# Open Neighborhood Seminar

### Harvard University Math Department

### February 8

**Speaker:**Melody Chan (Brown)**Title:**Counting Trees**Abstract:**I will talk about counting trees, in the sense of graph theory---although this can be adjusted to the demands of the audience. No prerequisites.

### February 15

**Speaker:**Rohini Ramadas (Harvard)**Title:**Dynamics of polynomials and rational functions**Abstract:**I will introduce polynomials as dynamical systems on the set of complex numbers and rational functions as dynamical systems on the Riemann Sphere (the complex numbers plus a point at infinity). I will discuss how the study of the dynamics of polynomials and rational functions leads to the study of higher dimensional dynamical systems. A famous example: If 'c' is a complex constant then the polynomial 'z goes to z^2+c' is a map from set of complex numbers to itself. How do the dynamics of this map depend on the complex number 'c'? This will be a very introductory talk.

### March 1

**Speaker:**Wes Cain (Harvard)**Title:**Delay Equations**Abstract:**Time delays are associated with instabilities and oscillations in a wide array of phenomena in nature (e.g., oscillations in insulin/glucose levels). The desire to understand such phenomena leads naturally into the study of delay differential equations (DDEs). We'll begin by discussing some basic DDE-related notions (mainly via examples), at a level accessible to students who have completed a second semester calculus course (Math 1b or equivalent). Then, we'll delve into the theory of stability of equilibria for DDEs, drawing upon notions covered in courses in the 110-level (analysis) offered through Harvard's math department. I'll close by discussing a research project that I hope to complete in collaboration with one or more undergraduates: a functional analytic framework for stability of a special class of DDE (namely, neutral DDEs).

### March 23 (Note the unusual day!)

**Speaker:**Kathryn Mann (Brown)**Title:**Ordering Groups**Abstract:**The integers come with an operation (+) and an ordering (<) preserved by this, in the sense that a < b implies a+c < b+c. In this talk, I’ll introduce you to some rich and beautiful structures that generalize this: the theories of left-orderable and circularly-orderable groups. These structures are interesting algebraically, but also have connections to questions in topology and dynamics, which is how I came to use them in my research.

### April 5

**Speaker:**Karen Edwards (Harvard)**Title:**A Hands-On Exploration of Non-Euclidean Geometries**Abstract:**Just when you think you've absorbed a fundamental truth of mathematics, it throws you a curveball. You think all triangles have an angle-sum of 180? Try again! Isn't the ratio of circumference to diameter always pi? Nope! Using paper, scissors, spheres and yarn we will discover what happens when you reject the Parallel Postulate.

### April 19

**Speaker:**Heather Macbeth (MIT)**Title:**Tangent developables**Abstract:**A piece of paper can be twisted into a cone, or a cylinder. Euler, in 1772, showed that there is a third kind of surface that can be made in this way: the "tangent developables." I will explain what tangent developables are, and show how to find the pieces of paper that twist into them.

**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)