Calendar
- 01December 1, 2020
Topological order, tensor networks and subfactors
We present recent progress on studies of 2-dimensional topological order in terms of tensor networks and its connections to subfactor theory. We explain how Drinfel’d centers and higher relative commutants naturally appear in this context and use of picture language in this study.
Zoom: https://harvard.zoom.us/j/779283357?pwd=MitXVm1pYUlJVzZqT3lwV2pCT1ZUQT09
- 01December 1, 2020
CMSA Computer Science for Mathematicians: Some extensions on argumentation frameworks via hypergraphs
The Dung Abstract Argumentation Framework (AAF) is an effective formalism for modelling disputes between two or more agents. Generally, the Dung AF is extended to include some unique interactions between agents. This has further been explained with the Bipolar Argumentation Framework (BAF). In the academic space, the use of AAF is highly signified. We can use the AF as a means to resolve disagreements that allows for the determination of a winning argument. In general, there can be imperfect ontologies that affect how reasoning is defined. Typical logic-based AFs apply to the incoherent/uncertain ontologies. However, Dung demonstrated a stable extension of AF to support an “acceptable standard of behavior”. This talk will align with present endeavors on extending the Dung AAF to consider the notion of conflict-freeness in relation to persistence over a hypergraph. With a generic type of argumentation, there are some methods that can exploit certain complex decision procedures. Argument and attack relations within the Dung AAF, thus are further defined to obtain a graphical formula of Kripke groundedness. The incorporating of multiple levels of knowledge aligns with a computational linguistics aspect for the defining of a classification criteria for AAF. In the construction, I will provide some treatment of ‘good’ model-theoretic properties that bridge AAF with Zarankiewicz’s problem to introduce how arguments are consistent with bipartite hypergraphs. The Zarankiewicz problem appears with the communication complexity on AF graphs. - 01December 1, 2020
Positroid varieties and q,t-Catalan numbers
Positroid varieties are subvarieties of the Grassmannian obtained by intersecting cyclic rotations of Schubert varieties. We show that the “top open positroid variety” has mixed Hodge polynomial given by the q,t-rational Catalan numbers (up to a simple factor). Unlike the Grassmannian, the cohomology of open positroid varieties is not pure.
The q,t-rational Catalan numbers satisfy remarkable symmetry and unimodality properties, and these arise from the Koszul duality phenomenon in the derived category of the flag variety, and from the curious Lefschetz phenomenon for cluster varieties. Our work is also related to knot homology and to the cohomology of compactified Jacobians.
This talk is based on joint work with Pavel Galashin.
Zoom: https://harvard.zoom.us/j/91794282895?pwd=VFZxRWdDQ0VNT0hsVTllR0JCQytoZz09
- 02December 2, 2020
CMSA Strongly Correlated Quantum Materials and High-Temperature Superconductors Series: Interplay between superconductivity and non-Fermi liquid at a quantum critical point in a metal
I discuss the interplay between non-Fermi liquid behaviour and pairing near a quantum-critical point (QCP) in a metal. These tendencies are intertwined in the sense that both originate from the same interaction mediated by gapless fluctuations of a critical order parameter. The two tendencies compete because fermionic incoherence destroys the Cooper logarithm, while the pairing eliminates scattering at low energies and restores fermionic coherence. I discuss this physics for a class of models with an effective dynamical interaction V (Ω) ~1/|Ω|^γ (the γ-model). This model describes, in particular, the pairing at a 2D Ising-nematic critical point in (γ=1/3), a 2D antiferromagnetic critical point (γ=1/2) and the pairing by an Einstein phonon with vanishing dressed Debye frequency (γ=2). I argue the pairing wins, unless the pairing component of the interaction is artificially reduced, but because of fermionic incoherence in the normal state, the system develops a pseudogap, preformed pairs behaviour in the temperature range between the onset of the pairing at Tp and the onset of phase coherence at the actual superconducting Tc. The ratio Tc/Tp decreases with γ and vanishes at γ =2. I present two complementary arguments of why this happens. One is the softening of longitudinal gap fluctuations, which become gapless at γ =2. Another is the emergence of a 1D array of dynamical vortices, whose number diverges at γ =2. I argue that once the number of vortices becomes infinite, quasiparticle energies effectively get quantized and do not get re-arranged in the presence of a small phase variation. I show that a new non-superconducting ground state emerges at γ >2.
- 02December 2, 2020
Joint Dept. of Mathematics and CMSA Random Matrix & Probability Theory Seminar: Thermodynamics of a hierarchical mixture of cubes
The talk discusses a toy model for phase transitions in mixtures of incompressible droplets. The model consists of non-overlapping hypercubes of side-lengths 2^j, j\in \N_0. Cubes belong to an admissible set such that if two cubes overlap, then one cube is contained in the other, a picture reminiscent of Mandelbrot’s fractal percolation model. I will present exact formulas for the entropy and pressure, discuss phase transitions from a fluid phase with small cubes towards a condensed phase with a macroscopic cube, and briefly sketch some broader questions on renormalization and cluster expansions that motivate the model. Based on arXiv:1909.09546 (J. Stat. Phys. 179 (2020), 309-340).