Titles and Abstracts (Spring 2018)

February

2 – Leon Liu (MUST)

Title: 27 Lines

Abstract: There are 27 lines on every smooth cubic surface. This is a very intriguing result from the field of algebraic geometry. In this talk, I will give the proof of this theorem.

 

6 – Christine Lee

Title: Knots vs. diagrams: The Jones polynomial and the Tait Conjecture

Abstract: Knots are embedded circles in the 3-dimensional space R^3, and knot diagrams are 2-dimensional projections of a knot on a plane which are convenient tools for visualizing knots. However, it is usually difficult to extract information about the knot from the diagram. We introduce the Jones polynomial, a quantum link invariant from the representation theory of quantum groups, and we show how the Jones polynomial easily proves the Tait conjecture concerning crossing numbers of diagrams of alternating knots.

 

13 – Daniel Freed

Title: Parallelism in flat and curved geometry

Abstract: Euclid’s parallel postulate, a central point of interest in geometry for centuries, is the basis of affine geometry. Most curved spaces do not admit global notions of parallelism, but there are weakened parallelism structures which pervade differential geometry.

 

20 – Scott Aaronson

Title: Complexity and Computability with Closed Timelike Curves

Abstract: In a seminal 1991 paper, David Deutsch proposed a formal model of closed timelike curves (CTCs), or time travel into the past, which used linear algebra and fixed points to “resolve the grandfather paradox.” In 2008, John Watrous and I showed that, under Deutsch’s model, both a classical computer and a quantum computer with access to a polynomial-size CTC could solve exactly the problems in the complexity class PSPACE. I’ll discuss that result as well as a recent extension by myself, Giulio Gueltrini, and Mohammad Bavarian to the setting of computability theory. Namely, a classical or quantum computer with access to a Deutschian CTC (with no bound on its size) could solve exactly the problems that are Turing-equivalent to the halting problem.

Papers available at:
http://www.scottaaronson.com/papers/ctc.pdf
http://www.scottaaronson.com/papers/ctchalt.pdf

 

23 – Juan Moreno

Title: Cohomology and Poincare Duality (MUST)

Abstract: Cohomology has widespread appearance throughout mathematics. This is, in part, due to its many guises. A particularly useful formulation is the de Rham theory for smooth manifolds. This formulation provides an excellent introduction to the topic of cohomology with little prerequisite knowledge (a solid calculus background, and basic topology and abstract algebra). We begin with a quick overview of differential forms in Euclidean space and move on to define the de Rham cohomology of smooth manifolds. We then present a Poincare Duality statement and examine some examples and consequences of the theorem. If time allows we will also introduce the Thom class of a vector bundle, which will lead to a nice geometric interpretation of Poincare duality.

 

March

9 – Tam Cheetham-West

Title: Birkhoff’s Ergodic Theorem (MUST)

Abstract: A discussion motivating the ergodic theorem of Birkhoff and presenting a proof of the theorem. An analog of this ergodic theorem which involves the free abelian group on two generators will also be discussed.

 

23 – Zachary Gardner

Title: Fermat’s Last Theorem for Regular Primes (MUST)

Abstract: The search for a proof of Fermat’s Last Theorem (FLT) helped spur much of the historical development of algebraic number theory (ANT). Ideals, class groups, and other core concepts of ANT all have connections to FLT. We will focus on a proof of FLT due to German mathematician Ernst Kummer for the case of regular primes – i.e., those primes p such that p does not divide the class number of the pth cyclotomic field.

 

27 – Dan Knopf

Title: Close connections between topology and curvature in low dimensions

Abstract: In dimensions two and three, there are beautiful relationships between the topology of manifolds and their geometry. I will explain these relationships, how they give rise to elegant classification theorems, and (time permitting) introduce the role played by “geometric heat equations” in their proofs. The talk will be expositional, hence won’t require any extensive background.

 

30 – Jeffrey Jiang

Title: The Laplace-Beltrami Operator: An Introduction to Hodge Theory (MUST)

Abstract: Hodge theory is a powerful tool to study cohomology in differential and algebraic geometry. I will define the Laplace-Beltrami Operator, which generalizes the standard Laplacian to differential forms, in addition to stating several important results regarding the de Rham cohomology groups of a smooth manifold.

 

April

6 – Leon Liu

Title: Category theory and Linear Algebra (MUST)

Abstract: Category theory was invented by Eilenberg and Mac Lane back in the 1940s. The definitions and notions in category theory are very straightforward, so why do people call category theory “abstract nonsense?” In this talk, I am first going to examine the definition of a category. We will then use category theory to connect many constructions across different areas of math, with examples focused on linear algebra and vector spaces.

 

10 – Pedro Morales

Title: Summability of Divergent Series

Abstract: I will talk about how can we extract numbers out of divergent series. A couple of these examples include the famous sum of positive integers being -1/12 and the sum of (-1)^n being 1/2. We will explore these and other examples in order to make mathematical sense of these results. We will extend the notion of convergence to be able to “sum” divergent series.

 

13 – Hunter Stufflebeam

Title: Dr. Levi-Civita, or: How I Learned to Stop Worrying and Love the Torsion (free) Connections (MUST)

Abstract: In this talk, I will introduce the basic machinery of smooth manifold theory and explore the concept of connections on tangent bundles of Riemannian manifolds. In this way, I hope to provide a quick yet digestible invitation to the wonderful worlds of differential topology and geometry. Instead of presenting a lot of theorems and proofs, I’ll take a more heuristic and conversational approach to keep the talk as light and friendly as possible. Time permitting, I will also demonstrate a cool visual result concerning torsion connections, and what this property means in math as well as physics.

 

20 – Ewin Tang

Title: The Polynomial Method (MUST)

Abstract: The polynomial method is a technique that has risen to prominence because of its ability to produce short, clean, elementary proofs that solve long-open conjectures in combinatorics. We will cover the basic idea underlying the polynomial method, present (in full!) these aforementioned proofs, and give some context for the importance of the results and the surprising nature of these proofs.

 

24 – Sam Gunningham

Title: An Invitation to Lie Groups and Representation Theory

Abstract: Lie (pronounced “Lee”) groups are the mathematical expression of continuous symmetry; as such, they play a key role in math, physics, chemistry, and engineering. The subject of representation theory is concerned with ways in which a Lie group may operate on a vector space, i.e. ways to “represent” the Lie group via matrices. Examples of Lie groups include:

  • The circle group SO(2) (aka U(1)), consisting of rotations about the origin in 2d space
  • The rotation group SO(3), consisting of rotations about the origin in 3d space, and its close cousin, the spin group Spin(3) (aka SU(2)).
  • The Lorentz and Poincare groups, which express symmetries of Minkowski space-time in relativity
  • The Heisenberg group, which plays a key role in Quantum mechanics
  • The exotic “exceptional” groups such as G_2 and E_8, which crop up in surprisingly diverse applications.

The talk will assume no prior knowledge of group theory (though familiarity with linear algebra will be helpful), and will focus on examples.