Speaker: Kameryn Williams (UHM)

Title: The universal algorithm, the $Sigma_1$-definable universal finite sequence, and set-theoretic potentialism

Abstract: As shown by Woodin, there is an algorithm which will computably enumerate any finite list you want, so long as you run it in the correct universe. More precisely, there is a Turing machine $p$, with the following properties: (1) Peano arithmetic proves that $p$ enumerates a finite sequence; (2) running $p$ in $mathbb N$ it enumerates the empty sequence; (3) for any finite sequence $s$ of natural numbers there is a model of arithmetic $M$ so that running $p$ in $M$ it enumerates $s$; (4) indeed, if $p$ enumerates $s$ running in $M$ and $t$ in $M$ is any finite sequence extending $s$, then there is an end-extension $N$ of $M$ so that running $p$ in $N$ it enumerates $t$. In this talk, I will discuss the universal algorithm, along with an analogue from set theory due to Hamkins, Welch, and myself, which we call the $Sigma_1$-definable universal finite sequence.

These results have applications to the philosophy of mathematics. Set-theoretic potentialism is the view that the universe of sets is never fully completed and rather we only have partial, ever widening access. This is similar to the Aristotelian view that there is no actual, completed infinite, but rather only the potential infinite. A potentialist system has a natural associated modal logic, where a statement is necessary at a world if it is true in all extensions. Using the $Sigma_1$-definable universal finite sequence we can calculate the modal validities of end-extensional set-theoretic potentialism. As I will discuss in this talk, the modal validities of this potentialist system are precisely the theory S4.

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.

Speaker: Phillip Wesolek (Weslyan U.)

Title: An invitation to totally disconnected locally compact groups

Abstract: Locally compact groups appear across mathematics; they arise as Galois groups in algebra, isometry groups in geometry, and full groups in dynamics. The study of locally compact groups splits into two cases: the connected groups and the totally disconnected groups. There is a rich and deep theory for the connected groups, which was developed over the last century. On the other hand, the study of the totally disconnected groups only seriously began in the last 30 years, and moreover, these groups today appear to admit an equally rich and deep theory. In this talk, we will begin by motivating the study of totally disconnected locally compact groups and presenting several examples. We will then discuss a natural dividing line in the theory and a fundamental decomposition theorem.