Acknowledgements and dedication

I have always enjoyed reading this section in the work of others, even where I have no intelligent opinion in the scientific merit of said work. The nature of this thesis demands I acknowledge the people who have steered, nudged, inspired, or directly pushed me into the fire. Despite my omission of unrelated joys, there are still many people to thank.

I recollect a gathering with Andrei and Emil. Emil asked semi-rhetorically, ''What problem shall we solve today?" Andrei replied fervently, "Everything." I hope to have imbibed some of his gung-ho, take-no-prisoner ambition. His standard for clarity and rigor in thinking and writing shall be my standard.  He taught me that coincidences were symmetries waiting to be unmasked -- this thesis describes our unmasking.

Chen Fang listened patiently to many of my errant speculations, until some of them eventually struck home. To him I owe the debt of levelling me up from slaying rats in the basement, and for accelerating my productivity. His experience in band theory steered many conversations into discoveries, and we drank together the wine of shared ideas, and sometimes just plain wine and too much of it. Our musings in wintery Aspen, while soaking in mountain-top, open-air jacuzzis, birthed Chap. 3.

My unlikely partnership with Wang Zhijun proved instrumental to this thesis. He arrived in Princeton with big dreams but little English -- our mode of communication was at first completely mathematical. Nevertheless, his insights in material chemistry complemented my abstract, group-theoretical perspective. He found the (classified) material class whose true beauty was veiled to us in the months that we prodded and probed. After a chance dinner at Szechuan House, he suggested that our hourglass fermion might be robust but didn't know why; in completing his thought I (or we!) discovered the hourglass-zigzag topology, which underlies Chap. 4.

I arrived in Princeton at Daniel Arovas' insistence, and in the ensuing five years he would frequently descend at Princeton as a whirlwind, whipping out his brand of wit and physical intuition. When I professed to being lost  in in existentialist crisis, he threatened to put a fork through my hand. He introduced me to elementary band representations, which sparked an insight detailed in the Summary and Outlook. His support throughout my career has been tremendous.

Nicolas Regnault is a giant. When he flexes, beautiful numerics stream out. Our conversations flow from parafermions to practical know-how, e.g., how to find a job without despairing, and how to sensitively acknowledge collaborators in talks -- by joking at the expense of the French. I partook so much of his intelligence, compassion and humor.

I have had the immense pleasure of being the resident shrink for band-topological advice. Belopolski often bursts through my office door to bounce off his latest ideas. He writes frenetically on the board, then peers back at me to register my reactions. Since my reactions are often not forthcoming, an awkward silence ensues whereupon my rising impatience chokes me to further silence. Many episodes occur in this fashion. I will miss many things about Belopolski, but I will not miss that searching look. Some semblance of ideas seep through, and it is still hopeful that one of them will bear fruit. Among proven collaborations with more-serious experimentalists, Nasser sold me a mystery whose definitive solution eluded me, though I have my speculation. Our joint exploration of the Cerium monopnictides uncovered a beautiful example of band-theoretical holography. Bismuth christened me to nature's fold, and instilled in me a deep-seated belief in band theory. Here marked my first rewarding partnership with an experimentalist, Ilya Drozdov, with whom many nights were spent understanding the scattering of Bloch waves.

Dedication:

My parents, Rugai and Connie, carried my burdens and actualized my evanescent dream.

My sisters, Charmelia, Nina, Sasha, Marcia, and Laetitia, withstood my long absences.

 

I wish it were not so. 

 

To Katie, with whom I shared the first tentative steps. 

I bear you, now, sure-footed.

Two cups of water for Erwin

The topic has only burned in your mind in the last five years, and well-stoked fires in the engine room spread quickly when unleashed.

If data analysis is indeed the art of cohomology, I think you will have much to teach me. I am tempted to title my next paper as 'Cohomological Insulators', to challenge the Topological Insulator. However, I must retain a cool head and not let the excitement overrun my judgement. My knowledge of cohomology is too narrow, I have learned to navigate to one island in an ocean of unknowns. It is rich mathematics I am delving into, I understand so little of even the Wikipedia page on it. Somehow I am going to publish a paper on it? My goal is to learn this subject better, even abstractly. Some abstractions are just waiting to be actualized.

It is amusing how Erwin talks about entropy (or free energy, as he later qualifies) and the second law, and then flippantly applies these concepts to the human body. Yet his example of the second law is two cups of water, one sugar-saturated and the other clear. Entropy increases when water migrates from the clear cup to the one saturated, so that the sugar has more space to roam. Here, entropy is well-defined locally for each cup, which maintains local equilibration at each step of the migration; the sum of the two entropies will increase. Quite right, Erwin!

A time for marriages

The revelatory marriage of group cohomology with symmetry-protected topological order brimmed with possibilites. Here was the simplest extension of group theory as we know it – in principle there are very many extensions that are possible, the question is which extension is a physical realization of natural reality. I studied group cohomology with this tantalizing promise of new frontiers, but ultimately failed at originality. I let this idea go.

Rediscovering group cohomology in the band theory of crystals felt a deliverance of my convictions. There is a richness to crystals that keeps surprising me. The symmetries of a crystal involve spatial transformations which are distinguished by a basic geometric property: rotations, reflections and inversions preserve the spatial origin, while screw rotations and glide reflections translate the origin by a fraction of the lattice period. This geometric distinction necessitates the extension of origin-preserving groups by spatial translations; I find that these groups can be further extended by momentum translations, thus placing real and momentum space on equal footing.

The excitement of my discovery spurred a frenetic writing episode that consumed me immediately after Japan. On completion, I further produced a two-hundred-page thesis in two days, submitted at 6am and one week overdue, then awoke at noon for a bachelor party which I organized in Los Angeles. Jimmy was preternaturally quick in observational deduction in the paniq room, leading us to emphatic victory. I had a uniquely communal casino experience with Jake and Jimmy on the ultimate hold ‘em table, where our only collective pressure was to beat the dealer. Nathan was so amused by my easy winnings that I gave him half of it. Eric has grown into the build of a horse, and sports a mane to complete the transformation. To Jake, he expressed his own brand of love: “Jake, if I were to give you an ipad, would you find any use of it?” With superlative composure, “Eric, I absolutely would.”

Two more days of wedding preparations and ceremony ensued, and being the best man carried ambassadorial duties that I undertook diligently. A formal title provided the impetus to mingle with different social elements, to become a bridge between two families and thus actualize Jake’s ambition for this wedding.

As weddings go, the caricatures were minimized to create room for honest fun. While writing the best man's toast, I felt crushingly inadequate, my vocabulary to describe love as feeble as vapor, the words were torn out of me, coughingly blood-spittled, and riddled with the anxiety of speaking in foreign languages. Nerves crippled the first half of my speech, but I recovered some dignity in the finish. I had difficulty reconciling the rapturous reception with my private struggle. Nathan, whose socially-sensitive criticism proved influential, was beaming as a proud parent. He appreciates the power of words publicly spoken, more than I could. 

Toast

Two years ago, Jake started a blog titled ‘The Seattle Life’. I don’t know if you’ve read his blog, but if you have, you might imagine it is less about Seattle, and would more appropriately be titled as: ‘Finally, Life with Kimberley, once again’. Indeed, his move to Seattle was a tremendous relief of tension from years of geographical separation.

During their first encounters, Jake would tell me stories about Kimberley. His telling carried a sense of helpless disbelief. Here was a girl so ideologically different from himself, and he cannot help but be drawn to her.

Yet, their connection is stronger than their differences, and it is a testament to both their characters that we are here today. Neither character is overpowered, theirs is a proud marriage of two strong individuals.

They have achieved an uncommon peace that is less about compromise, and more about developing their unique pathways of communication, of both emotions and ideas. Underlying this peace is a deep yearning to truly understand each other, to have their thoughts and joy become numerically one.

On this note, would you join me in wishing Jake, the love of my life, and Kimberley, the love of his life, their well-deserved happiness?

Slapstick

I should have submitted my thesis on July 1st, by the departmental requirements. However, I have not started writing my thesis, because I have the ambition of writing one last mammoth paper and including that in my thesis. This last paper is also about quantum hourglasses, and is extension of the first paper. It's a hard paper to write, not least because I am trying to introduce a mathematical field (group cohomology) to the theory of crystals. 

We have went so long without a word, I actually started and finished a book by your recommendation: what is life? Admittedly a very slim book, and my accomplishment pales against your voracious appetite.

Many ideas here that I want to discuss with you. I am rather shocked that entropy has much less applicability than I thought it to have. Something about Erwin's last chapter intrigued me. His two premises are that:

I am in control of my actions.

Physical bodies obey the laws of physics. 

His conclusion: I am god, who controls my body according to the laws of nature. 

Reading this felt like being in a slapstick comedy, where I wasn't sure if actors were acting. Some of his other comments on consciousness seem much more believable, and I particularly liked his metaphor of consciousness as a canvas of sensory experiences. 

I will get back to you about skyping. I am very close to finishing this last paper, it feels like a matter of one to two days. However, my estimations have proven in the past to be very poor, and even if my estimation were correct I do not know if I have one or two days. The 'pride of publishing many articles' may be my downfall.