Graduate Colloquium at University of South Carolina
The graduate colloquium allows any member of the mathematics department, be it student or professor, to give talks ranging in levels from expository to expert. This provides an opportunity to bridge ideas, learn, and practice public speaking. It builds a sense of community with the graduate department and establishes valuable experience for all of its members. Regarding this semester, there will be panels for graduate course advertising, professors pitching their research to our community, and graduate students sharing their work. For location with time, please feel free to email me (see front page).
Schedule
 9/20/22

 Speaker: Gabrielle Tauscheck “An Introduction to Distance Critical Graphs”

 Abstract: Graham and Pollak discovered that the determinant of the distance matrix of any tree is given by $(n1)(2)^{n2}$ independent of the structure of the tree. The simplicity of this equation’s proof relies on the fact that every tree has at least two pendant vertices and deleting these vertices does not change the remaining entries of the distance matrix. We vary this property to instead consider graphs that possess the characteristic such that deleting a vertex does change at least one of the remaining entries of its distance matrix. We call these graphs “distance critical” and present our results so far.
 9/27/22

 Speaker: Mohamed Wafik “Construction of Polynomials with Prescribed Divisibility Conditions on the Critical Orbit”

 Abstract: Let $f_{d,c}(x)=x^d+c\in\mathbb{Q}[x]$, $d\ge 2$. We write $f_{d,c}^n$ for $\underset{\text{n times}}{\underbrace{f_{d,c}\circ f_{d,c}\circ\dots\circ f_{d,c}}}$. The critical orbit of $f_{d,c}(x)$ is the set $O_{f_{d,c}}(0):={f_{d,c}^n(0):n\geq 0}$. For a sequence ${a_n:n\geq 0}$, a primitive prime divisor for $a_k$ is a prime dividing $a_k$ but not $a_n$ for any $1\leq n < k$. A result of H. Krieger asserts that if the critical orbit $O_{f_{d,c}}(0)$ is infinite, then each element in $O_{f_{d,c}}(0)$ has at least one primitive prime divisor except possibly for 23 elements. In addition, under certain conditions, R. Jones proved that the density of primitive prime divisors appearing in any orbit of $f_{d,c}(x)$ is always 0. Inspired by the previous results, we display an upper bound on the count of primitive prime divisors of a fixed iteration $f_{d,c}^n(0)$. Fixing $n\geq 2$, we also, under certain assumptions, calculate the density of primes that can appear as primitive prime divisors of $f_{2,c}^n(0)$ for some $c\in\mathbb{Q}$. Further, we show that there is no uniform upper bound on the count of primitive prime divisors of $f_{d,c}^n(0)$ that does not depend on $c$. In particular, given $N>0$, there is $c\in\mathbb{Q}$ such that $f_{d,c}^n(0)$ has at least $N$ primitive prime divisors
 10/4/22


 10/11/22


 10/18/22


 10/25/22


 11/1/22


 11/8/22


 11/15/22


 11/22/22

 11/29/22

