Talks

In this talk, I will discuss recent results for charge density wave (CDW) and pair density wave (PDW) instabilities on the kagome lattice near a van Hove singularity. I then discuss the relevance of these results to experimental observations in kagome metals such as CsV_3Sb_5.

The ultranodal superconducting state exhibits very unusual physical properties since it has a strongly enhanced low energy density of states compared to a nodal state. This is due to the existence of a so-called Bogoliubov Fermi surface which is topologically protected and can emerge in a multiband system if a spin singlet pairing gap coexists with a nonunitary interband triplet component. Starting from a microscopic model, I will discuss how such a ultranodal state can be stabilized and examine signatures in the low temperature specific heat, tunneling spectroscopy and spin-relaxation rate pointing towards the existence of Bogoliubov Fermi surfaces. It turns out that FeSe doped with S seems to exhibit a number of these features and might be a strong candidate material to study consequences of Bogoliubov Fermi surfaces.

The phase diagram of extremal black holes in supergravity is surprisingly rich. In some regimes, quantum effects are so strong that they dominate. On the supersymmetric locus, there is a large ground state degeneracy, protected by a gap. Throughout, there is an intricate classical interplay between charge and rotation that gives rise to instability via various mechanisms, including superradiance and superconductivity. The talk highlights examples from black holes in AdS(3) and AdS(5).

Space group symmetries and multipole moments allowed by them can be used to not only classify different phases but also to predict the macroscopic response induced by structural or electronic order parameters. Similarly, magnetic space groups can be utilized to predict macroscopic response induced by magnetic orders. In this talk, we apply similar ideas to possible charge density wave orders in AV3Sb5 Kagome materials with time-reversal symmetry breaking imaginary charge density waves. After showing that charge density wave orders can be represented by magnetic irreducible representations of the space group, we tabulate the different phases induced by them, and predict experimental signatures therein. In particular, we focus on piezomagnetism and spontaneous gyrotropic birefringence and show that these two complimentary tensors can differentiate between most possible phases in Kagome compounds.

Let Sg be a closed surface of genus g ≥ 2. The curve graph corresponding to Sg, denoted by C(Sg), is a 1-dimensional simplicial complex whose vertices are isotopy classes of essential closed curves on Sg and two vertices share an edge if they represent mutually disjoint curves. Little is known about curves which are at a distance n ≥ 4 apart in C(Sg). This is primarily because the local infinitude of the vertices in C(Sg) hinders the calculation of distances in C(Sg).

In this talk, we will look at a family of pairs of curves on Sg which are at a distance 4 apart in C(Sg). These curves are created using Dehn twists. As an application, we will deduce an upper bound on the minimal intersection number of curves at a distance 4 apart in C(Sg). Finally, we will look at an example of a pair of curves on S2 which are at a distance 5 apart in C(S2).

A random walk on a Fuchsian group (that is, a lattice in SL(2,R)) gives a random walk in the hyperbolic plane. Under mild conditions, a typical sample path for such a walk converges to the circle at infinity. The distribution of the limits of sample paths define a stationary measure on the circle. This talk will survey the landscape of results related to the study of such stationary measures on the circle and in analogous contexts.

By works of Agol, Bergeron–Wise, Dufour, and Kahn–Markovic, among others, it is known that fundamental groups of fibred hyperbolic 3-manifolds (and those of closed hyperbolic 3-manifolds in general) act geometrically on CAT(0) cube complexes and thus these manifolds possess strong group theoretic and topological properties. However, it is unknown if these results extend to the fundamental groups of all hyperbolic surface bundles over finite graphs (in other words, arbitrary hyperbolic surface-by-free groups). In this talk, we will describe how, after modifying these bundles by drilling (that is, removing small tubular neighbourhoods of) enough simple closed curves in specific fibres, the corresponding fundamental groups become hyperbolic relative to the introduced tori subgroups, and as a consequence, act geometrically on CAT(0) cube complexes. This is joint work with Mahan Mj.

Let F be a finite-rank free group with rank greater than 2, Q be a finitely generated subgroup of the outer automorphism group of F, and consider a short exact sequence
1 → F → EQ → Q → 1.
My talk will discuss some known results and some open questions regarding the geometry of EQ, given information about Q. There is a somewhat decent understanding of the geometry of EQ when Q is cyclic, where we have necessary and sufficient conditions on Q to determine if EQ is hyperbolic or relatively hyperbolic or thick. One key idea we will look into is – why EQ is relatively hyperbolic when Q is an infinite cyclic group generated by an exponentially growing outer automorphism of F and then discuss some difficulties one faces while extending such results to the cases when Q is not cyclic.

A group G is called acylindrically hyperbolic if it admits a non-elementary acylindrical action on a hyperbolic space. It includes many examples of interest, e.g., nonelementary hyperbolic and relatively hyperbolic groups, all but finitely many mapping class groups of punctured closed surfaces, most 3-manifold groups and Out(Fn) for n > 1. Although BS(p, q) is essentially never acylindrically hyperbolic we will see in this talk that Out(BS(p, q)) is acylindrically hyperbolic for non-solvable Baumslag-Solitar groups, and explore further properties of the group using its acylindricity. This is joint work with Daxun Wang.

In geometric group theory, a group is studied via its action on some metric spaces, often with unbounded orbit. It is then natural to investigate the growth of orbit points, group elements, and conjugacy classes. Furthermore, the pioneering works of Patterson and Sullivan and subsequent development connected this growth problem with the dynamics on the boundary and the geodesic flow on the space. In this mini-course, we will study the growth problem in hyperbolic groups and CAT(0) groups via a combinatorial approach. The flexible nature of our approach allows many generalizations.

In week 2, we will compute the growth rate of certain subsets/subgroups of the ambient group. For example, refer to the following articles:
 1. I. Gekhtman and W. Yang, Counting conjugacy classes in groups with contracting elements. (2022)
 2. V. Erlandsson and J. Souto, Counting geodesics of given commutator length. (2023)

Basic references:

Introduction to hyperbolic groups, by Davide Spriano h...

I will give the basics on property(T), and explain the characterization in terms of actions on median spaces. I will discuss CAT(0) cubical complexes as examples of median spaces, and discuss groups acting on those objects as having a strong negation of property(T).
Possible problems for 2nd week:

Read ‘Spectral interpretations of property(T)’ by Yann Ollivier, with a generalization in mind. http://www.yann-ollivier.org/rech/publs/aut_spec_T.pdf
Study the orbits of the action of discrete cocompact subgroup P in SL^(2,R) on a median space (viewing P as a subgroup of the mapping class group of a surface prevents a proper action on a CAT(0) space). This will involve reading ‘Geometries of 3-manifolds’ by Peter Scott https://deepblue.lib.umich.edu/bitstream/handle/2027.42/135276/blms0401....

Pre-requisites: Bridson-Haefliger Part I, Part II Sections 1, 2 (ideally also 6,8,10), Part III H