Undergrad Seminar: Gideon Zamba @ 402
Jan 17 @ 3:00 pm – 4:00 pm

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.

Colloquium: Ian Marquette (U. of Queensland)
Jan 18 @ 3:30 pm – 4:30 pm

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.

Kameryn Williams: Logic seminar @ Keller 313
Jan 24 @ 2:30 pm – 3:20 pm

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 $w$.

Colloquium: Phillip Wesolek (Wesleyan U. )
Jan 25 @ 3:30 pm – 4:30 pm

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.

Number Theory Seminar @ Keller 301
Jan 31 @ 3:00 pm – 3:50 pm

Speaker: Amita Malik
Title: Zeros of the derivatives of the completed Riemann zeta function

For the completed Riemann zeta function $xi(s)$, it is known that the Riemann Hypothesis for $xi(s)$ implies the Riemann hypothesis for higher order derivatives $xi^{(m)}(s)$ where $m$ is any positive integer. In this talk, we discuss the distribution of the fractional parts of the sequence $(alpha gamma_m)$ where $alpha$ is any fixed non-zero real number and $gamma_m$ runs over imaginary parts of zeros of $xi^{(m)}(s)$. This involves obtaining horizontal distribution of zeros such as zero density estimate and explicit formula type results for the zeros of $xi^{(m)}(s)$.

Undergraduate Seminar:Yuji Nakatsukasa, National Institute for Informatics, Japan @ 402
Jan 31 @ 3:00 pm – 4:00 pm

Title: Computing matrix eigenvalues

Speaker: Yuji Nakatsukasa, National Institute for Informatics, Japan


The numerical linear algebra community solves two main problems: linear
systems, and eigenvalue problems. They are both vastly important; it
would not be too far-fetched to say that most (continuous) problems in
scientific computing eventually boil down to one or both of these.

This talk focuses on eigenvalue problems. I will first describe some of
its applications, such as Google’s PageRank, PCA, finding zeros and
poles of functions, and nonlinear and global optimization. I will then turn to
algorithms for computing eigenvalues, namely the classical QR
algorithm—which is still the basis for state-of-the-art. I will
emphasize that the underlying mathematics is (together with the power
method and numerical stability analysis) rational approximation theory.

Colloquium: Pamela Harris (Williams) @ Keller 401
Feb 1 @ 3:30 pm – 4:30 pm

Speaker: Pamela Harris (Williams College)

Title: Kostant’s partition function

Abstract: In this talk we introduce Kostant’s partition function which counts the number of ways to represent a particular weight (vector) as a nonnegative integral sum of positive roots of a Lie algebra (a finite set of vectors). We provide two fundamental uses for this function. The first is associated with the computation of weight multiplicities in finite-dimensional irreducible representations of classical Lie algebras and the second is in the computation of volumes of flow polytopes. We provide some recent results in the representation theory setting and state a direction of ongoing research related to the computation of the volume of a new flow polytope associated to a Caracol diagram.

Colloquium: Lee Altenberg (ICS UH Mānoa)
Feb 15 @ 3:30 pm – 4:30 pm

Speaker: Lee Altenberg (ICS UH Mānoa)
TITLE: Application of Dual Convexity to the Spectral Bound of Resolvent Positive Operators

Donsker and Varadhan (1975) developed a variational expression for the spectral bound of generators of strongly continuous positive semigroups, which Karlin (1981) applied to finite matrices to show that the spectral radius of product [(1-m) I + m P]D decreases monotonically in m, where P is a stochastic matrix and D a positive diagonal matrix. This result has key applications in evolutionary and population dynamics. Simultaneously, Cohen (1981) showed that the spectral bound of essentially nonnegative matrices is convex in the diagonal elements. Kato (1982) extended Cohen’s result to resolvent positive operators. A “dual convexity” lemma shows that Cohen’s and Karlin’s results are actually equivalent, and via Kato’s theorem, allows extension of Karlin’s theorem to all resolvent positive operators, showing that d/dm s(L m + V) <= s(L), where s is the spectral bound, L is a resolvent positive operator, and V is an operator of multiplication. The motivation behind this work is that it unifies many results in reaction diffusion theory and shows generally that increased diffusion in the presence of local variation in decay or growth rates will decrease asymptotic growth rates.