After the Annie’s talk last week (see below for the abstract!), we will enjoy some more combinatorics.
Jennifer will give a talk about “Convex Reflexive Lattice Polygons and the Number 12”, revealing how a simple equality can be connected to a deep result from another side of mathematics, which is called algebraic geometry. Here is her abstract:
A lattice polygon is a polygon with integral vertices in R^2. This means that its vertices are of the form (a, b), where a and b are integers. A reflexive lattice polygon has the property that the origin (0,0) is the unique integral interior point of the polygon. These polygons have duals, which are also reflexive lattice polygons. Today I will present a curious relation between convex reflexive lattice polygons, their duals, and the number 12. I will explain how this relation has deep connections with toric varieties, forming a bridge between two beautiful areas in mathematics: combinatorics and algebraic geometry.
What is the maximum number of edges in a graph on n vertices without triangles? Mantel’s answer in 1907 that at most half of the edges can be present started a new field: extremal combinatorics. More generally, what is the maximum number of edges in a n-vertex graph that does not contain any subgraph isomorphic to H? What about if you consider hypergraphs instead of graphs? I will introduce the technique of sums of squares and discuss how it can be used to attack such problems.
We had a very fun time last week to playing handbells and figuring out mathematical patterns in it. Here, answering some questions about change ringing activities, I share the informations from Leland Kusmer:
Handbells (at UMass):
Tuesdays, 4 – 6pm, ILC N417
Towerbells (at Smith):
Mondays, 7 – 9pm; Wednesdays, 5 – 7pm
If you would like to enjoy some bells ringing and see mathematical backgrounds of it, join us in the math club this week. William from the Physic Department will give a talk about “Group Theory in Change Ringing”. His abstract is below:
Group Theory in Change Ringing
Change Ringing is a musical/mathematical team sport which consists of ringing a set of tuned bells in a controlled manner to produce variations in the order of the bells. These sets of permutations, when seen through the lens of abstract algebra, form mathematical Groups. We will construct the basic ringing groups, examine their structure and properties, and demonstrate how they are rung using hand bells.
Check out the abstract below.
NUMBERS WE CANNOT NAME
The number line seems simple. It ain’t. The riddles of the continuum have bugged Chinese philosophers, Argentine poets, Greek trolls, and 20+ centuries of mathematicians. We’ll explore the puzzles of the irrational, the transcendental, and the non-computable. Bad drawings copiously featured.
We are very happy to restart the math club with the talk given by Shelby Cox, our own undergraduate student, after the spring break. Please see the abstract below.
Counting Curves in the Plane
Curve counting goes all the way back to Euclid, who asked how many lines pass through 2 given points in the plane. More generally, we can ask how many curves of degree d pass through 3d-1 points. In this talk, I will present a beautiful and well-known proof for the case when d=3, which uses methods from Topology, Geometry, and Algebraic Geometry. The talk will focus on introducing concepts like the Euler characteristic, projective space, and blow-ups, which can help us answer long-standing questions, like: how many pentagons are on a soccer ball? In particular, the talk should be accessible to most math majors.
Here is the rescheduled GaYee’s math club talk tomorrow about “Seven Bridges of Königsberg”. See her abstract below.
Long ago in a little town on a river in Prussia, the people wondered: can you visit every part of the city, crossing all seven bridges only once? This is called the Seven Bridges of Königsberg problem which was popular in the 18th century. In this talk I will introduce basic graph theory and Euler’s solution to this problem. Using the result, we will also explore other related puzzles and its application.
The math club offers a pleasant place to take a break in busy exam weeks. Floyd Williams gives a talk for us. For the abstract, see below:
A tractroid ( or pseudosphere ) is a surface generated by revolving a special curve ( a tractrix ) about an axis. By a clever change of variables one can find a nice realization of this surface as a certain “vaccum space”, which mathematicians call an orbifold -the latter being defined by a simple relation between points. This vaccum space is similar in nature to a 2-dimensional black hole with mass degenerating to zero. Thankfully however these objects are “not” the same (otherwise there’s no life on earth ), and thankfully no knowledge of black holes is needed for the talk-which mainly involves calculus.
We will meet in LGRT 1634 at 5 pm just as usual.
The abstract of the talk is as follows:
The Radon transform is a way of transforming one function into another which has applications in mathematics and image processing. I will define the Radon transform as well as the closely related X-ray transform. I will recall the more famous Fourier transform, and then discuss the connection between these operators (which seems to have given rise to the popularity of the Radon transform in medical imaging)
It was really unfortunate that we had to cancel the math club last week due to the snow. We will reschedule GaYee’s talk at some time this semester, so don’t miss it!
This week, we are happy to have Eric Sommers as a speaker. He will talk about “An Introduction to Representation Theory”.
Here is his abstract:
I’ll give a survey of the main ideas behind the representation theory of finite groups. Representation theory is the study of the ways in which a group can be represented by a set of matrices. Knowing some linear algebra (Math 235) is helpful, but not required. It is not necessary to know what a group is– I will give the definition. This is a beautiful subject, accessible to all math majors, but not something that shows up in our undergraduate curriculum (although our friends in chemistry do see it).