Given two arbitrary knots (tangled up strings with their ends tied together), how can we (easily) tell if they are different or not? In general, this problem is extremely difficult to answer, and has led to the development of a variety of knot invariants. In this talk, we will examine geometric invariants of knots. These are knot invariants that arise from examining the (often hyperbolic) geometry of the space surrounding a knot. Historically, these invariants have been extremely useful in helping classify knots. However, it is possible to construct large sets of knots that are geometrically similar, that is, all the knots in such a set are different, yet these knots have a number of geometric invariants in common. We will give two examples of geometrically similar sets of knots and raise some interesting questions about geometric invariants.
The relationship between embedded surfaces and their knotted boundaries has been one of the main topics of knot theory for much of the last half century. This talk focuses on a particular case, namely whether a given knot in the three-sphere can be the boundary of a Mobius band embedded in the four-ball, B^4. We will discuss a new example of a knot which does not bound a Mobius band in B^4, and describe how the d-invariant of Heegaard-Floer theory is used to obstruct this and other knots from bounding Mobius bands in B^4.
In enumerative combinatorics, the study of permutation patterns blossomed in the 1980s with the Stanley-Wilf conjecture. In this talk, I will introduce the basic concept of permutation patterns and some approachable examples as well as the final result of my doctoral research.
An abelian group is decomposable if it can be written as a direct sum of two (or more) nontrivial subgroups. Otherwise it is indecomposable. The only indecomposable torsion groups are cyclic groups of the form $\mathbb{Z}(p^n)$, where $p$ is prime (as well as Pr\"ufer groups, $\mathbb{Z}(p^\infty)$). Mathematicians have been unable to describe the class of decomposable torsion-free groups. As it turns out, this problem is analytic complete (it cannot be characterized by a first-order formula). We show this by comparing it to the problem of determining whether an infinitely branching tree in $\omega^{<\omega}$ has an infinite path.
Finite Difference (FD) schemes have been used widely in computing approximations for partial differential equations for wave propagation, as they are simple, flexible and robust. However, even for stable and accurate schemes, waves in the numerical schemes can propagate at different wave speeds than in the true medium. This phenomenon is called numerical dispersion error. Traditionally, FD schemes are designed by forcing accuracy conditions, and in spite of the advantages mentioned above, such schemes suffer from numerical dispersion errors.
Traditionally, two ways have been used for the purpose of reducing dispersion error: increasing the sampling rate and using higher order accuracy. More recently, Finkelstein and Kastner (2007, 2008) propose a unified methodology for deriving new schemes that can accommodate arbitrary requirements for reduced phase or group velocity dispersion errors, defined over any region in the frequency domain. Such schemes are based on enforcing exact phase or group velocity at certain preset wavenumbers. This method has been shown to reduce dispersion errors at large wavenumbers.
In this talk, we discuss the construction and behaviors of FD schemes designed to have reduced numerical dispersion error. We prove that the system of equations to select the coefficients in a centered FD scheme for second order wave equations with specified order of accuracy and exact phase velocity at preset wavenumbers can always be solved.
Furthermore, from the existence of such schemes, we can show that schemes which reduce the dispersion error uniformly in an interval of the frequency domain can be constructed from a Remez algorithm. In these new schemes we propose, we can also specify wavenumbers where the exact phase or group dispersion relation can be satisfied. For an incoming signal consisting of waves of different wavenumbers, our schemes can give more accurate wave propagation speeds. Furthermore, when we apply our schemes in two dimensional media, we can obtain schemes that give small dispersion error at all propagation angles.
In numerically solving linear partial differential equations, carefully formed matrices are powered up as the algorithm iterates the solution through time. Depending upon the eigenvalues, the solution either blows up to non-numerical values, stabilizes with bounded oscillations, or stabilizes free of oscillations. This talk presents comparisons of eigenvalue conditions through extended von Neumann stability analysis of standard numerical methods for foundational partial differential equations. Known lower eigenvalue bounds for stability are not optimal, so computational testing is necessary. We will compare several implementations to automate detection of solution behavior for testing theoretical conjectures. Along the way, we will pause to enjoy some surprisingly beautiful figures resulting from such analysis and computation.