Search Talks

Over the three lectures and the tutorial, I want to explore some problems in enumerative algebra that can be understood by associating lattices to semistandard Young tableaux. I will introduce a family of rational functions called Hall--Littlewood--Schubert series (HLS series), which was recently introduced by Christopher Voll and myself. The enumerative problems we discuss will be solved by judicious substitutions of the variables of the HLS series.

The series of three short lectures provides an introduction to the subject of subgroup growth and related directions of research in asymptotic group theory. For instance, we will discuss polynomial subgroup growth and look at subgroup zeta functions of finitely generated nilpotent groups. We also cover methods from the theory of compact p-adic Lie groups, which have applications to subgroup growth and representation growth.

The series of three short lectures provides an introduction to the subject of subgroup growth and related directions of research in asymptotic group theory. For instance, we will discuss polynomial subgroup growth and look at subgroup zeta functions of finitely generated nilpotent groups. We also cover methods from the theory of compact p-adic Lie groups, which have applications to subgroup growth and representation growth.

The series of three short lectures provides an introduction to the subject of subgroup growth and related directions of research in asymptotic group theory. For instance, we will discuss polynomial subgroup growth and look at subgroup zeta functions of finitely generated nilpotent groups. We also cover methods from the theory of compact p-adic Lie groups, which have applications to subgroup growth and representation growth.

Our theoretical investigation explores a feasible route to engineer the two-dimensional (2D) Kitaev model of first-order topological superconductivity (TSC) introducing a magnetic spin texture. The main outcome of 2D Kitaev’s model is that a px + py type superconductor can exhibit a gapless topological superconducting phase in bulk hosting non-dispersive Majorana flat edge mode (MFEM) at the boundary. Our proposed general minimal model Hamiltonian is suitable to describe magnet/superconductor heterostructures. It reveals robust MFEM within the emergent gap of Shiba bands, spatially localised at the edges of a 2D magnetic domain of spin- spiral. We finally verify this concept from real material perspectives by considering Mn (Cr) monolayer grown on an s-wave superconducting substrate, Nb(110) under strain (Nb(001)). In both the 2D cases, the antiferromagnetic spin-spiral solutions exhibit robust MFEM at certain domain edges. This approach, particularly when the MFEM appears in the TSC phase for such heterostructure materials, offers significant prospect to extend the realm of TSC in 2D. Very recently, we expand this theoretical framework for engineering a 2D second-order topological superconductor (SOTSC) by utilizing a heterostructure: incorporating noncollinear magnetic textures between an s-wave superconductor and a 2D quantum spin Hall insulator. It stabilizes the SOTSC phase within the Shiba band, resulting in Majorana corner modes (MCMs) at the four corners of a 2D domain. The calculated non-zero quadrupole moment characterizes the bulk higher-order topology. Analytically calculated effective pairings in the bulk illuminate the microscopic behaviour of the SOTSC. Such first and second order Majorana modes are believed to be the building blocks for the fault-tolerant topological quantum computation.

Reference: Phys. Rev. B (Letter) 109, L041409 (2024) .
Phys. Rev. B (Letter) 109, L121301 (2024).

The effects of disorder and chaos on quantum many-body systems can be superficially similar, yet their interplay has not been sufficiently explored. We study this using an all to all interacting spin chain with disordered interacting term in presence of periodic kicks. The disorder free version of this model shows regular and chaotic dynamics within permutation symmetric subspace as the interaction strength is increased. When the disorder is increased, we find a transition from a dynamics within permutation symmetric subspace to full Hilbert space where the expectation values of various operators are given by random matrix theory in full Hilbert space. Interestingly, finite size scaling predicts a continuous phase transition at a critical disorder strength.