What should you expect in intercollegiate classes?  What can you do to get the most out of them?  In this session, experienced class tutors will share their thoughts, including advice about online classes. 

All undergraduate and masters students welcome, especially Part B and MSc students attending intercollegiate classes. 

One of the prevailing paradigms in data analysis involves comparing groups of samples to statistically infer features that discriminate them. However, many modern applications do not fit well into this paradigm because samples cannot be naturally arranged into discrete groups. In such instances, graph techniques can be used to rank features according to their degree of consistency with an underlying metric structure without the need to cluster the samples. Here, we extend graph methods for feature selection to abstract simplicial complexes and present a general framework for clustering-independent analysis. Combinatorial Laplacian scores take into account the topology spanned by the data and reduce to the ordinary Laplacian score when restricted to graphs. We show the utility of this framework with several applications to the analysis of gene expression and multi-modal cancer data. Our results provide a unifying perspective on topological data analysis and manifold learning approaches to the analysis of point clouds.

In recent years a fruitful interplay has been unfolding between quantum chaos and black holes. In the first part of the talk, I provide a sampler of these developments. Next, we study the fate of the black hole interior at late times in simple models of quantum gravity that have dual descriptions in terms of Random Matrix Theory. We find that the volume of the interior grows linearly at early times and then, due to non-perturbative effects, saturates at a time and towards a value that are exponentially large in the entropy of the black hole. This provides a confirmation of the complexity equals volume proposal of Susskind, since in chaotic systems complexity is also expected to exhibit the same behavior.

Since the mid 1980s, there have been hints of a connection between 2-dimensional field theories and elliptic cohomology. This lead to Stolz and Teichner's conjectured geometric model for the universal elliptic cohomology theory of topological modular forms (TMF) for which cocycles are 2-dimensional (supersymmetric) field theories. Properties of these field theories lead to the expected integrality and modularity properties of classes in TMF. However, the abundant torsion in TMF has always been mysterious from the field theory point of view. In this talk, we will describe a map from 2-dimensional field theories to a cohomology theory that approximates TMF. This map affords a cocycle description of certain torsion classes. In particular, we will explain how a choice of anomaly cancelation for the supersymmetric sigma model with target $S^3$ determines a cocycle representative of the generator of $\pi_3(TMF)=\mathbb{Z}/24$.