Persistent homology is a central object of study in applied topology. It offers a flexible framework for defining invariants, called barcodes, of point cloud data and of real valued functions. Many of the key results of the last several years in the theory of persistent homology have been formulated in terms of a metric on barcodes called the bottleneck distance. There is a multi-parameter generalization of persistent homology, called multi-dimensional persistent homology, which is naturally suited to the study of noisy point cloud data.
We will discuss the notion of loops in linguistic structures, mainly in dictionaries. In a simplified view, a dictionary is a graph that links every word (vertex) to a set of alternative words (the definition) which in turn point to further descendants. Iterating through definitions, one may loop back to the original word. We will examine possible links between such definitional loops and the emergence of new concepts during the evolution of languages. Potential relation to living systems will be briefly discussed.
Our present framework for physics is difficult to modify without destroying its marvelous, successful properties. This provides a strong check on theoretical speculations and helps guide us to a small set of candidates for new laws. In this talk, Nima Arkani-Hamed, Professor in the School of Natural Sciences, illustrates these ideas in action by explaining why theoretical physicists knew the Higgs boson had to exist long before it was discovered at the Large Hadron Collider in July 2012. While the discovery of the Higgs is a triumph for both experimental and theoretical physics, its existence opens up a set of profound conceptual paradoxes, whose resolution is likely to involve radical new ideas. The talk concludes with a description of possible avenues of attack on these mysteries, and what we might learn from the LHC in this decade.
In this lecture, Juan Maldacena, Professor in the School of Natural Sciences, describes the theoretical ideas, developed in the 1960s and '70s, that led to the prediction of the Higgs boson, the particle that appears to have been discovered over the summer in 2012. The forces of nature are based on beautiful symmetries. Maldacena explains why the Higgs mechanism is necessary to avoid some of the naive consequences of these symmetries and to explain various features of elementary particles.
A knot is more or less what you think it is—a tangled mess of string in ordinary three-dimensional space. In the twentieth century, mathematicians developed a rich and deep theory of knots. And surprisingly, as Edward Witten, Charles Simonyi Professor in the School of Natural Sciences, explains in this lecture, it turned out that many of the most interesting ideas about knots have their roots in quantum physics.