A common pastime of geometric group theorists is to try and derive algebraic information about a group from the geometric properties of its Cayley graphs. One of the most classical demonstrations of this can be seen in the work of Maschke (1896) in characterising those finite groups with planar Cayley graphs. Since then, much work has been done on this topic. In this talk, I will attempt to survey some results in this area, and show that the class group with planar Cayley graphs is QI-rigid.

The virtual Haken theorem is one of the most influential recent results in 3-manifold theory. The statement dates back to Waldhausen, who conjectured that every aspherical closed 3-manifold has a finite cover containing an essential embedded closed surface. The proof is usually attributed to Agol, although his virtual special theorem is only the last piece of the puzzle. This talk is dedicated to the unsung heroes of virtual Haken, the mathematicians whose invaluable work helped turning this conjecture into a theorem. We will trace the history of a mathematical thread that connects Thurston-Perelman's geometrisation to Agol's final contribution, surveying Kahn-Markovic's surface subgroup theorem, Bergeron-Wise's cubulation of 3-manifold groups, Haglund-Wise's special cube complexes, Wise's work on quasi-convex hierarchies and Agol-Groves-Manning's weak separation theorem.

We will look at the vanishing of group cohomology from the perspective of Kazhdan’s property (T). We will investigate an analogue of this property for any degree, introduced by U. Bader and P. W. Nowak in 2020 and describe a method of proving these properties with computers.

We will give an introduction to hyperbolic cusps and their Dehn fillings. In particular, we will give a brief survey of quantitive results in the field. To motivate this work, we will sketch how these techniques are used for studying the classical question of characteristic slopes on knots.

This talk is concerned with the question of generating the homology of a covering space by lifts of simple closed curves (from topological viewpoint), and generating the first homology of a subgroup by powers of elements outside certain filtrations (from group-theoretic viewpoint). I will sketch Malestein's and Putman's construction of examples of branched covers where lifts of scc's span a proper subspace. I will discuss the relation of their proof to the Magnus embedding, and present recent results on similar embeddings of surface groups which facilitate extending their theorems to unbranched covers.

There has been recent advances in understanding the microscopic origin of the Bekenstein-Hawking entropy of supersymmetric ant de Sitter (AdS) black holes using holography and localization applied to the dual quantum field theory. In this talk, after a brief overview of the general picture, I will propose a BPS partition function -- based on gluing elementary objects called gravitational blocks -- for known AdS black holes with arbitrary rotation and generic magnetic and electric charges. I will then show that the attractor equations and the Bekenstein-Hawking entropy can be obtained from an extremization principle.

Nonlinear filtering is a central mathematical tool in understanding how we process information. Unfortunately, the equations involved are often high dimensional, and therefore, in practical applications, approximate filters are often employed in place of the optimal filter. The error introduced by using these approximations is generally poorly understood. In this talk we will see how, in the case where the underlying process is a continuous-time, finite-state Markov Chain, results on the stability of filters can be strengthened to yield bounds for the error between the optimal filter and a general approximate filter.  We will then consider the 'projection filter', a low dimensional approximation of the filtering equation originally due to D. Brigo and collaborators, and show that its error is indeed well-controlled. The talk is based on joint work with Sam Cohen.

Stochastic portfolio theory is a framework to study large equity markets over long time horizons. In such settings investors are often confined to trading in an “open market” setup consisting of only assets with high capitalizations. In this work we relax previously studied notions of open markets and develop a tractable framework for them under mild structural conditions on the market.

Within this framework we also introduce a large parametric class of processes, which we call rank Jacobi processes. They produce a stable capital distribution curve consistent with empirical observations. Moreover, there are explicit expressions for the growth-optimal portfolio, and they are also shown to serve as worst-case models for a robust asymptotic growth problem under model ambiguity.

Time permitting, I will also present an extended class of models and illustrate calibration results to CRSP Equity Data.

This talk is based on joint work with Martin Larsson.