, this was the last special case of the Strong Perfect Graph Conjecture that was solved before the general one.

Ingrid Daubechies, Duke University

Mathematicians can help Art Historians and Art Conservators in studying and help understand art works, their manufacture process and their state of conservation.

The presentation will review several instances of such collaborations in the last decade or so, and then focus on one particular example: virtual cradle removal.

Between the 12th to the 17th century, European artists typically painted on wooden boards. To remediate or prevent structural or insect damage, conservators in the 19th and first half of the 20th century first thinned the panels to a few mm, and then strengthened the much thinner wood structures by (permanently) attaching to their backs hardwood lattices called cradles. These cradles are highly visible in X-ray images of the paintings. X-rays of paintings are a useful tool for art conservators and art historians to study the condition of a painting, as well as the techniques used by the artist and subsequent restorers. The cradling artifacts obstruct a clear “reading” of the X-rays by these experts. We now can remove these artifacts automatically, using a variety of mathematical tools, including Bayesian algorithms.

Jill Pipher, Brown University

In this talk, we survey some of the many modern developments in Fourier analysis that arise through the representation of functions by a discrete wavelet basis such as the Haar or Walsh system, as opposed to the traditional basis of sine and cosine waves. The Haar basis is the simplest example of a \wavelet” representation, and also gives rise to simple martingales. For many decades, dyadic analysis, via discrete or Haar basis representation, has been viewed as providing a toy example of the complicated structure of continuous function spaces. But in recent years, many problems about continuous function spaces in harmonic analysis have been effectively reduced to their discrete analogs. These reductions have resulted in solutions to old conjectures as well as applications to metric structures important in graph theory and computer science.

Katrin Wehrheim, UC Berkeley

I will introduce string diagrams for 2-categories to illuminate connections between low dimensional topology, symplectic geometry, and the analysis of pseudoholomorphic curves.  On the one hand, the construction of topological invariants (eg. Heegard Floer homology for 3-manifolds) via symplectic geometry can be understood in terms of isomorphisms in a symplectic 2-category. Here the crucial algebraic identity corresponds to strip shrinking in a string diagram, which represents an adiabatic limit between different types of elliptic PDEs. On the other hand, a novel singularity formation in this adiabatic limit is naturally represented as generalized string diagram, which in turn gives rise to new algebraic structures.