Posted November 1, 2017
Undergraduate Student Colloquium
1:30 pm – 2:30 pm Lockett 284
Peter Nelson, University of Waterloo
Squaring the square
Is it possible to decompose a square into smaller squares of different sizes? The solution to this problem, which has surprising links to graph theory, linear algebra and even physics, was discovered by four undergraduate students at Cambridge University in the 1930\'s. I will tell the interesting mathematical story that led to this discovery.
Posted March 14, 2018
Undergraduate Student Colloquium
10:30 am – 11:20 am tba
Renling Jin, College of Charleston
A taste of Logic -- from the reasoning of a thief to a painles proof of th incompleteness theorem of Godel.
We will present some fun part of mathematical logic including a puzzle, a true paradox, and a fake paradox. The discussion will lead to Godel's Incompleteness Theorem. Godel's Incompleteness Theorem is well-known but difficult to proof. We will present a heuristic proof of the theorem which should be sufficient to understand the idea of the rigorous proof of the theorem.
Refreshments will be served in the Keisler Lounge at 10am. Posted April 6, 2018
Last modified April 9, 2018
Undergraduate Student Colloquium
3:30 pm Lockett 239
Ken Goodearl, UCSB
How fast does a group or an algebra grow?
Abstract: An algebraic object ``grows" from a set $X$ of generators as larger and larger combinations of those generators are taken. In the case of a group $G$, this means taking longer and longer products of generators and their inverses. For an algebra $A$ (a ring containing a field), it means taking linear combinations of longer and longer products of the generators. The growth rate of $G$ is the rate at which the number of elements that can be obtained as products of at most $n$ generators and their inverses grows with increasing $n$. The growth rate of $A$ amounts to counting dimensions of subspaces spanned by products of at most $n$ generators. These rates of growth provide important measures for the ``complexities" of $G$ and $A$, respectively. They may be given by a polynomial function or an exponential function, but there are quite a few surprises -- rates like a polynomial with degree $\sqrt 5$ can occur, or rates in between polynomial and exponential functions, whereas some other potential rates are ruled out. We will discuss the basic ideas of growth for groups and algebras; the distillation of growth rate into a ``dimension" for algebras; and the values that this dimension can take.
Refreshments will be served in the Keisler Lounge at 3:00 pm.
Posted November 13, 2018
Undergraduate Student Colloquium
3:30 pm – 4:20 pm Lockett 114
Joeseph E. Bonin, George Washington University
What do lattice paths have to do with matrices, and what is beyond both?
Abstract: A lattice path is a sequence of east and north steps, each of unit length, that describes a walk in the plane between points with integer coordinates. While such walks are geometric objects, there is a subtler geometry that we can associate with certain sets of lattice paths. Considering such sets of lattice paths will lead us to examine set systems and transversals, their matrix representations, and geometric configurations in which we put points freely in the faces of a simplex (e.g., a triangle or a tetrahedron). Matroid theory treats these and other abstract geometric configurations. We will use concrete examples from lattice paths to explore some basic ideas in matroid theory and some of the many intriguing problems in this field.
Posted March 21, 2019
Last modified March 2, 2021
Undergraduate Student Colloquium
3:30 pm – 4:30 pm Lockett 9
Gilles Francfort, Université Paris XIII and New York University
Spring Brake
I wish to demonstrate that minimization is a natural notion when dealing with even simple mechanical systems. The talk will revolve mainly around a simple spring brake combination which will in turn illustrate how the search for minimizers tells us things are never as simple as first thought. All that will be needed for a correct understanding of the material are basic notions of convexity, continuity as well as some familiarity with integration by parts.
Refreshments will be served at 3:00PM in the Keisler lounge.
Posted April 10, 2019
Undergraduate Student Colloquium
10:30 am – 11:30 am Lockett 113
Peter Jorgensen, Newcastle University
Knots
Abstract: Knots are everyday objects, but they are also studied in mathematics. They were originally envisaged as models for atoms by Lord Kelvin, and have been studied by increasingly sophisticated mathematical methods for more than 100 years.
Two knots are considered to be "the same" if one can be manipulated to give the other without breaking the string. The natural question of whether two given knots are the same turns out to be highly non-trivial; indeed, this is the central question of Knot Theory.
The talk is a walk through some aspects of this fascinating area of pure mathematics.
Refreshments will be served in the Keisler Lounge from 10 to 10:30 am.
Posted October 15, 2019
Last modified October 28, 2019
Undergraduate Student Colloquium
9:30 am – 10:30 am Allen Hall 123
Keith Conrad, University of Connecticut
Applications of Divergence of the Harmonic Series
Abstract: The harmonic series is the sum of all reciprocals $1 + 1/2 + 1/3 + 1/4 + ...$, and a famous counterintuitive result in calculus is that the harmonic series diverges even though its general term tends to 0. This role for the harmonic series is often the only way students see the harmonic series appear in math classes. However, the divergence of the harmonic series turns out to have applications to topics in math besides calculus and to events in your daily experience. By the end of this talk you will see several reasons that the divergence of the harmonic series should be intuitively reasonable.
Posted October 15, 2019
Last modified November 14, 2019
Undergraduate Student Colloquium
1:30 pm – 2:20 pm Lockett Hall 241
Steven Leth, University of Northern Colorado
Fixed Points of Continuous Functions in the plane
Abstract: If we crumple up a map of Colorado, then place that crumpled map on top of an identical map, some point on the top map is directly over the same location on the lower map. This might not be true if we use a map of Louisiana or Michigan, which have "disconnected" portions. Also, we must be careful to not tear the map while we are crumpling it. This is an example of a consequence of the famous Brouwer Fixed Point Theorem. We will examine this beautiful mathematical result, and demonstrate how it follows from a simple combinatorial theorem about coloring vertices of triangles. The extent to which the Fixed Point Theorem can be generalized is still unsolved, and we will briefly discuss that as well.
Posted October 24, 2022
Undergraduate Student Colloquium
3:30 pm – 4:30 pm Lockett 232
Hal Schenck, Auburn University
Combinatorics and Commutative Algebra
This talk will give an overview of the spectacular success of algebraic methods in studying problems in discrete geometry and combinatorics. First we'll discuss the face vector (number of vertices, edges, etc.) of a convex polytope and recall Euler's famous formula for polytopes of dimension 3. Then we'll discuss graded rings, focusing on polynomial rings and quotients. Associated to a simplicial polytope P (every face is "like" a triangle) is a graded ring called the Stanley-Reisner ring, which "remembers" everything about P, and gives a beautiful algebra/combinatorics dictionary. I will sketch Stanley's solution to a famous conjecture using this machinery, and also touch on connections between P and objects from algebraic geometry (toric varieties). No prior knowledge of any of the terms above will be assumed or needed for the talk.