Oct

25

2014

Oct

27

2014

Given a finite set of n elements, how many possible topologies can be put on this set? While the number of finite topologies has been calculated for values up to n=18, there is no known formula (explicit or recursive) the number of topologies on a set of n elements in terms of n. In this talk we will look at the connection of this problem with open combinatorial problems in graph theory and order theory. We will also discuss a foundational theorem in this area which relates the number of total topologies on a finite set with the number of topologies satisfying the T_0 separation axiom.

Oct

27

2014

For the past 15 years I have been involved with research that seeks to understand how students can transition from more informal reasoning to more formal, mathematical reasoning. As part of this NSF funded research, we have developed curriculum to help students make this transition. In my current project our team is developing instructional materials appropriate for a first college course in linear algebra including the topics of (1) span and linear independence, (2) matrix multiplication and linear transformations, and (3) eigen theory, change of basis and diagonalization.

Oct

28

2014

For any permutations $\sigma \in S_n$ and $\pi \in S_k$, we say that $\sigma$ contains the pattern $\pi$ if $\sigma$ has a subsequence that is order isomorphic to $\pi$. In this case, we refer to $\pi$ as a pattern of length $k.$ We may endow a pattern $\pi=\pi_1\pi_2\cdots\pi_k$ with proximity restrictions, that is, requiring certain elements of the pattern to correspond to adjacent elements in the permutation $\sigma$. If the elements $\pi_i,\pi_{i+1}$ of the pattern $\pi$ are required to correspond to adjacent elements of $\sigma$, then we place an underscore beneath $\pi_i,\pi_{i+1}$.

Oct

30

2014

Progress toward the development of a safe and effective treatment for AIDS has been slow because the Human Immunodeficiency Virus has the ability to mutate its own structure. This mutation enables the virus to become resistant to previously effective drug therapies. Apart from the traditional role of preventing progression from HIV to AIDS Antiretroviral drugs have an additional clinical benefit of substantially reducing infectiousness thus making them an important strategy in the fight against HIV.