Thesis defence (hybrid):
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In this talk, we will describe two independent results based on my thesis. The first is on a unique factorization property for tensor products of parabolic Verma modules. More precisely, we give a
necessary and sufficient condition for when products of characters of two collections of parabolic Verma modules (and their restrictions to some subalgebras of the Cartan subalgebra) are equal. The second part is on the factorization phenomenon of flagged skew Schur polynomials upon twisting the variables by roots of unity. These flagged skew Schur polynomials include an interesting family of Demazure characters as a special case.
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In [Radchenko & Villegas, Bull. Lond. Math. Soc., 2021], Radchenko and Villegas proved that a finite simple graph
G is chordal if and only if the inverse of the independence polynomial of G is Horn hypergeometric. In this talk,
we present a new proof of their result using some elementary combinatorial methods and also generalize their
result to POE graphs that could possibly have a countably infinite number of vertices. Our proof is based on the
connection between the independence polynomial of G with the multi-colored chromatic polynomials of G which was
established first in [Arunkumar, Kus, and Venkatesh, J. Algebra, 2018].
This talk, split into two parts, will be on how symmetry principles can help us identify new phases of matter.
In the first part (based on [1,2]), I will introduce a novel system of classical 'fractons' which exhibit a remarkable set of features defying conventional expectations from almost all known classical mechanical systems. This includes the disassociation of velocity and momenta, the appearance of attractors seemingly in violation of Liouville's theorem, the formation of crystalline structures, evading the famous theorem by Hohenberg, Mermin, Wagner and Coleman and the failure of statistical mechanics. I will show that all this is a robust feature of underlying symmetries which enforces 'Machian' dynamics wherein isolated particles are immobile, and motion requires the assistance of proximate particles.
In the second part of my talk (based on [3,4]), I will discuss how new gapless phases emerge when unbroken microscopic symmetries manifest themselves in distinct ways on infrared fields. I will demonstrate the existence of several `symmetry-enriched' critical phases in the phase diagram of well-known coupled spin ladders which have been missed in all previous study. One of these is also a gapless topological phase and hosts protected edge modes of a distinct nature compared to their gapped counterparts.
[1] Phys. Rev. B 109, 054313 (2024)
[2] arXiv:2312.02271
[3] Phys. Rev. Lett. 130, 256401 (2023)
[4] Phys. Rev. B 108, 245135 (2023)