Harvard University Math Department

Talk begins at 4:30, reception begins at 5:30, snacks provided.

**Speaker:**Alissa Crans (Loyola Marymount)**Title:**Matrices, Reflections, and Knots, oh my!**Abstract:**Motivated by questions arising in starkly different contexts, quandles have been discovered and rediscovered over the past century. The axioms defining a quandle simultaneously encode the three Reidemeister moves from classical knot theory and capture the essential properties of conjugation. Thus, on the one hand, quandles are a fruitful source of applications to knots and knotted surfaces; in particular they provide a complete invariant of knots. On the other, they inspire independent interest as algebraic structures; for instance, the set of operation preserving maps from one quandle to another admits a natural quandle structure in a large class of cases. We will illustrate the history of this theory through numerous examples and survey recent developments.

**Speaker:**Martin Nowak (Harvard)**Title:**The Mathematics of Cooperation**Abstract:**Cooperation means that one individual pays a cost for another to receive a benefit. Cooperation can be at variance with natural selection: Why should you help a competitor? Yet cooperation is abundant in nature and is an important component of evolutionary innovation. Cooperation can be seen as the master architect of evolution and as the third fundamental principle of evolution beside mutation and selection. I will present mathematical principles of cooperation.

**Speaker:**Joe Harris (Harvard)**Title:**Poncelet's theorem and the birth of modern algebraic geometry**Abstract:**In the early 19th century, Poncelet asked a simple question: given two ellipses in the plane, one inside the other, does there exist a polygon inscribed in the outer one and circumscribed about the inner? In this talk, I'll describe the (somewhat surprising) answer, and also describe how consideration of this simple problem led to two major changes in how we do algebraic geometry: the introduction of projective space, and the use of complex numbers.

**Speaker:**Bill Dunham (Harvard)**Title:**A Recipe for π**Abstract:**In recognition of the upcoming “Pi Day, 2019,” we begin with a quick history of this illustrious constant, mentioning contributions from Archimedes, Newton, Leibniz, and Euler and carrying the story to the present day, where the value of π has been computed to trillions of places. Then, to add a bit of mathematics to the tale, we examine a result from Euler that features this constant in a starring role. In 1783, he sought the exact sum of the infinite seriestan(

His solution was devilishly clever in an 18th-century kind of way and should bring our celebration of π to a fitting end.^{π}⁄_{4})+^{1}⁄_{2}tan(^{π}⁄_{8})+^{1}⁄_{4}tan(^{π}⁄_{16})+^{1}⁄_{8}tan(^{π}⁄_{32})+...

**Speaker:**John Mackey (Carnegie Mellon)**Title:**Tournaments having the most cycles**Abstract:**In celebration of March Madness, we will consider the problem of maximizing the number of cycles in round robin tournaments. A round robin tournament is a set of matches in which each contestant plays all other contestants. A cycle is a sequence of contestants, each beating the one that follows it, with the last beating the first. This problem is about three-fourths solved. Open problems and connections with the zeta function will be discussed.

**Speaker:**Lauren Williams (Harvard)**Title:**Combinatorics of shallow water waves (via the KP hierarchy)**Abstract:**I'll give a gentle introduction to the Grassmannian, and explain how its points give rise to soliton solutions of the KP equation (which model shallow water waves). I'll then describe how the combinatorics of the positive Grassmannian and its cell decomposition allows one to answer questions about shallow water waves.

**Speaker:**Alissa Crans (Loyola Marymount)**Title:**Matrices, Reflections, and Knots, oh my!**Abstract:**Motivated by questions arising in starkly different contexts, quandles have been discovered and rediscovered over the past century. The axioms defining a quandle simultaneously encode the three Reidemeister moves from classical knot theory and capture the essential properties of conjugation. Thus, on the one hand, quandles are a fruitful source of applications to knots and knotted surfaces; in particular they provide a complete invariant of knots. On the other, they inspire independent interest as algebraic structures; for instance, the set of operation preserving maps from one quandle to another admits a natural quandle structure in a large class of cases. We will illustrate the history of this theory through numerous examples and survey recent developments.

**Mailing list:** You can subscribe to the seminar mailing list here.

**Organizers:** Ana Balibanu (ana@math.harvard.edu) and Alison Miller (abmiller@math.harvard.edu)