Hilbert space is big. No, not big like the how the Earth is big compared to you. Rather, Hilbert space is astronomically big. Actually, that’s not quite right either. It’s bigger than that. I guess the best adverb I can use is that it’s mathematically big. In a Hilbert space, you tend to have a lot of room to maneuver. (To read more about that, check out my essay, “The Curse of Dimensionality”.) In the…

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As a quantum theorist, I spend a lot of time thinking about high-dimensional spaces. These are the playgrounds for quantum many-body systems, and they are vast. The technical name is a Hilbert space, and it’s the space of complex vectors with the additional structure of a way to put vectors together (called an inner product). Hilbert space is big (see “The Curse of Dimensionality”), but the usable area for quantum theory is often much smaller.…

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How to pick a random unit vector. As a quantum theorist, I spend a lot of time thinking about high-dimensional spaces. These are the playgrounds for quantum many-body systems, and they are vast. The technical name is a Hilbert space, and it’s the space of complex vectors with the additional structure of a way to put vectors together (called an inner product). Hilbert space is big (see “The Curse of Dimensionality”), but the usable area…

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From combinatorics to many-body quantum systems. If there’s one field of mathematics that everyone encounters in their daily life, I would argue that it’s combinatorics (with perhaps geometry being the other one). The rules of combinatorics cast a shadow over our lives. They affect how we make decisions and form the scaffolding for how options in our lives are displayed to us. In this essay, I want to explore the idea which is known as…

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One of my favourite mathematical pieces of writing is Flatland, by Edwin Abbott Abbott (the book is in the public domain, so you can download it from Wikipedia). Published over a century ago, it’s a story1 involving residents (Flatlanders) who live in a two-dimensional world. Without giving too much of the story away (because you should seriously read it!), the inhabitants find themselves shocked when a strange shape dips into their world. That other “shape”…

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When I hear the word “quantum”, I think of all the misconceptions and crazy ideas people associate with it in a lot of popular media. Physicists are great (and terrible) at coming up with names, and the word “quantum” is such an example of a word with a lot of baggage attached. Pair it with the word “computer”, however, and the misconceptions skyrocket, sometimes turning into full-blown hype. The reality (at the time of this…

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In my final year of undergrad, I had a plan: go to the university near my house, begin my master’s degree, and eventually do a PhD. It was nice, simple, and straightforward. Not having a ton of people around me applying for graduate school, I wasn’t aware of how big a deal the choice of institution was, nor the fact that some people apply to ten or more schools (often for those looking to go…

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If I’m having an argument, I tend to use phrases that include the word “you” or “they”. What I’m doing is projecting what I think a person is feeling into my own words. In essence, I’m taking what I think is important in their disagreement and only addressing that. The problem is that I’m not letting the person describe their own experience. Instead, I’m asserting what I think they are saying, and basing my response…

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As a physicist, mathematics is the language I speak. It’s what I use to discuss physics, and I’m familiar with many of the tools that mathematicians learn. The tools range from the fields of calculus, probability, statistics, linear algebra, differential equations, graph theory, and many others. In particular, solving differential equations is like the bread and butter of physics, so I’m versed in this area. Knowing how to untangle the Schrödinger equation or the field…

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