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We will study groups acting geometrically on infinite polyhedral complexes. A good example to think of is the fundamental group of a surface Sg acting on the universal cover of a triangulated Sg. We will define, discuss, and state some properties of two very useful invariants of these spaces/actions: the ℓ2-homology of the space and the (Gromov) boundary of the space when it is hyperbolic. There will be many examples, which will hopefully lead to some intuition about these invariants. Possible second week problem based on Matt Clay’s paper ‘ℓ2-homology of the free group’.

Pre-requisites: Background that might be helpful includes some topology and covering space theory (for example Hatcher chapter 1), familiarity with simplicial homology, CW complexes, geometry of H2.

If G is a group, its outer-automorphism group Out(G) is obtained from Aut(G) by quotienting out inner automorphisms, that are conjugations by elements of G. It is natural to ask methods and invariants to discuss whether two elements of Out(G) are conjugate. Important examples are GLn(Z) as automorphism group of Zn, Mapping Class Groups as outer-automorphism groups of surface groups, and outer-automorphism groups of finitely generated free groups. The case of the free group of rank 2: Out(F2) is isomorphic to GL(2, Z) and the classification of its conjugacy classes is classical. In rank 3, it is well known that interesting features appear, and they illustrate the rich theory of train tracks, laminations, geometry of suspensions, and structure of the polynomially subgroups, associated to an automorphism. With Francaviglia, Martino, and Touikan, we produced a solution to the conjugacy problem in Out(F3), which is the aim of this mini-course.

References:

‘Introduction to group the...

We will study groups acting geometrically on infinite polyhedral complexes. A good example to think of is the fundamental group of a surface Sg acting on the universal cover of a triangulated Sg. We will define, discuss, and state some properties of two very useful invariants of these spaces/actions: the ℓ2-homology of the space and the (Gromov) boundary of the space when it is hyperbolic. There will be many examples, which will hopefully lead to some intuition about these invariants. Possible second week problem based on Matt Clay’s paper ‘ℓ2-homology of the free group’.

Pre-requisites: Background that might be helpful includes some topology and covering space theory (for example Hatcher chapter 1), familiarity with simplicial homology, CW complexes, geometry of H2.

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

The planar Hall effect in 3D systems is an effective probe for their Berry curvature, topology, and electronic properties. However, the Berry curvature-induced conventional planar Hall effect is forbidden in 2D systems as the out-of-plane Berry curvature cannot couple to the band velocity of the electrons moving in the 2D plane. Here, we demonstrate a unique 2D planar Hall effect (2DPHE) originating from the hidden planar components of the Berry curvature and orbital magnetic moment in quasi-2D materials. We identify all planar band geometric contributions to 2DPHE and classify their crystalline symmetry restrictions. Using gated bilayer graphene as an example, we show that in addition to capturing the hidden band geometric effects, 2DPHE is also sensitive to the Lifshitz transitions. 

Reference: arXiv 2405.00379 (2024)

In this talk, I will present a detailed experimental study of the particle- hole (PH) symmetry of the the abelian and a putative non-abelian Fractional Quantum Hall (FQH) state about half filling in a multiband system. Specifically, we focus on the lowest Landau level of the monolayer-like band of Bernal stacked trilayer graphene (TLG). In pristine TLG, the excitation energy gaps, Lande g factor, effective mass, and disorder broadening of the FQH states is the same as their hole-
conjugate counterpart. This precise PH symmetry stems from the lattice mirror symmetry that precludes Landau-level mixing. Introducing a non- zero displacement field D disrupts this mirror symmetry, facilitating the interaction and hybridization between the NM = 0 of monolayer-like and NB = 2 bilayer-like Landau levels. This band hybridization eventually leads to a violation of the particle-hole symmetry of the FQH states. We find the one-third and two-fifth FQH states to be more robust against Landau level m...

Transition metal oxides exhibit a rich variety of collective phenomena already in the bulk due to a strong interplay of lattice, charge, spin and orbital degrees of freedom. Oxide interfaces open new possibilities for applications in electronics devices or for energy conversion due to the emergence of novel electronic phases that are not available the bulk constituents. In this talk I will review the insight obtained from density functional theory (DFT) calculations with an on-site Coulomb term how this novel functionality can be steered by a set of control parameters. Several examples will be discussed: (i) the prediction of topological Chern insulating phases and competition with Mott insulating states in oxide superlattices with a honeycomb [1] and dice pattern [2]; the emergence of a spin-polarized two-dimensional electron gas at the EuTiO 3 (001) surface and in LaAlO 3 /EuTiO 3 /SrTiO 3 (001) [3] (ii) the role of the film geometry [4] and the interface structure [5] in infinite-la...

We will first present an introduction to quantum geometry of electrons in periodic structures in terms of Berry phases and curvature. We show that the coupling of phonons with electrons can have nontrivial consequences to quantum geometry of electronic structure, which manifests as oscillations in the Berry curvature dipole and hence have observable nonlinear Hall signatures. Using these, we introduce a vibrational spectroscopy based on Geometry of Quantum Electronic Structure (GQuES) making specific predictions for the transport and radiative GQuES spectra of 2D materials. On a related topic, we demonstrate emergence of nontrivial Berry curvature in graphene from its interaction with monolayer of CrTe2, and how their heterostructure can be used to develop an Anomalous Hall Transistor.

Circularly polarized lattice vibrations carry angular momentum and lead to magnetic responses in applied magnetic fields or when resonantly driven with ultrashort laser pulses. The phonons associated with such vibrations are known as chiral phonons. On the basis of purely circular ionic motion, these phonons are expected to carry a magnetic moment of the order of a few nuclear magnetons. However, some recent experiments have demonstrated a phonon magnetic moment of the order of a few Bohr magnetons. This kind of giant magnetic response points towards the electronic contribution to the magnetic moment of phonons. Many diverse mechanisms have been discovered for this enhanced magnetic response of chiral phonons. The orbital-lattice coupling is one such mechanism where low-energy electronic excitations on a magnetic ion hybridize with phonons and endow a large magnetic moment to phonons. In this talk, I'll present a microscopic model for the effective magnetic moments of chiral phonons ba...

We have studied 2DEG formed at the interface of two oxide perovskite insulators; one is LaVO3 (LVO) which is a Mott insulator and another one is KTaO3 (KTO) which is a band insulator. Our experimental collaborators in the group of Prof. Suvankar Chakraverty from INST, Mohali have created LVO/KTO interface for the first time and observed metallic behaviour at this interface with one order of magnitude higher electron density and very high mobility of the 2DEG. The reason for 2DEG formation at the interface of two insulators/semiconductors was not clear. Our DFT calculations showed that LVO and KTO bulk materials are insulating, but the LVO/KTO interface is metallic which is an emergent phenomena and existence of parabolic bands crossing the Fermi level indicates source of free electrons at the interface. In this LVO/KTO heterostructure, both the individual parts are polar, consisting of alternating charged layers. Our DFT calculations show, to avoid polar catastrophe, “electronic recons...

Bulk-boundary correspondence licenses us to probe the bulk topological order by studying the transport properties of the edge modes. However, edge modes in a fractional quantum Hall (FQH) state can undergo edge reconstruction and, on top of that, can be in the coherent regime or exhibit varying degrees of charge and thermal equilibrations, giving rise to a zoo of intriguing scenarios. This can happen even in many abelian cases (like ν = 2/3), as well as non-abelian cases (like ν = 5/2). 5/2 has been particularly a focal point of both theoretical and experimental studies as it hosts non-abelian quasiparticles, a proposed basis for topological quantum computation. I will discuss how shot noise can provide a path to resolution and how its application can go beyond the quantum Hall regime to other systems like graphene quantum Hall states, Kitaev magnets, fractional Chern insulators in Twisted Bilayer graphene, and more.