# ESP Biography

## DAVID GRABOVSKY, Columbia senior studying math and physics

Major: Physics, Mathematics

College/Employer: Columbia University

Not Available.

## Past Classes

(Clicking a class title will bring you to the course's section of the corresponding course catalog)

M772: All of Linear Algebra in Splash Fall 2018 (Oct. 28, 2018)
This class will be my attempt to do the entirety of an advanced undergraduate course in linear algebra in less than two hours. We will develop the theory of finite-dimensional vector spaces, starting with vectors and linear combinations and moving on to linear transformations, isomorphisms, and the change of basis. Then we'll introduce eigenvectors and eigenvalues and discuss inner products as an excuse to define linear functionals and the dual space. Finally, we will finish off with the crowning glory of linear algebra, the spectral theorem. The goal of the class is to show you a wild ride through linear algebra, and to "give away the trade secrets" of a beautiful subject.

M773: False Theorems and Fake Proofs in Splash Fall 2018 (Oct. 28, 2018)
In this introductory “math” class, we will “prove” some entirely false claims in surprisingly convincing ways. For example, we'll show that 1 is the largest natural number, but that all natural numbers are equal to zero. Come find out why the Pythagorean theorem is a lot simpler than you thought (spoiler: $$c = a+b$$), why all functions are equal to zero, and why $$1 = 2$$. (I'll offer as many proofs of this "fact" as I have time for!) The emphasis will be on making math ridiculous and funny, as well as on discovering what really makes standard methods of proof tick. Come ready to participate!

S777: Poetry for Physicists in Splash Fall 2018 (Oct. 28, 2018)
In this class, I will wax poetic on the beauty of physics by writing down an inordinate number of equations. Forgoing technical derivations in favor of physical argument, I will try to describe the glorious apparatus of modern theoretical physics, from classical mechanics to the quantum theory of fields. We will start by scrapping Newton's laws (because they're boring), and we'll spend most of the class discussing two beautiful reformulations of classical mechanics. Among our poetic vistas will be Hamilton's least-action principle, the Euler-Lagrange equations of motion, Noether's theorem relating symmetries to conservation laws, canonical conjugate momenta, Hamilton's equations of motion, Poisson brackets, and Liouville's theorem. Time permitting, we may also discuss classical field theory or symplectic geometry. At the end of class, I will try to convince you that classical and quantum physics are really the same, and that commutators, Heisenberg's uncertainty principle, and path integrals are all really part of the same poetic physical structure.

M649: All of Linear Algebra in Splash Spring 18 (Mar. 31, 2018)
This class will be my attempt to do the entirety of an advanced undergraduate course in linear algebra in less than two hours. We will develop the theory of finite-dimensional vector spaces, starting with vectors, linear combinations, and bases. We'll talk about the kernel and image of linear transformations before moving onto isomorphisms and changes of basis. We will introduce eigenvectors, discuss inner products, linear functionals, and the dual space, and finish off with self-adjoint operators and the spectral theorem. The goal of the class is to show you a wild ride through linear algebra, and to "give away the trade secrets" of a beautiful subject.

M650: False Theorems and Fake Proofs in Splash Spring 18 (Mar. 31, 2018)
In this introductory “math” class, we will “prove” some entirely false claims in surprisingly convincing ways. For example, we'll show that 1 is the largest natural number, but that all natural numbers are equal to zero. Come find out why the Pythagorean theorem is a lot simpler than you thought (spoiler: $$c = a+b$$), why all functions are equal to zero, and why $$1 = 2$$. (I'll offer as many proofs of this "fact" as I have time for!) The emphasis will be on making math ridiculous and funny, as well as on discovering what really makes standard methods of proof tick. Come ready to participate!

S652: Poetry for Physicists in Splash Spring 18 (Mar. 31, 2018)
In this class, I will wax poetic on the beauty of physics by writing down an inordinate number of equations. Forgoing technical derivations in favor of physical argument, I will try to describe the glorious apparatus of modern theoretical physics, from classical mechanics to the quantum theory of fields. We will start by scrapping Newton's laws and postulating a principle of stationary action to govern the dynamics of classical particles. After writing down the Euler-Lagrange equations of motion, we will discuss the truly poetic theorem of Noether relating symmetries to conservation laws. We will then rewrite mechanics in the Hamiltonian formalism and pause to discuss phase space. Next, we will change gears and give the basic postulates of quantum mechanics. We will solve the harmonic oscillator, and then discuss unitary dynamics in the Schrödinger and Heisenberg pictures. Having laid down our "basic" formalism, we will start the second hour by making the transition to classical field theory, where Noether's theorem reappears more powerful than ever. We will embark on the project of quantizing the free scalar field, which will turn out to satisfy the Klein-Gordon equation. Time permitting, I will also talk about interacting quantum field theories, Feynman diagrams, and the path integral formulation of QFT.

M568: All of Linear Algebra in Splash Fall 2017 (Nov. 04, 2017)
This class will be my attempt to do the entirety of an advanced undergraduate course in linear algebra in less than two hours. We will develop the theory of finite-dimensional vector spaces, starting with vectors, linear combinations, and bases. We'll talk about the kernel and image of linear transformations before moving onto isomorphisms and changes of basis. We will introduce eigenvectors, discuss inner products, linear functionals, and the dual space, and finish off with the self-adjoint operators and the spectral theorem. The goal of the class is to show you a wild ride through linear algebra, and to "give away the trade secrets" of a beautiful subject. After this class, consider taking my "All of Quantum Mechanics" sequence for a spectacular application of the ideas developed in this class.

M576: False Theorems and Fake Proofs in Splash Fall 2017 (Nov. 04, 2017)
In this introductory “math” class, we will “prove” some entirely false results in surprisingly convincing ways. For example, we'll show that 1 is the largest natural number, but that all natural numbers are equal to zero. Come find out why the Pythagorean theorem is a lot simpler than you thought (spoiler: $$c = a+b$$), how all functions integrate to zero, and as many proofs that $$1 = 2$$ as I have time for. The emphasis will be as much on making math hilarious as on introducing the methods of “proof” that make these results tick—and why they’re wrong. After this class, consider taking Theo's "Things you Think should be False but Aren't" for some more serious examples of the counterintuitive element in mathematics.

S581: All of Quantum Mechanics (Part I) in Splash Fall 2017 (Nov. 04, 2017)
Part I of an intense introduction to one of the most ridiculous and ridiculously beautiful physical theories ever invented. We will cover Hilbert space and quantum states, Dirac's bra-ket notation, observables and hermitian operators, quantum measurement and collapse, the spectral theorem, probabilities and the Born rule, and commuting observables. We will conclude with the Stern-Gerlach experiment as a stunning demonstration of the quantum formalism.

S582: All of Quantum Mechanics (Part II) in Splash Fall 2017 (Nov. 04, 2017)
Part II of an intense introduction to one of the most ridiculous and ridiculously beautiful physical theories ever invented. We will start with the double-slit experiment, cover quantum wavefunctions, position and momentum eigenstates, the canonical commutator identity, and conclude with a derivation of the Schrodinger equation. This class is not to be taken without first taking Part I of this sequence! After this class, consider taking Ben Church's and Ryan Abbott's "Relativistic Quantum Theory" for a serious introduction to relativity of quantum mechanics.

M501: False Theorems and Fake Proofs in Splash Spring 2017 (Mar. 25, 2017)
In this introductory “math” class, we will “prove” some entirely false results in surprisingly convincing ways. For example, we’ll show that 1 is the largest natural number, that it’s actually equal to both -1 and 2, and that all natural numbers are a whole lot less than a million. Come find out why the Pythagorean theorem is a lot simpler than you thought (spoiler: $$c = a+b$$), why all infinite sets are the same, and more. The emphasis will be as much on making math hilarious as on introducing the methods of “proof” that make these results tick—and why they’re wrong.

M502: All of Linear Algebra in Splash Spring 2017 (Mar. 25, 2017)
This class will be my attempt to do the entirety of an undergraduate course in linear algebra in less than an hour. We will rapidly develop the theory of finite-dimensional vector spaces, starting with vectors and the notions of linear combination, basis, and dimension. Moving on to linear transformations, we will discuss the kernel, image, rank, and nullity of linear maps before diving into determinants and invertibility. From there, we’ll talk about change of basis, introduce eigenvectors, and finish off with self-adjoint operators and the mathematical fireworks of the spectral theorem. The goal of the class is to introduce a new way of thinking about mathematical structures, and to show you a wild ride through a beautiful subject.

M503: Groups and Representations in Splash Spring 2017 (Mar. 25, 2017)
Group theory and linear algebra are two of the most stunningly beautiful areas of mathematics, and representation theory is what happens when you put them together. In this class, we will start by introducing groups and some cool stuff you can do with them (e.g. homomorphisms, group actions, etc.) before moving to vector spaces and discussing invariant subspaces and maps. We'll define representations, talk briefly about irreducibility, and then prove the cute but powerful lemma of Schur. If there's time, we can then move on to direct sums and discuss the decomposition of representations into irreducible pieces.

S413: Quantum Mechanics I: A Mathematical Perspective in Splash Fall 2016 (Nov. 05, 2016)
In this class, we will attempt to build quantum mechanics up from the ground. We start with the mathematical formalism that describes wave functions, the strange inhabitants of an infinite-dimensional world known as the Hilbert space $$L^2$$, as well as a special class of the linear operators that roam this space. We will go on to discuss the physical meaning of this formalism, along the way exploring probability and the measurement problem. And for the grand finale, we will prove Heisenberg's uncertainty principle and show, using the spectral theorem, that no two incompatible observables admit a simultaneous eigenbasis. While this class is not for the faint of heart and relies on some pretty powerful mathematical machinery, it does not assume any previous experience with quantum mechanics, or even physics for that matter. We have a lot to cover and will be moving fast, but I hope you will enjoy the ride! This class is the first in a two-part sequence of quantum mechanics. The second class approaches quantum mechanics from a completely different perspective and covers completely different material focused on the Schrödinger equation, its origins, solutions, and limitations. Students are encouraged to take both classes, but each one is self-contained and can be taken independently.

S414: Quantum Mechanics II: A Physical Perspective in Splash Fall 2016 (Nov. 05, 2016)
In this class, we will attempt to invent quantum mechanics. We begin with empty space, governed classically by Maxwell's equations. We will derive the wave equation and start shooting electromagnetic plane waves at a conducting surface, discovering (with Einstein's help) the photoelectric effect. On a whim, we will reformulate the wave equation in terms of its energy and momentum; by considering a massive particle from this perspective, we will find ourselves face to face with the Schrödinger equation. For the rest of the class, we will discuss some consequences, solutions, and limitations of the equation. As a grand finale—time permitting—we will break free of Schrödinger's non-relativistic limit and discover the Klein-Gordon equation, which describes all relativistic spinless particles. This class is certainly not for the faint of heart, and relies on powerful mathematical machinery in addition to a considerable amount of classical physics. We have a lot to cover and will be moving fast, but I hope you will enjoy the ride! This class is the second in a two-part sequence of quantum mechanics. The first class approaches quantum mechanics from a completely different perspective and covers completely different material focused on the mathematical formalism governing quantum mechanics. Students are encouraged to take both classes, but each one is self-contained and can be taken independently.

M312: How to Break Math in Splash Splash Fall 2015 (Nov. 14, 2015)
You read the title correctly. In this fast-paced mathematical adventure, we will start by destroying the notions of truth and self-consistency in mathematical systems through Russell's Paradox and Gödel's Incompleteness Theorems. Next, we will do away with counting and size by measuring infinities against each other. Finally, we will gather what's left of mathematics just to demolish it, ending class with an astounding proof that a single ball can turn into two. Needless to say, this class is not for the faint of heart. There is a good bit of mathematical abstraction, but if you are willing to join me in deconstructing the foundations of everything we know, I think you will enjoy the ride and perhaps get a glimpse of the true beauty of mathematics.