Having run the Ross Mathematics Program online in Summer 2020, we will continue to build the online Ross community with some outreach events during the academic year. All these events will take place in our Zoom room, and are open to the broader mathematical community, so please feel free to invite your friends.
Introduction to Algebraic TopologyJohn Nicholson
Weekdays January 25 - 28, 2021 at 6pm Eastern daily
Algebraic topology begins with the realisation that fundamental questions about spaces like manifolds and cell complexes are best understood using methods from abstract algebra such as groups and rings. This course will be a fast-paced and intuitive introduction to the basic techniques of algebraic topology with a particular focus on the Euler characteristic and the fundamental group. The final goal will be to apply these techniques to the classification of 2-dimensional cell complexes, which are spaces made up of vertices, edges and faces.
We will not assume any previous knowledge of abstract algebra and topology but some familiarity with groups, metric spaces and continuous functions would be helpful.
Conway's Topograph to Construct Solutions to Fermat's Christmas Problem.Steven Gubkin
Friday, February 12, 2021 at 6pm Eastern
A binary quadratic form is a function of the form f: ℤ² → ℤ of the form f(x,y) = ax²+bxy+cy². Such a form is determined by its values on a “lax superbasis” of ℤ² (a trio of vectors v⃗, w⃗, v⃗+w⃗ any two of which span ℤ², mod a sign). We can visualize all of the lax superbases using a “Conway Topograph”. Walking around on this topograph for the quadratic form x²+y² gives us an algorithm to decompose any odd prime congruent to 1 mod 4 into a sum of two squares.
Residual finiteness and decision problemsProf. Mark Pengitore
Tuesday, November 17, 2020 at 6pm Eastern
In this talk, we will introduce residually finiteness and its connections to decision problems in finitely generated groups Residual finiteness and decision problems.
Perfect numbers, like perfect people, are very rareProf. Paul Pollack
Wednesday, November 11, 2020 at 6pm Eastern
A natural number n is called perfect if the sum of the proper divisors of n is n itself. For example, 28 is perfect, since 28 = 1 + 2 + 4 + 7 + 14. We will survey what is known about the count of perfect numbers up to a given height x, as x grows. Time permitting, we will prove the theorem of Hornfeck and Wirsing that the perfect numbers are rarer than the kth powers, for any fixed positive integer k.
Games, Graphs, Groups, and Geometric SeriesProf. Rachel Skipper
Tuesday, October 27, 2020 at 6pm Eastern
In this talk, we will discuss the game “The Towers of Hanoi” which comes from a legend about monks who have divine orders to move 64 golden disks among three pegs. The legend says that when the monks have moved the last disk, the world will end and asks the question: when? We’ll answer this question and see how the “Hanoi Tower group” connects the game to the Sierpinski triangle.
ProofgrammingProf. Jim Fowler
Thursday, October 22, 2020 at 6pm Eastern
People use computers to help with computations, but can computers also help us write proofs? This talk is a friendly introduction to proof assistants, that is, to computer programs which help humans write proofs and verify their correctness.
The underlying machinery is “type theory” but rather than a survey, we’ll see some examples. This is a “live” talk in the sense that we’ll see real examples running on a real computer, specifically using Agda. By the end, you will perhaps believe that writing proofs and writing programs are not so different after all.
Office hours, or Ask Me AnythingProf. Tim All
Wednesday, October 14, 2020 at 6pm Eastern
Join me for some informal office hours. We can discuss Ross problem sets, mathematics I’ve been thinking about and/or teaching recently, or anything else; digressions, diversions, and distractions are very welcome! Hope to see you there.
QuaternionsProf. Daniel Shapiro
Thursday, October 08, 2020 at 7pm Eastern
The plane is represented as ordered pairs (a, b) of real numbers. By defining a multiplication operation on those pairs, we create the field 𝗖 of complex numbers. That multiplication can be viewed geometrically, yielding rotations and dilations of the plane. Many geometric problems can be recast as simpler statements within 𝗖. In the 19th century, the complex numbers bacame a useful tool in studying problems of plane geometry.
In the 1840s, Hamilton hoped to find a similar algebraic structure underlying 3-dimensional space. After many failed attempts, he realized that the “correct” algebra is 4-dimensional. His system 𝗛 of quaternions has deep links with geometry in 3 and 4 dimensions. During the past 175 years, that 4-dimensional algebra has provided major motivations for subsequent work in algebra and geometry.
In this lecture we will discuss Hamilton’s ideas.
Click here to download the slides.