Fall 2022

Putnam Competition Postmortem

Speaker: Noam Elkies, Harvard University

Abstract: Solutions, outlines, and/or mathematical context will be given for some of the problems from the Putnam exam on December 3, starting with problems that most lend themselves to discussion.

Pushing a Camel through the Eye of a Needle

Speaker: Maira Khan

Abstract: I will present some ideas from the proof of Gromov's Non-Squeezing Theorem with emphasis on a more dynamical view. I hope to give some intuition about symplectic "width" and deformations of manifolds in phase space. 

The Ax-Grothendieck Theorem

Speaker: Hari Iyer

Abstract: Results in mathematics with simple statements often have surprising proofs. This talk will present such an example, applying structures from set theory to solve a cute problem about polynomial maps between vector spaces.

Breakdown of the Einstein field equations and space-time singularities

Speaker: Oswaldo Vazquez

Abstract: General Relativity (GR) is our best theoretical framework for describing large scale phenomena such as supernovae, rotating black holes, planetary orbits, etc. The theory models space and time as a 4-dimensional semi-Riemannian manifold M and posits that the force of gravity is manifested through the curvature of M. The equations of GR are powerful but not perfect as they can lead to singular solutions containing blow-ups of geometric invariants. The most famous GR singularity is that which occurs at the center of a (static) spherically symmetric black hole. In theory, there can also exist singularities that are not necessarily hidden under an event horizon (these pathological singularities are called "naked") . It is then natural to ask: what are the conditions that allow space-time to have such singularities? In this week's Math Table, I will give a heuristic overview on how one may tackle the singularity formation problem as well as recent progress I have made with Dr. Puskar Mondal.

Extended Promotion and Quasi-tangled Labelings 

Speaker: Eliot Hodges

Abstract: In 2022, Defant and Kravitz introduced extended promotion (denoted ∂ ), a map that acts on the set of labelings of a partially ordered set. Extended promotion is a generalization of Schützenberger's promotion operator, a well-studied map that permutes the set of linear extensions of a poset. A remarkable fact about promotion is the following: if L is a labeling of an n-element poset, then ∂^{n-1}(L) is a linear extension. This result allows us to regard promotion as a sorting operator on the set of all labelings of a finite poset. In this talk, I will introduce extended promotion and discuss some research on promotion I did at the Duluth REU. In particular, the talk will conclude with an outline of the enumeration of the quasi-tangled labelings of a poset as well as the statement of some open problems and further directions of inquiry.

Hyperbolic Knotoids

Speaker: Dora Woodrfuff

Abstract: In the history of hyperbolic geometry, two revolutionary observations stand out: first, that hyperbolic space (most classically, space in which an alternative to Euclid's Parallel Postulate holds) exists, and secondly, that many spaces turn out to be hyperbolic. In the history of knot theory, there are two corresponding observations: first, that one can define a meaningful 'volume' for some types of knots and secondly, that the class of knots for which you can do this is surprisingly large. Combining these two threads gives us 'hyperbolic knot theory.' This exciting field is rich with astonishing, powerful results, connections with areas as far reaching as number theory or sphere packing, and even applications to chemistry. In this talk, I will introduce hyperbolic knot theory, share some of its highlights, and conclude with some research I conducted this summer at the SMALL Knot Theory REU. 

Jobs Panel

Please join us for a panel discussion about what careers and jobs you can pursue with a background in mathematics. Our panelists are 

• Tyler Piazza, a quantitative analyst,

• Siva Muthupalaniappan, a clinical research fellow at Beth Israel Deaconess Medical Center,

• Heemyung Hwang, at Zagaran Software Development, and

• Aftab Chitalwala, at Weiss Asset Management.

Sailing, calculus and conformal maps

Speaker: Mike Hopkins, Harvard University

Abstract: In the mid 1500s the Flemish cartographer Gerardus Mercator introduced a revolutionary idea into the construction of maps used for navigation. This talk will cover some of the wonderful mathematics in Mercator's idea, and the surprising story of the formula for the integral of the secant.   

After the talk, there will be light refreshments!