About the Undergraduate Math Colloquium
This year, we have started a colloquium for undergraduates. The aim of the colloquium is for undergraduates to present their math research or any math topic they find intriguing, and to attend talks exploring different kinds of math.
The math talks are given by our fellow classmates at UofT every Monday night at 6pm Toronto time on Zoom: https://utoronto.zoom.us/j/87436990443, Passcode: 593045.
If you would like to give a talk, please email email@example.com to schedule your talk with us!
Teaching proof with technology to enhance understanding in undergraduate mathematics (part 1)
Teaching proof with technology to enhance understanding in undergraduate mathematics (part 2)
To be updated.
Teaching proof with technology to enhance understanding in undergraduate mathematics (part 3)
To be updated.
Past Talks - in (almost) reverse chronological order
The shapes formed by compressing thin structures with heterogeneous rigidity (part 1) - Carlos Moguel
Thin structures are objects with one or two dimensions smaller than the other ones, e.g., a piece of paper (thickness way smaller than width and length). Under lateral compression, it is energetically favourable for flat-thin structures to undergo out-of-plane deformations or buckling when the compressive force reaches a critical value (think of compressing a ruler with your hands!). The parameter quantifying the energetic cost of this bending and fixing the compression required to buckle the structure is what I loosely call “rigidity”. Many theories have been developed to explain this buckling effect in different scenarios, but it is common to assume the rigidity is constant, i.e., isotropic materials. Motivated by elastic deformations of biological membranes in cells, we will explore the space of shapes predicted by two of the simplest physical theories for buckling of thin structures with a non-constant rigidity (i.e., anisotropic materials).
In this first talk we will consider one-dimensional models. Our starting point will be the well-known Euler Buckling problem studied by Leonhard Euler in the 18th century, when the rigidity is constant. Afterwards, we will study how the buckled shape changes when we employ a non-constant rigidity using a mix of geometry, numerical methods, analysis, and physical intuition.
Prerequisites: If you know what derivatives are, you will be able to grasp the big picture (which is great!). Familiarity with geometry of curves (MAT363) and differential equations (MAT244) would be helpful, but I will introduce whatever we need as we go. Background in physics or biology is not required.
The shapes formed by compressing thin structures with heterogeneous rigidity (part 2) – Carlos Moguel
In this second talk we will consider two-dimensional fluid membranes, i.e., membranes that cannot support shearing. Our starting point will be the Helfrich Hamiltonian, an extensively used model for cell membranes where a constant rigidity is usually assumed. Afterwards, we will consider the case with a non-constant rigidity, where a mix of numerical methods and studying different initial plate shapes is required. Time permitting (and if the audience is interested!), I will talk about an application of Riemannian geometry to study this case. This is very much work in progress.
Eigenvalues, puzzles, and intersections of subspaces – Dr. Allen Knutson
Q1. How many lines touch four generic lines in space? (Two.) How about 5-planes in 18-space that intersect a generic 10-plane in at least dimension 2, while also intersecting a 7-plane in at least dimension 3, etc.
Q2. If we know the eigenvalues of two nxn symmetric matrices, and we sum the matrices, what can we say about the eigenvalues of the sum? (For example, the largest eigenvalue of the sum is at most the sum of the two largest eigenvalues.) It turns out that both questions can be answered by the same jigsaw-style “puzzles” invented by Terry Tao and me.
I’ll explain how the two questions are connected, how the puzzles work, and what open questions remain.
Access code: 9VY0@iL.hh
Virtual Ring Routing – Amalrose Vayalinkal
Virtual Ring Routing (VRR) schemes define a routing algorithm for communication between devices by establishing a virtual network overlay given a physical network of N devices. Using graphs to model the physical network, we introduce the algorithm and explore the pros and cons of VRR. Time permitting, we take a closer look at the simpler case where the physical network is also a ring (circle) and discuss future directions. This work is part of an NSERC USRA this summer under the supervision of Professor Almut Burchard.
Prereqs: basic probability theory is recommended but not required.
Quantum walk and friendship on graphs – Neo Yin
What is quantum walk on graphs? Easy. You take a graph, which is a few vertices and edges connecting some of them. You take a matrix that encodes information about this graph and which point is connected to which (specifically the Laplacian matrix). You exponentiate t, a real number for the time, scalar multiplies that matrix, and get a smooth path, in t, in the space of matrices. This path is the quantum walk.
Why care? This has something to do with quantum computation, and the graph has something to do with quantum circuits. On some graphs, a pair of vertices may become “friends” and “catch up” with each other periodically. This kind of friendship and catching up makes quantum computer scientists very happy, excited, and full of joy for reasons I know very little about.
I will mathematically define this “friendship” (you can abbreviate using the first two letters: “FR” which, coincidentally, is the acronym for “fractional revival”). I will also present some cool results we know about on which graphs and for which value of time t this friendship can happen.
Prereqs: basic Linear algebra.