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 $w$.

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.

Speaker: Amita Malik

Title: Zeros of the derivatives of the completed Riemann zeta function

Abstract:

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 $xi^{(m)}(s)$.

Title: Computing matrix eigenvalues

Speaker: Yuji Nakatsukasa, National Institute for Informatics, Japan

Abstract:

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.

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.

Speaker: Lee Altenberg (ICS UH Mānoa)

TITLE: Application of Dual Convexity to the Spectral Bound of Resolvent Positive Operators

ABSTRACT:

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.

Title: Active matter invasion of a viscous fluid and a no-flow theorem

Abstract: Suspensions of swimmers or active particles in fluids exhibit incredibly rich behavior, from organization on length scales much longer than the individual particle size to mixing flows and negative viscosities. We will discuss the dynamics of hydrodynamically interacting motile and non-motile stress-generating particles as they invade a surrounding viscous fluid, modeled by equations which couple particle motions and viscous fluid flow. Depending on the nature of their self-propulsion, colonies of swimmers can either exhibit a dramatic splay, or instead a cascade of transverse concentration instabilities, governed at small times by an equation which also describes the Saffman-Taylor instability in a Hele-Shaw cell, or Rayleigh-Taylor instability in two-dimensional flow through a porous medium. Analysis of concentrated distributions of particles matches the results of full numerical simulations. Along the way we will prove a very surprising “no-flow theorem”: particle distributions initially isotropic in orientation lose isotropy immediately but in such a way that results in no fluid flow anywhere and at any time.

Devin Murray

Title: Introduction to right-angled Artin groups