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I am keeping this diary so that we all can see what I'm studying in mathematics. Also, you can look at my pages Recent Changes in Research and Recent Changes in Exposition. In particular, I listen to videos while I exercise or do mundane chores. Diary 2021.02.22 I am spilling the guts of quantum mechanics with combinatorial interpretations of orthogonal polynomials that arise in solutions of the Schroedinger equation. 2020.01.21 Nobel Prize laureate Duncan Haldane gives an accessible talk on Topological Quantum Matter, Entanglement, and the Second Quantum Revolution. (The Indian Institute of Technology Roorkee has a whole series of great talks.) Haldane's talk was a good introduction for me to physical effects that are some of the prerequisites to understanding the tenfold way. I was also intrigued that the Hall effect might be related to symmetric functions and, in particular, Hall-Littlewood polynomials but these are named after Edwin Hall and Philip Hall, respectively. The concept of fractional charge or fractional quanta is important, for example, in a chain of people holding hands where the hands at either end are free and possibly entangled. Similarly, the https://en.wikipedia.org/wiki/Fractional_quantum_Hall_effect fractional quantum Hall effect may express divisions of everything. Fractionally charged quasiparticles exhibit neither bosonic nor fermionic but rather anyonic statistics. This also relates to topological order in the zero-temperature phase of matter. 2020.01.11 I listened more to the The Tenfold Way by Vijay Shenoy (Part 2 of 4) about the tenfold way, the difference between the one ordinary (usual, linear) and three non-ordinary symmetries. It was helpful to see the chart and also to realize that the labels seem to be related to the classical Lie families. He talked about Hamiltonians but that part I didn't understand and should return to. 2020.01.10 At the NYCT After Meetup, Wenbo said he was interested in simplicial sets and quasicategories as he is realizing that categories are too sterile for modeling emergence. That makes sense to me from participating in Oliver's study group in Category theory and Statistics. I had sketched an adjunction for modeling statistics from which it became clear that categories need to be graded, which is to say, we need to be able to associate a probability to a morphism so that we can consider morphisms of a particular validity. In talking with Wenbo, I also realized that we need to be able to model nonassociativity, especially for composition of perspectives, and the probabilities would allow us to speak of associativity up to a certain probability. In emergence, systems or subsystems would "pop out" when they satisfied the requirements of category theory. I told Wenbo about the conceptual frameworks, the difference between interpreting choice syntactically-asymmetrically as with observerful simplexes, or semantically-symetrically as with observerless coordinate systems. I related this to the problem of the collapse of the wave function when nature measures itself. This suggested to me that the collapse must occur when a coordinate system is introduced and thereby separates the observer from the observed. As a subsystem emerges, at a certain point it is robust enough to indicate a coordinate system, which an observer can then leverage to indicate, for example, that something did not happen. Thus the establishment of the coordinate system should be that which creates space and perhaps time, on the one hand, and that which collapses the wave function, on the other hand. Thus this would be the place to look for a relation between general relativity and quantum mechanics. Wenbo is interested in wreath products. He shared a related excerpt from The Topos of Music by Guerino Mazzola. It had to do with time, and also to Greimas's theory of time in narrative. 2020.01.08 I'm interested in Bott Periodicity. The most helpful videos are the ones about the Tenfold Way in condensed matter research. I'm listening to The Tenfold Way by Vijay Shenoy (Part 2 of 4). The previous episode was a wonderful overview that I learned a lot from. In this episode it seems that he had to dumb it down, go back to basics, what is a group. But that's all right by me. |

Šis puslapis paskutinį kartą keistas March 08, 2021, at 12:46 PM