Fall 2024

September 4 (MT/ONS joint opening talk)

September 18

• Speaker: Joe Harris (Harvard)
• Title: Poncelet’s theorem
• Abstract: Imagine nested two ellipses C, D ⊂ R²  in the ordinary real plane. Poncelet asks: are there any n-sided polygons in the plane inscribed in C and circumscribed about D?
  
  Poncelet’s theorem gives a striking answer to this question: it asserts that, depending on C and D, there are either no such polygons or a continuously varying family of them. In this talk I’ll describe a modern proof of the theorem, in which we work in complex projective space and use the topology of associated loci. There’s also a less well-known Poncelet-type theorem for polyhedra in 3-space, which I’ll describe.

October 2

• Speaker: Sameera Vemulapalli (Harvard)
• Title: Plane quartics, bitangents, and symmetric determinantal representations
• Abstract: It is a beautiful fact that a general plane quartic in the complex projective plane has 28 bitangents. How do we write down these bitangents? It is also a beautiful fact that a general plane quartic in the complex projective plane has 36 symmetric determinantal representations (i.e. there are 36 inequivalent ways to write the equation of the quartic as a determinant of a symmetric 4×4 matrix whose entries are linear forms in 3 variables). 
  
  This talk will go over some of the classical theory of plane quartics, including relationships between bitangents and symmetric determinantal representations, theta characteristics, configurations of 8 points in P^3, and more.

October 16

• Speaker: Ben Knudsen (Northeastern)
• Title: The complexity of collision-free motion planning on graphs
• Abstract: The problem of multiple occupancy in a space is captured by the topology of its configuration spaces. The problem of navigation within a space is tied to its topological complexity, a numerical invariant introduced by Farber to measure the constraints in motion planning imposed by the shape of the ambient space. Combining these ideas, one is led to the study of the topological complexity of configuration spaces as a measure of the difficulty of collision-free motion planning. Thinking of autonomous vehicles and automated factories, it is natural to take the workspace to be a graph, and the complexity in this setting was described by Farber in 2005 in a conjecture with surprising and counterintuitive consequences. After a gentle introduction to some ideas around topological complexity, configuration spaces, and graph braid groups, I will describe a recent proof of this conjecture.

October 30

• Speaker: Thomas Brazelton (Harvard)
• Title: Are lines even real?
• Abstract: Fix four pieces of string, tied to opposite ends of the room. How many other pieces of string can you stretch across the room which touch the initial four you started out with? The 19th century enumerative geometer will tell you there are exactly two pieces of string satisfying this property, i.e. there are two lines meeting four lines in three-space. This, unfortunately, is a lie, because by a “line” this geometer means a copy of the complex projective line, rather than a literal string. We will discuss conditions where this lie is no longer a lie – that is, when solutions to problems of this flavor are actually defined over the reals. In low dimensions we need no tools more complicated than linear algebra, although in higher dimensions, methods derive from all corners of mathematics: from topology to algebra to analysis. Time pending we will discuss recent explosive progress in this area in the past few decades, and a suite of conjectures that guide current thinking about “reality” in enumerative geometry.

November 13

• Speaker: Carolyn Abbott (Brandeis)
• Title: From algebra to geometry and back again
• Abstract:  A group is an algebraic object, but it also possible (and useful!) to study the geometry of a group. To do this, we need to understand what a group “looks like.”  In this talk, I will introduce a way to draw a picture of a group and discuss how to translate between the geometry of this picture and the algebra of the group.  In particular, I’ll describe how various kinds of curvature on the geometric side affect the algebraic structure of the group.