University of Oxford

Asymptotic dimension was introduced by Gromov as an invariant of finitely generated groups. It can be shown that if two metric spaces are quasi-isometric then they have the same asymptotic dimension. In 1998, the asymptotic dimension achieved particular prominence in geometric group theory after a paper of Guoliang Yu, which proved the Novikov conjecture for groups with finite asymptotic dimension. Unfortunately, not all finitely generated groups have finite asymptotic dimension. 

In this talk, we will introduce some basic tools to compute the asymptotic dimension of groups. We will also find upper bounds for the asymptotic dimension of a few well-known classes of finitely generated groups, such as hyperbolic groups, and if time permits, we will see why one-relator groups have asymptotic dimension at most two.

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 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.

For many stationary-state PDEs, solutions can be shown to satisfy certain key identities or structures, with physical interpretations such as the dissipation of energy. By reformulating these systems in terms of new auxiliary functions, finite-element models can ensure these structures also hold exactly for the numerical solutions. This approach is known to improve the solutions' accuracy and reliability.

In this talk, we extend this auxiliary function approach to the transient case through a finite-element-in-time interpretation. This allows us to develop novel structure-preserving timesteppers for various transient problems, including the Navier–Stokes and MHD equations, up to arbitrary order in time.

In this talk, we introduce Newton-MR variants for solving nonconvex optimization problems. Unlike the overwhelming majority of Newton-type methods, which rely on conjugate gradient method as the primary workhorse for their respective sub-problems, Newton-MR employs minimum residual (MINRES) method. With certain useful monotonicity properties of MINRES as well as its inherent ability to detect non-positive curvature directions as soon as they arise, we show that our algorithms come with desirable properties including the optimal first and second-order worst-case complexities. Numerical examples demonstrate the performance of our proposed algorithms.

The SYZ conjecture is a geometric way of understanding mirror symmetry via the existence of dual special Lagrangian fibrations on mirror Calabi-Yau manifolds. Motivated by this conjecture, it is expected that $G_2$ and $Spin(7)$-manifolds admit calibrated fibrations as well. I will explain how to construct examples of a weaker type of fibration on compact $Spin(7)$-manifolds obtained via gluing, and give a hint as to why the stronger fibrations are still elusive. The key ingredient is the stability of the weak fibration property under deformation of the ambient $Spin(7)$-structure.

In this talk, I give a statement of the “Galois to automorphic” direction of categorical geometric Langlands. I will describe the Galois and automorphic side, the Hecke action on both sides, and the definition of Hecke eigensheaves. On the way, I hope to give motivation for the various objects at play : the stack of $G^L$ local systems on the fixed curve $X$, the stack of $G$ bundles on $X$, $D$-modules, arc groups, loop groups, the affine Grassmannian, and geometric Satake.