Cindy Blois, Thurs Jan 31, 4:30-6:00, BA6180
Title: Path-Ordered Exponentials
Abstract: In this talk, we will look at calculus through a new lens as we approach the definition of the path-ordered exponential. We will see that the path-ordered exponential arises naturally in the solution of first-order linear systems of ODEs (with variable coefficients) and is also a stepping stone toward path integrals in quantum physics.
99% of this talk will be accessible to undergraduates that have some experience with introductory analysis and linear algebra. However, 1% will be accessible to no one, because it will be utter nonsense.
Yu-Wen Hsu, Wed Jan 30, 5:00-6:30, BA6180
Title: The Power of Geometric Evolution Flow
Abstract: What numerical characteristics do geometric shapes have? You will probably think of the length of a curve, or the volume of a 3-dimensional shape, or an angle between two directions in space. In this talk we will discuss a characteristic of curves called curvature, which measures the sharpness of a turn while moving along a path.
It turns out that this characteristic controls the process of curve evolution, called the CSF, “curve shortening flow”. We will talk about some important work that was done in the 80s on curves in R^2 and see how the CSF can reshape a curve.
Steve Bennoun, Thurs Jan 24, 4:30-6:00 pm, BA6180. Pizza.
Title: A Peek into Category Theory
Abstract: Have you ever heard about categories? What are they? What are they useful for?
We will start the following question: take two groups G and H and look at the set of homomorphisms between them. Does this set have a group structure? To answer this question, we will introduce some basic notions of category theory. We will reformulate our question in categorical terms and then see how categories help us solve it. Along the way, we’ll define what categories and functors are and look at examples. We’ll actually see that many mathematical objects we know are categories.
Lindsey Shorser, Thurs Jan 17, 12:30-2:00 pm, BA6180. Pizza.
Title: Categories: What, Why, and How – An Introduction
Abstract: Category theory can be seen as a language that allows common features to be abstracted out of mathematical objects such as sets, vector spaces, groups, topological spaces, etc. In this talk, we will look at the flavour of categorical thinking with same basic definitions and examples. Then we will discuss commutative diagrams and why homological algebra and category theory are inextricably intertwined.
Fabian Parsch, Wed Jan 16, 5:10 – 6:00, BA6183. Pizza begins at 4:30.
Title: Fantastic Nets and Where to Find Them
Take a sheet of paper and draw three points in the plane. Now connect them with curves, but try to make the total length of all curves as short as possible. After some experimentation, you will probably arrive at a good guess of a best solution. But how do you know this is in fact a best solution? How do you know it is the only best solution? How do you even know there is a best solution? What if you have to connect four points instead? Five? n?
These are the kind of questions that are answered by the concept of geodesic nets.
The best thing about geodesic nets is that they are very easy to define, as we will see in the talk. The worst thing about geodesic nets is that not much is known about their behaviour, even in the simplest case: connecting points on the flat plane. If we look at other surfaces (like the sphere or a torus), things only get worse.
In this talk we will define what geodesic nets are, talk about some known results and discover some “animals“ in the zoo of nets together. Time permitting, we will also have a look where geodesic nets appear in architecture and material science.
The talk has only minimal prerequisites (see below). As long as you bring along an intuitive understanding of geometry, you are ready for take-off!
Euclidean Geometry, Linear Algebra
Helpful (but not necessary!)
Elementary Surfaces (Sphere, Torus), Geodesics, Curvature
You have been inside the Roger‘s Centre while the roof was closed.
Bringing your Laptop for some experiments is encouraged!
Asif Zaman, Fri Jan 11, 5:00 – 6:30 BA6183
Title: The multiplication table problem
The $N$ by $N$ multiplication table is formed by multiplying two integers from 1 to $N$. The numbers in the table are between 1 and $N^2$ but how many distinct numbers are in this table? By mixing ideas from probability and number theory, Erdös proved that very few numbers up to $N^2$ appear. I’ll describe the remarkably elegant proof and explore related probabilistic ideas when studying the anatomy of integers.
Tyler Kloefkorn, Thurs Jan 10, 4:30 – 6:00 BA6180
Title: Patterns in the Periodic Table of Finite Elements
Abstract: The finite element method is used to find numerical solutions to partial differential equations, with boundary conditions, on various domains discretized into simple geometric elements (e.g., simplicies and cubes). This method is applied in a variety of contexts, including animation, image processing, computational electromagnetism, atmospheric motion, and fluid mechanics. Often, finite element computations in these areas (and others) can be ad hoc. Accordingly, the Periodic Table of Finite Elements was recently constructed to summarize useful information for careful implementation of the finite element method. In this talk and with input from attendees, we will broadly describe the finite element method and the Periodic Table of Finite Elements, and we will look for interesting algebraic patterns found within the Table.
The math department is hosting a series of hiring talks given by professor candidates, designed for undergrads. There will be pizza. We will post further information, and possibly make event pages for these individually, but this is data on time and place and name:
Thurs Jan 10 Kloefkorn, 4:30 – 6:00 BA6180
Fri Jan 11 Zaman, 5:00 – 6:30 BA6183
Wed Jan 16 Parsch, 4:30 – 6:00 BA6183
Thurs Jan 17 Shorser, 12:30-2:00 BA6180
Thurs Jan 24 Bennoun , 4:30 – 6:00 BA6180
Fri Jan 25 Hoell, 5:00 – 6:30 BA6183
Wed Jan 30 Hsu, 5:00 – 6:30 BA6180
Thur Jan 31 Blois, 4:30 – 6:00 BA6180
Our next talk will be this Thursday October 4th in BA 2135 at 6:00.
Prof. Almut Burchard is presenting on
A shape optimization problem related with flocking
Given a large number of particles where each pair
experiences a force determined by their distance.
When can a flock form, and what should it look like?
Suppose that any two particles attract when they are
far apart, and repel each other when they are close. One
may conjecture that a sufficiently large number of
particles would organize itself in a round flock; if the
number of particles is too small, they would disperse. I will
describe recent progress towards this conjecture,
and mention open problems.
Pizza in the grad lounge at 5:30!
Math Union Talk: Symmetry
Professor Joe Repka
Friday Sep 28, 5:30-6:30pm, BA1190
Pizza following the talk