Spring 2018
The Leech Lattice
Speaker: Xavier Gonzalez, Harvard Undergraduate
Date: 04/24/2018
Abstract: The Leech lattice has been in the news as M. Viazovska won the 2018 Breakthrough Prize in part for helping show that the Leech lattice is the most efficient way to pack 24-dimensional spheres into 24-dimensional space. The Leech lattice, however, appears in many other settings, with connections to error-correcting codes, finite simple groups, the "cannonball problem,'' Lorentzian inner products, and string theory. This talk will introduce the Leech lattice as an exceptional even and self-dual lattice in 24 dimensions and discuss its connection to the (also exceptional) binary Golay code before stating its connection to the exceptional Moonshine vertex operator algebra. Prerequisites:} No background in lattices or error-correcting codes is needed, but familiarity with linear algebra will be helpful.
Preventing Period Doubling
Speaker: Wes Cain, Harvard University
Date: 04/17/2018
Abstract: To paraphrase a former Harvard student, "Two roads diverged in a wood, and I ... didn't really like either option, so I decided to plow a new road between the other two, smoothly extending the path I was already on." In this talk I'll discuss (i) period-doubling bifurcation in dynamical systems; (ii) why period-doubling bifurcations in nature can be catastrophic; (iii) the mathematics behind one cute method of preventing period-doubling bifurcations; and (iv) the relevance of the quote in the preceding sentence. Prerequisites: Students who are completing (or have completed) 21b will have no trouble understanding this talk in its entirety. Even students who haven't completed 21a/b will understand quite a lot, especially if they've ever experimented with recursively-defined sequences (e.g., the Fibonacci sequence)
Cycles in Tournaments
Speaker: John Mackey, Carnegie Mellon
Date: 04/10/2018
Abstract: When n people play each other in a tournament (each person playing the other n-1 people), we'd like to rank the participants after knowing the results. Cycles are an impediment to such a ranking. If A beats B, B beats C, and C beats A, then how should A, B and C be ranked relative to one another? Intuitively, when n is odd, one would expect that the number of length k cycles would be maximized in a tournament where each person defeats (is defeated by) (n-1)/2 other people. There are some bumps in the road in proving this, but they are great bumps, involving Bernoulli numbers, eigenvalues, and the Riemann Hypothesis for Tournaments. We'll discuss the current state of what is known and conjectured. No prerequisite knowledge is required to attend this talk.
Penrose Tilings
Speaker: Philip Engel, Harvard University
Date: 04/03/2018
Abstract: In the 1970's, Roger Penrose discovered two tiles which have an amazing property: Any way to tile the two-dimensional plane by them is aperiodic—you can't drag the tiling so that it overlays itself. On the other hand, an arbitrary large region will repeat infinitely many times! We'll talk about some of the patterns that emerge, and describe a method discovered by de Bruijn to construct all the Penrose tilings.
The Transfinite Subway: Counting to Infinity and Beyond
Speaker: Sebastien Vasey, Harvard University
Date: 03/27/2018
Abstract: It is a little known fact that there is a subway line connecting Logan airport to the Hilbert hotel. This line has infinitely-many, and in fact uncountably-many, stops. At each station, exactly one passenger gets off (if the train is not empty), and a countable infinity of passengers get on board. How many passengers will eventually get to the Hilbert hotel? In this talk, we will make the question precise and find the surprising answer to this puzzle! This will be done by sketching the development of a rigorous theory of "infinite numbers", called ordinals, that go beyond the natural numbers while keeping many of their properties. Time permitting, we will investigate connections with more classical problems, such as generating Borel sets of reals and studying the automorphism tower of a group.
Additional Resources: The transfinite subway: counting to infinity and beyond
What Information Do You Need to Recover the Homotopy Type of a Space?
Speaker: Brabeeba Mien Wang, Harvard Undergraduate
Date: 03/20/2018
Abstract: One of the main goals of algebraic topology is to study the homotopy type of a space X using easily computable algebraic invariants. The fundamental group is one such example. Given two spaces X, Y, if they are homotopy equivalent to each other, then these groups are the same. But the converse is generally not true. A natural question to ask is how much more information do we need to recover the homotopy type of a space. In this talk, I will introduce singular cochains and the singular cohomology and show that by adding on extra algebraic structure on them we can distinguish more spaces from each other, and with enough structure we can even recover the entire homotopy type at prime p.
How to Integrate in π+i Dimensions
Speaker: Davis Lazowski, Harvard Undergraduate
Date: 03/06/2018
Abstract: Ever wanted to integrate in imaginary or transcendental dimensions? Well, for a large class of functions, you can! In this talk, I'll describe a delightfully simple procedure to extend the integral to the entire complex plane. Then we'll spend some time considering the properties of the integral as a function of the number of dimensions. Applications to physics will be mentioned. The talk will require no more than knowledge of integration. However, some complex analysis might be helpful to understand what's formally going on.
The Hopf Invariant
Speaker: Reuben Stern, Harvard Undergraduate
Date: 02/27/2018
Abstract: In this talk, I'll tell a story of a great success in algebraic topology, and explain how it led to the development of an enormous amount of important mathematics. The Hopf invariant is a number attached to certain continuous maps between spheres. A major question during the first half of the 20th century was: for which spheres do there exist maps of Hopf invariant 1? This question was answered spectacularly by J.F. Adams in 1960; a much shorter proof was published by Adams and Atiyah in 1966. The shorter proof seems somehow far away from Adams' original one: how did he get to the new one from the old? There's much interesting material to discuss. After a discussion of classical results involving the Hopf invariant, I'll outline the historical progression from 1960 to 1966, and then summarize very broadly the developments from the 1970s to the present study of homotopy theory. This talk is meant for students of many levels, even those who have little background in algebra or topology. If you have some vague idea of what a ``space'' is, I'm sure you'll get something out of this! Simultaneously, those students with a little knowledge of cohomology and homotopy groups will understand more deeply the material.
Surgery with Poincaré Manifold as an Application
Speaker: Frid Fu, Harvard Undergraduate
Date: 02/20/2018
Abstract: In this talk, we will familiarize ourselves with ``surgery'', a powerful operation on 3-manifolds. Like an actual surgery, it cuts unwanted stuff and makes life easier. To be more specific, it cuts knots and turns them into circles. Moreover, surgery enables us to manipulate 3-manifolds even when we have no idea what the manifolds actually look like (which, I think, is the common feature of good mathematical devices---that they enable computation before understanding). As an application, we show that two descriptions of the Poincaré manifold coincide: one is a surgery description and the other comes from a singularity of an algebraic variety.
Polishing Euler’s Gem
Speaker: Oliver Knill, Harvard University
Date: 02/06/2018
Abstract: Just before Euler's day (2/7/18) it is appropriate to look at one of the most beautiful formulas in Mathematics, the so called Euler Gem: V - E + F = 2. Less well known is that this formula has historically been proven wrong again and again, as counter examples have turned up, like Kepler solids. This was such an embarrassment for Mathematics, that Imre Lakatos suggested that theorems do to "evolve" like species, maybe never really reaching their ultimate final form. In this talk we aim to refute Lakatos' hypothesis and give a crystal clear proof of the formula in arbitrary dimensions: the Euler characteristic of a discrete sphere is 1 + (-1)d. The Euler Gem case is d = 2. Understanding the proof does not require Euclidean space and is inductive. The key is to define in a precise way, and entirely combinatorially, what a discrete d-dimensional sphere is. The quest to do that combinatorially has only started in the 20th century with Hermann Weyl, and Euler plays also here an important role as he initiated linking topology with combinatorics. If time permits, the classification of all the Platonic solids in arbitrary dimension will be derived. While known in the case d = 2 already in ancient Greece, the higher dimensional case has been tackled first by rather unusual mathematicians: by Ludwig Schläfli, who was a high school teacher at first or by Alicia Boole Stott, the princess of Polytopia.
Additional Resources: Polishing Euler's Gem