Discrete Geometry & Combinatorics Seminar: Spring 2014
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Discrete Versions of Continuous Mathematics
| April 9
- One of the more interesting ways to develop discrete mathematics is to do so under the influence of a continuous analog. In this talk I will give some examples of this idea with particular attention on the additive and multiplicative versions of discrete calculus.
The Unit Distance Graph Problem and the Axiom of Choice
| April 16
- Here's the unit distance graph problem:
"What is the smallest number of colors needed to color the plane, so that no two points of the same color are distance 1 apart?"
On one hand, it is not difficult to get some rudimentary bounds on this problem: in the first few minutes of our talk, we will show that we need at least four colors and at most seven. The surprising thing about this problem is that these simple bounds are the best currently known for this problem, despite over a half-century of attacks from some of the best minds in combinatorics!
In this talk, we will discuss some of the history behind these attempts. In particular, we will discuss results of Shelah and Soifer that suggest that the axiom of choice (somehow) plays a key role in answering this problem.
Cryptography in the Braid Group
| April 30
- Since the introduction of algorithms such as RSA and the Diffie-Hellman key exchange in the 1970s, modern cryptography has been dominated by the exploration of computationally difficult problems. The most popular of these subjects - prime factorization, discrete logarithms, and elliptic curves - originate from number theory and thus reside in abelian groups. In the hopes of developing more robust options, several non-abelian cryptosystems have been proposed, with focus on the braid group.
In this talk, we will discuss some existing braid-based algorithms, their weaknesses, and directions for future research. We will not assume any background in either cryptography or braids.
| June 4
- Matroids and oriented matroids arise as generalizations of certain combinatorial, algebraic, and geometric structures. In the talk, we will see various ways to describe (oriented) matroids.