Graduate Student Advisory Committee (GSAC) Colloquium Schedule:

Welcome back!

First week of classes.

Special Positions of Frameworks

A framework is given by a collection of rigid bodies in space and set of geometric constraints among these rigid bodies; rigid geometric constraint systems arise in engineering, computer aided design (CAD), and computational biology. In this talk we will be interested in frameworks which, generically, admit no non-trivial motions and have no redundant constraints. We study the configuration space of the underlying structure graph of these minimally rigid structures. We outline a combinatorial approach for analyzing the zeros of a polynomial, which arises as the determinant of a rigidity matrix associated to the structure graph. The zeros of this polynomial correspond to “special positions” of the framework.

The Princess Problem and Extensions

The princess problem (also known as the secretary problem) is a famous problem in optimal stopping theory. Suppose that a princess is visited by n suitors. What strategy should she use to pick the best suitor, if she cannot bring back suitors she has already rejected? In this talk, we will derive the optimal strategy for the princess problem, then discuss variants of the problem and extensions.

NOTE: Special date and time: Thursday Jan 28, 4:00–5:00 pm, JWB 335

Mathematics in Data Science

Many graduate students make the decision not to continue along the academic track, but the transition from academia to industry can be stressful and scary. I want to share my experience going from grad school to being a data scientist, and some tips for making the transition to industry as easy as possible if you choose that route. I will also tell you about a neat bit of mathematics called the "kernel trick" that allows some machine learning classification algorithms to model data with complex, non-linear decision boundaries will little extra computational cost.

NOTE: Special room: LCB 219 (Tuesday Feb 2, 4:35–5:35 pm)

References:

• Brian Mann's slides: (link).
• Slides were written in markdown and converted to beamer using pandoc.

Topology of syntax

Topological methods can be used to investigate big sets of data, and shed light on the structures and relations between these data. We present one of these methods, persistent homology. It allows to detect “holes” in the distribution of the data. We'll focus on one example, and consider the Indo-European language tree. A study of the main characteristic of Indo-European languages provides a data set which shows nontrivial persistent homology of hard interpretation. We will explore the topology of the language tree and compare it with the one of the data set, trying to understand its linguistic meaning.

NOTE: Special date and time: Thursday Feb 11, 4:00–5:00 pm, JWB 335

Wave-Ice Interaction in the Marginal Ice Zone

The Marginal Ice Zone (MIZ) in the Southern Ocean around the continent of Antarctica can be defined as the part of the ice cover which is close enough to the open ocean boundary to be affected by its presence. One way in which the open ocean affects the MIZ is through wave-ice interaction. Waves coming in from the turbulent southern ocean break up the ice. The ice floes in turn damp and limit the wavelengths that can exist in the MIZ. This wave-ice interaction is important for determining the thickness, floe sizes and extent of ice in the MIZ, all of which are important factors in the interaction of the ice cover with the global climate system. In this talk I will present a model for gravity wave propagation in the MIZ in which the ice is viewed as a composite with an effective visco-elasticity sitting atop an inviscid ocean. The condition for a solution to the resulting equations yields a dispersion relation which depends on the effective parameter. I will also highlight possible applications of this model to better understanding the dynamics of the MIZ.

NOTE: Special room: LCB 219 (Tuesday Feb 16, 4:35–5:35 pm)

Diffusion on Fractals

Diffusion on fractals has been used to model anomalous diffusion, a behavior which is difficult to investigate despite having been observed in many physical systems. We will present use of fractals in describing anomalous diffusion in geometrically complicated environments. A treatment of diffusion on static and random fractals, Sierpinski Carpets, will be developed and illustrated with numerical results.

Community Structure in Complex Networks

In recent years, network theory has been used extensively to study systems in a wide range of fields including sociology, biology, and physics. In this talk we will be interested in networks with communities, groups of vertices that are relatively densely connected to each other but sparsely connected to other dense groups in the network. First we will discuss techniques for detecting communities in complex networks then we will use these techniques to study networks arising in various scientific fields.

Spectral theory of the Laplace-Beltrami operator

In this talk we will discuss interactions between differential equations, algebra, and geometry. We will use representation theoretic results describing actions of groups on function spaces to analyze the spectrum of certain differential operators. This line of research illuminates several deep connections between solutions of differential equations and symmetries of geometric objects, and has been utilized in some of the greatest mathematical achievements of the twentieth century.

Spring break!

Decay of Correlations: Probabilistic Methods in Dynamics

A dynamical system is a deterministic rule describing a change over time. These systems arise naturally in both pure and applied contexts. Despite the lack of randomness, there is a surprising correspondence between dynamics and probability. However, many theorems from probability can only be proven for strongly chaotic systems. One measure of chaos in a system is given by decay of correlations. We will discuss how rapid decay of correlations can be used to prove a dynamical central limit theorem.

References

• Slides for this talk are online (link)

Polynomial Dynamical Systems over Finite Fields

Although continuous models are ubiquitous and invaluable throughout applied math (particularly in mathematical biology), they suffer from a number of limitations. In this talk, I will present an alternative, discrete time and discrete state modeling framework called polynomial dynamical systems (PDS) over finite fields. Rather than classical dynamical systems theory being the tool for analysis of these objects, we will instead utilize algebraic tools for understanding their behavior. For instance, the attractors of the system, a feature of interest in both continuous and discrete systems, correspond to a specific variety of an ideal that can be computed directly (and efficiently) using Gröbner bases. If time permits, we'll do some live Macaulay2 coding and discuss some results in special cases.

References

• Slides for this talk are online (link)

Uncertainty Quantification and the Kalman Filter

In this talk I will introduce some fundamental concepts and methods of uncertainty quantification and their applications, with an emphasis on the Kalman filter. The Kalman filter is a real-time data assimilation algorithm. At every time step it combines model prophesy with the latest observational data, based on their respective estimated uncertainty, to produce a more accurate prediction. Applications of Kalman filters abound, from virtual reality, to weather forecasting, to modeling the central nervous system's control of movement.

“Failing” to Learn

During the first few years of graduate school, one often becomes comfortable solving difficult problems in a matter of weeks. This paradigm usually changes significantly when one's focus shifts toward new research. Successful research is generally preceded by a large amount of failed attempts and it is uncommon to start a new project and proceed directly to a conclusion without any setbacks. In this talk I will give a broad overview of some of the research I have done, but in the context of my failed attempts and setbacks, and the ways I overcame them. I will discuss some applied math research topics including imaging with waves, polycrystalline materials modeling, and general PDE theory, as well as some of the tools used in these fields. However, this talk assumes only basic knowledge of vector spaces and multivariable calculus, primarily gradients and integrals.

Elliptic and Hyperbolic Trigonometry

Classical trigonometry is a subject well-known to many, but there are other spaces on which one can consider the relation between the sides and angles of a triangle. In this talk I will focus on elliptic and hyperbolic trigonometry, i.e. trigonometry on a sphere and a negatively curved space respectively. There are some remarkable similarities between these two spaces and their trigonometry, which I will try to explain.

Organizational Meeting

We will organize the GSAC subcommittees and establish contacts for prospective and incoming students. Please attend!