Jan 29, 2024

This morning’s lectures were both presented by Professor Jacob Shapiro, Princeton Professor of mathematical physics. Professor Shapiro explained that since his degree and current area of study are both in physics, his talks will be physics-related. They turned out to be very informative and densely packed with information!

The first lecture started with the observations that physics can be used to predict the future, and to study things that are very small and things that are very large, so that the method of study is what distinguishes physics from other areas of inquiry more than the object of study. A summary of the method is: start with a hypothesis (which does not rule out the use of intuition or even guessing in coming up with a hypothesis), then run an experiment to test the hypothesis. (Generally speaking, it is difficult to prove a hypothesis, so usually the goal is to try to disprove it.)

Theoretical physics uses math to make predictions - for example to predict where an object will be at any moment in time given a starting position and velocity. Classical or Newtonian mechanics was designed to make these predictions, but in order to make them scientists needed a whole new mathematical theory to do so, and thus they invented calculus.

In classical mechanics what is unknown is not a single number, but a function that a researcher is trying to find. What is a function? One way to think of it is as a picture or graph; a more abstract way is that it is a rule or assignment: the domain is the starting set of values, and the function is the rule for how to assign each value a match in the target set, called the co-domain. Note that the starting.set doesn’t have to consist of numbers! A function could, for example, be mapping from US Presidents to the political parties they represented. Another way to think of a function is as a formula. (Professor Shapiro pointed out at this point that if there is more than one starting number that gives you the same result, it is not a one to one function, so you can’t trace back from a particular answer output to a specific input (for example, if the function is x^2 + 4)).

Trajectory (usually represented by the Greek letter gamma) is a function of special importance in classical mechanics. Trajectory specifies location or position in space, and thus requires a coordinate system; we generally use a Cartesian system with x, y, and z projections. If each axis is considered separately, trajectory is really three functions

What is calculus? A way to imagine what will happen, for example seeing that 1/x approaches 0. Calculus has two parts to it: 1) a rate of change in a velocity/speed (the derivative) and 2) the accumulation of a quantity such as distance (the integral).

One math tool that is essential to calculus is the limit, the answer to the hypothetical question: what will happen if the pattern continues? For example consider the pattern 1, 1, 1, 1, … which has a limit of 1, or 1, 2, 3, … which has a limit of infinity (one way of saying a limit doesn’t really exist, since any given entry you can imagine is finite). What about 1, -1, 1, -1, …? The limit doesn’t exist because the function is “indecisive.” What about 1/n vs 1/(n^2)? Both go to a limit of zero, though one does so faster than the other, which is sometimes important.

A derivative is the instantaneous rate of change or instantaneous velocity, which we can think of as a limit of a function involving a tiny change in x. For example if f(x)=1, the derivative of f(x) is 0. But if the function is a parabola, such as f(x)=x^2, when h (the tiny change) is 0, the derivative is 2x.

The basic axiom of classical mechanics is that if you have the trajectory function (gamma), the derivative of gamma (gamma dot) is velocity, and the derivative of the derivative is acceleration (gamma with two dots). Newton realized that force = mass x acceleration, or mass x gamma with two dots

Part 2

After the whirlwind tour of classical mechanics and basic calculus Professor Shapiro led the students through in his first lecture before the morning break, his second lecture today was on Connecting the Dots, aka Topology.The lecture started with a key conceptual distinction: “how many?” is a question that can be answered with counting numbers (discrete), vs questions like “how much does it weigh?” or “how long?” or “how big?”: these are all questions that can be answered with continuous calculations.

A brief discussion of the basic units used to measure length, time, weight, temperature, electricity, speed, and mass gave way to the important question: How do we measure something? For a long time, the way to measure length was based on a standard meter stick kept in France as the universal reference object, from which other meter sticks were created. The length of anything else was measured by comparison to a meter stick, and the answer was not restricted to counting numbers, so it could be considered a continuous measurement.

What about a situation in which you could find a discrete answer but might prefer not to? For example, how many grains are in a bowl of rice? You would be able to count the grains, but how would that help? Much more useful would be to get the weight of the bowl of rice in kg. (There also might be different sizes to grains - so even when a discrete answer is possible, a continuous answer might be more accurate as well as more useful practically - for super small things like atoms and molecules a discrete count might be very useful, but not if necessarily so at larger scales.)

What is quantum mechanics? A quantum refers to a basic (discrete) quantity. For an illustration, consider the following experiment: shine light on a hydrogen atom and measure what comes out. You will see that only certain frequencies, such as -13.6ev, -3.4ev, or -1.5ev, are possible.

What is the distinction between classical and quantum mechanics? Classical is about big, macroscopic, many and therefore continuous measurements, whereas a quantum is about small, microscopic, few and therefore discrete ones. In modern physics, since the 1980’s, we see the two mix - with macroscopic/big/many systems that are still quantized.

In 1979 von Klitzing in Germany did an experiment in which he ran electricity through materials restricted to two-dimensional space (gallium arsenic) at very low temperatures with high magnetic fields (hard to do!). Voltage was applied in the y direction and current moved in the x direction, and von Klitzing plotted conductivity (sigma) y vs density of electrons x. The expected result based on classical mechanics would be an upward sloping line, but because of quantum mechanics the graph was more of a step graph at integer values. This experiment redefined electrical conductivity units - basis for quantum computing - several Nobel prizes were won over the birth of topology in physics from 1985-2000s!

After lunch, students were treated to a more hands-on presentation by Paul Ellis on modular origami and platonic solids. Working with Sonobe units Professor Ellis taught them to create (basic origami building blocks that can be combined to create larger structures), students found that they could make a cube from six units. Cubes are the only Platonic solid students could make with basic Sonobe units as they exist, because they can only form right angles. But by folding the middle square along the diagonal in the Sonobe units, students could get a building block of four triangles. Using 12 of these slightly modified Sonobe units students collaborated to make a combination cube and octahedron (the cube and octahedron are said to be “dual” of each other).

The presentation ended with a brief discussion of an issue - why are only five Platonic solids possible? Professor Ellis started the explanation with a review of the formulas for the interior and exterior angles of a polygon, and left it to the students to figure out for themselves why those formulas are relevant.