2018–2019 Mathematics Seminar
Date: Wednesday, November 14 at 11:30am
Speaker: Moshe Cohen, Vassar
Title: An introduction to line arrangements and the search for Zariski pairs
Abstract: A line arrangement is a finite collection of lines in the plane. We consider the projective plane where every pair of lines intersects exactly once, so that parallel lines intersect at a point "at infinity" (and then we need to include a half-circle of points "at infinity").
We can study a projective line arrangement using algebra and geometry by looking at equations of lines as in high school algebra. We can study this using combinatorics by looking at the points that are intersections of lines. We can study this using topology by looking at the complement -- the leftover space. We can ask if the combinatorial information forecasts the topological information of the complement.
When this does not occur, that is, when the combinatorics does not predict the topology, we obtain two different geometric arrangements; we call this a Zariski pair. There is no such pair of up to nine lines. Examples have been found with thirteen lines by Rybnikov in 1998 and with twelve lines by Guerville-Balle in 2014. Together with Amram, Sun, Teicher, Ye, and Zarkh, we investigate arrangements of ten lines. Together with four undergraduate students, we investigate arrangements of eleven lines.
This talk is accessible to those without backgrounds in combinatorics, topology, or algebraic geometry.
Date: Friday, February 8 at 3:30pm
Speaker: Brian Katz, Augustana College
Title: How do mathematicians believe?
Abstract: Love it or hate it, many people believe that mathematics gives humans access to a kind of truth that is more absolute and universal than other disciplines. If this claim is true, we must ask: what makes the origins and processes of mathematics special and how can our messy, biological brains connect to the absolute? If the claim is false, then what becomes of truth in mathematics? In this seminar, we will discuss beliefs about truth and how they play out in the mathematics classroom, trying to understand this thing we call the Liberal Arts.
Date: Friday, September 21 at 3:30pm
Speaker: Russell Miller, Queens College, CUNY
Title: Noncomputable Functions and Unsolvable Problems
Abstract: The Turing machine, as defined by Alan Turing in 1936, is widely agreed to be an accurate if slightly idealized representation of what we mean when we speak today of "a computer." The definition is rigorous and has proven extraordinarily useful in mathematical logic. Unexpectedly, though, it also gave rise to the notion of a noncomputable function: a function (often just from N to N) which must exist, according to reasonable axiomatizations of mathematics, yet which cannot be computed by any Turing machine.
We will investigate these ideas from a slightly different angle, considering functions on the real numbers R instead of on the naturals. Here it is possible to impart a reasonably intuitive understanding of how an innocent-seeming function could fail to be computable. We will not spend any time on the details of Turing’s definition, since it is safe to assume that today’s students have a good intuition about what a computer is. Instead, we will discuss which functions on R can be computed, and how the representation of the real numbers (as decimals or other ways) may be relevant.
Date: Friday, September 28 at 3:30pm
Speaker: Corrin Clarkson, Courant Institute, NYU
Title: A pictorial introduction to the curve complex
Abstract: Curves on surfaces play an important role in low dimensional topology. The curve complex is a way of geometrically encoding the relationships between such curves. I will use pictures to describe the construction of this complex and then relate some interesting facts about its structure.
Date: Friday, October 19 at 3:30pm
Speaker: Chris Hanusa, Queens College, CUNY
Title: The Making Of Mathematical Art
Abstract: In this talk I'll be sharing some recent 2D and 3D art derived from mathematical concepts and created using a computer and 3D printing techniques. I'll discuss my inspiration, my methods, and the math that lies behind a number of my pieces. Come learn how it works and help me to generate new mathematical art right before your eyes!
Date: Wednesday, October 24 at 11:30am
Speaker: Nick Bragman, Hofstra University
Title: Probabilistic determinants of sign pattern matrices
Abstract: Sign pattern matrices are matrices where the only possible entries are +, – and 0. We say that two sign pattern matrices are equivalent if they can be obtained from one another solely through signature equivalence, where sign pattern matrix B is equivalent to A if B = AS, for some signature matrix S. We begin by focusing on the probability of nonnegative determinant for each equivalence class, intertwining ideas connected to (–1,1)-matrices and (0,1)-matrices. We use this numerical analysis to quickly identify many matrices with probability 1/2 of nonnegative determinant, and show that this is not exhaustive via failure of the converse.
This research has been conducted throughout the past year at both Hofstra University and The College of William & Mary's REU.
Date: Friday, November 9 at 3:30pm
Speaker: Robert Rand, University of Maryland
Title: Provably Correct Quantum Programming
Abstract: Quantum computing is hard, not only because of the challenges of building quantum devices, but also due to the challenges of programming them and having our programs run as intended. In this talk we introduce QWIRE, a tool that allows us to write quantum programs and mathematically prove that they have the desired behavior. These proofs are mechanically checked by the Coq proof assistant, guaranteeing that our programs meets their specifications.