Title: TBA

Title: Computing the Witten-Reshetikhin-Turaev Invariant of 3-Manifolds

Applied Mathematics in Action through Biostatistics

Gideon K. D. Zamba, PhD.

Professor of Biostatistics

Professor of Radiology and Nuclear Medicine

The University of Iowa

Applied mathematics is a field of constant adaptability to the world’s contingencies. Such

adaptability requires a solid training and a keen understanding of theoretical and pure

mathematical thinking—as the activity of applied thinking is vitally connected to research

in pure mathematics. One such applied mathematical field is the field of statistics. As the

world continues to rely more on data for inference and decision making, statistics and

associated data-driven fields have gained an increased recognition. The purpose of this talk

is to educate the audience about the field of statistics, about statistical involvements, and

provide examples of settings where statistical theory finds an application and where real-

world applications call for new statistical developments. The presentation further provides

some general guidance on the mathematical and computational skills needed for a

successful graduate work in Statistics or Biostatistics.

Title: Higher order superintegrability, Painlevé transcendents and representations of polynomial algebras

Abstract: I will review results on classiﬁcation of quantum superintegrable systems on two-dimensional Euclidean space allowing separation of variables in Cartesian coordinates and possessing an extra integral of third or fourth order. The exotic quantum potential satisfy a nonlinear ODE and have been shown to exhibit the Painlevé property. I will also present diﬀerent constructions of higher order superintegrable Hamiltonians involving Painlev´e transcendents using four types of building blocks which consist of 1D Hamiltonians allowing operators of the type Abelian, Heisenberg, Conformal or Ladder. Their integrals generate ﬁnitely generated polynomial algebras and representations can be exploited to calculate the energy spectrum. I will point out that for certain cases associated with exceptional orthogonal polynomials, these algebraic structures do not allow to calculate the full spectrum and degeneracies. I will describe how other sets of integrals can be build and used to provide a complete solution.

Title: Amalgamating generic reals, a surgical approach

Location: Keller Hall 313

Speaker: Kameryn Williams, UHM

The material in this talk is an adaptation of joint work with Miha Habič, Joel David Hamkins, Lukas Daniel Klausner, and Jonathan Verner, transforming set theoretic results into a computability theoretic context.

Let $\mathcal D$ be the collection of dense subsets of the full binary tree coming from a fixed countable Turing ideal. In this talk we are interested in properties of $\mathcal D$-generic reals, those reals $x$ so that every $D \in \mathcal D$ is met by an initial segment of $x$. To be more specific the main question is the following. Fix a real $z$ which cannot be computed by any $\mathcal D$-generic. Can we craft a family of $\mathcal D$-generic reals so that we have precise control over which subfamilies of generic reals together compute $z$?

I will illustrate a specific of this phenomenon as a warm up. I will show that given any $\mathcal D$-generic $x$ there is another $\mathcal D$-generic $y$ so that $x \oplus y$ can compute $z$. That is, neither $x$ nor $y$ can compute $z$ on their own, but together they can.

The main result for the talk then gives a uniform affirmative answer for finite families. Namely, I will show that for any finite set $I = \{0, \ldots, n-1\}$ there are mutual $\mathcal D$-generic reals $x_0, \ldots, x_{n-1}$ which can be surgically modified to witness any desired pattern for computing $z$. More formally, there is a real $y$ so that given any $\mathcal A \subseteq \mathcal P(I)$ which is closed under superset and contains no singletons, that there is a single real $w_\mathcal{A}$ so that the family of grafts $x_k \wr_y w_\mathcal{A}$ for $k \in A \subseteq I$ can compute $z$ if and only if $A \in \mathcal A$. Here, $x \wr_y w$ is a surgical modification of $x$, using $y$ to guide where to replace bits from $x$ with those from $w$.