# Graduate Student Seminar

The Graduate Student Seminar meets Fridays from 1:30–2:30pm in room MB 124. The seminar is organized by Uwe Kaiser (ukaiser@boisestate.edu). Everybody is invited to join!

**September 6**: Zach Teitler: Waring rank of a homogeneous polynomial

**Abstract**: If F is a polynomial of degree d, the Waring rank of F is the

least number of terms in an expression for F as a linear combination of dth

powers. For example, F = xy can be written as a linear combination of squares: (1/4)(x+y)^2+(1/4)(x-y)^2. The problems of finding the rank of a given polynomial and studying rank in general have applications throughout statistics, engineering, and the sciences, such as in signal processing and computational complexity. These problems have been studied using algebraic geometry. For example, J.J. Sylvester gave a lower bound for rank in terms of “catalecticant” matrices in the mid-19th century.

I will give an introduction to Waring rank, including some recent results and still open questions, applications of Waring rank to engineering and complexity theory, and also including an introduction to algebraic geometry.

**September 13**: Donna Calhoun and Michal Kopera: Adaptive mesh refinement for geophysical applications

**Abstract**: Oceanographers and atmospheric scientists need large scale computational models for understand weather and climate processes in the atmosphere and ocean. Emergency planners rely on computational modeling to understand and predict the impact of natural disasters such as tsunamis, storm surge, hurricanes, volcanic eruptions, forest fires and flooding events on affected communities.

A key feature of geophysical applications is that the phenomena of interest take place over large spatial domains and over long durations. As a result, a key requirement of these large scale models is being able to use computational resources (processors, memory, hard disk space) efficiently. One approach to doing this is to use adaptive mesh methods. These methods are able to generate computational meshes that adapt locally to the features of interest. For example, using adaptive methods, we can focus computational resources in areas where the ocean activity is most detailed (near coast lines, for example) or where a ash from a volcanic eruption is most concentrated.

Professors Donna Calhoun and Michal Kopera both develop geophysical models using adaptive mesh methods. A common feature in their research is the use of quadtree/octree code p4est (www.p4est.org). Michal will discuss his computational ocean model NUMO, and Donna will discuss her code ForestClaw for modeling natural hazards modeling.

Donna and Michal will also interesting problems for potential graduate students to work on.

**September 27**: Randall Holmes: Implementing Zermelo’s axiomatics and proof of the well-ordering theorem in Lestrade, a dependent types theorem prover

**Abstract**: Over a few weeks in the summer, we implemented much of the content of Zermelo’s important set theory papers of 1908, including the original axiomatization of Zermelo set theory, the precursor of our current set theory ZFC, and Zermelo’s proof of the Well-Ordering Theorem, under our dependent type theory based theorem proving system Lestrade. We will give an overview of this work.

**October 4**: Marion Scheepers: From genome maintenance to mathematics

**Abstract**: Sorting is ubiquitous. We briefly discuss a permutation sorting operation that is active in the genome maintenance algorithms of ciliates. Not all permutations are sortable by this sorting operation. Analysis of the class of non-sortable permutations suggests that this permutation sorting scenario is just a glimpse of a more pervasive mathematical structure. In this talk we feature some key findings and some directions for research.

**October 11**: Jens Harlander: Combinatorial Topology and Group Theory

**Abstract**: In my talk I will introduce the mathematical area I have been working in over the years with other mathematicians and mathematics graduate students. In combinatorial topology we impose a notion of finiteness onto space which allows arguments by counting. For example a graph drawn on a piece of paper consists of uncountably many points. But it will (most likely) have a finite number of vertices v and a finite number of edges e. If you count v-e+r, where r is the number or regions you see on the paper, you will always get 2. A combinatorial topologist would phrase this as “the Euler characteristic of the 2-sphere is 2”. The important thing is not the graph, but the fact that it is drawn on a 2-sphere (a piece of paper with the boundary identified to a point), and not, say, on the boundary of a torus. In combinatorial topology we use finite structures to assign algebraic objects to spaces, such as numbers or groups. This translation into algebra is at the heart of the matter of the subject.

**October 18**: Allison Arnold-Roksandich: Fermat’s Last Theorem

**Abstract**: In 1637, Pierre de Fermat proposed that a^n+b^n=c^n had no positive integer solutions for a, b, and c when n is greater than 2. It is also claimed that he said to have a “truly marvelous proof” of this statement but the margins [of the document where he made the statement] were too small. This proof was never published, and instead Fermat’s Last Theorem became a problem that mathematicians would attempt to solve for centuries. In 1995, Andrew Wiles formally published the first successful proof of Fermat’s Last Theorem. This talk aims to look at the history of this problem, in particular some of the early attempts made to prove the theorem, and the foundations of Wiles’s proof.

**October 25**: Leming Qu: Skew-t Copula-Based Semiparametric Markov Chains

**Abstract**: Without specifying a time series structure, a multivariate Markov processes in discrete time is modelled by a multivariate Markov family of dependence function and the one-dimensional flows of marginal distributions. Such models tackle simultaneously temporal dependence and contemporaneous dependence between time series. A specific parametric form of stationary copula, namely skew-t copula, is assumed. Skew-t copulas are capable of modeling asymmetry, skewness, and heavy tails. An empirical study with daily returns for three stock indices shows that the skew-t copula Markov model provides a better fit than the t-copula Markov model, and the skew-t copula model without Markov property.

**November 1**: Jodi Mead: Inverse methods and imaging

**Abstract**: Abstract: Inverse problems arise when we use data to create a mathematical model that can be used to predict or reach conclusions about a system of interest. For example, inverse problems arise when imaging the subsurface of the earth or when reconstructing a photographic image. Inverse problems are ill-posed either because there is no solution, infinitely many solutions or small changes in the data produce dramatically different results. These issues can be overcome by adding regularization or additional information about the mathematical model. I will discuss various ways of adding additional information to an inverse problem that make it well-posed and present examples in imaging and geophysics.

**November 8**: Liljana Babinkostova: The Next Generation of Cryptography

**Abstract**: Modern public key protocols, such as RSA and elliptic curve cryptography (ECC), will be rendered insecure by Shor’s algorithm when large-scale quantum computers are built. Grover’s algorithm will reduce the security of symmetric key cryptography and the complexity of finding a pre-image of existing hash functions. Moreover, the majority of current cryptographic algorithms were designed for desktop/server environments, many of these algorithms do not fit into the arena of devices with constrained resources.

The National Institute of Standards and Technology (NIST) has initiated a process to solicit, evaluate, and standardize: (1) lightweight cryptographic algorithms that are suitable for use in constrained environments and (2) quantum-resistant public-key cryptographic algorithms. In this talk we present recent developments in these areas of cryptography and state some open problems.

**November 15**: Barbara Zubik-Kowal: Error bounds for nonlinear systems of integro-differential equations

**Abstract**: In this talk, we introduce theorems on error bounds for approximate solutions of integro-differential equations applied in neuroscience. We present error bounds derived under general assumptions as well as those derived by making use of the properties of the kernels, which are applied in the given systems. Finally, we conclude that the approximate solutions converge rapidly to the exact solution on arbitrarily long time intervals.

**November 22**: Jaechoul Lee: Trend assessment for climatological time series

**Abstract**: This talk presents methods to estimate a long-term trend in climatological time series data. The methods use statistical models that allow for important features in climatological data such as seasonality, support set, autocorrelation, and mean level shift changepoints. A likelihood objective function is developed for the models and is used to estimate model parameters. Genetic algorithms are used to optimize a minimum descriptive length model selection criterion that estimates the changepoint numbers and locations. The methods are applied to United States monthly maximum and minimum temperatures and daily snow depths recorded near Warm Lake, Idaho.