Seminars archive

In 1946 Erdos asked to determine or estimate the minimum number of distinct distances determined by an n-element planar point set V. He showed that a square integer lattice determines \Theta(n/\sqrt{log n}) distinct distances, and conjectured that any n-element point set determines at least n^{1−o(1)} distinct distances. In 2010-2015, Guth and Katz answered Erdos’s question by proving that any n-element planar point set determines at least \Omega(n/log n) distinct distances. In this talk, we consider a variant of this problem by Erdos and Gyarfas. For integers n, p, q with p \geq q \geq 2, determine the minimum number D(n,p,q) of distinct distances determined by a planar n-element point set V with the property that any p points from V determine at least q distinct distance. In a recent paper, Fox, Pach and Suk prove that when q = {p \choose 2} - p + 6, D(n,p,q) is always at least n^{8/7 - o(1)}. We will discuss a recent improvement of their result and some new bounds for a related (graph theoretic) Ramsey problem of Erdos and Shelah which arise. This is joint work with Adam Sheffer.

In this thesis, the author proves theorems on the existence and mapping properties of tropical stacks that arise from result concerning toric Artin stacks. The author also provides a generalization, using the same ideas from the toric Artin stacks, of recent work involving analytic stacks and their tropicalizations. The author also proves results on the notion of a tropical jet space.

The data, information, knowledge and wisdom (DIKW) model has been widely used in data science fields to generate a comprehensive view of each domain. It provides a hierarchical representation of the understanding of the domain knowledge; the DIKW model can reveal insights in systems biology by integrating different types of –omics data to form a comprehensive understanding.\\ \\The foundation of systems biology is mining genomics data with machine learning. As the use of high-throughput, next-generation sequencing (NGS) applications grows, research in genomics enters the “big data” era. NGS applications can be divided into two major categories, short-read and long-read techniques, which are based on the principle differences in generating “reads”. A “read” is the fundamental element of genomic information. Short-read applications have been widely applied in several fields of genomics research, while long-read applications just came to market in 2011. Long-read applications have shown the potential to handle several areas of genomic questions. However, obtaining a well-defined genome still has a number of challenges in malaria systems biology research, and these challenges block researchers’ understanding the mechanism of the malaria disease progression.\\ \\To tackle these challenges, we built a novel long-read NGS pipeline with third party modules and modified them to solve complicated Plasmodium genome assembly questions. These techniques provided a solution where traditional, short-read technologies could not because of the Plasmodium genome’s highly repetitive nature. We also implemented infrastructure to solve data management difficulties and developed several novel and robust pipelines to process and analyze the data. We host this pipeline along with other third party applications for data quality control, generic data visualization and data management tools. Our pipeline is also scalable and flexible to combine different technologies (long reads and short reads) to assemble the Plasmodium genome and conduct downstream annotations.\\ \\This dissertation describes an overview of –omics research in the big data era and reveals the possibility of applying DIKW models through mining genomics data. A detailed discussion on how to apply our platform to solve questions, including multiple Plasmodium genome assemblies and annotations, and an initial discussion of applying machine learning approaches in a host-pathogen transcriptome analysis and its data mining applications are also provided.

Skyline, also known as Maxima in computational geometry or Pareto in business management field, is important for many applications involving multi-criteria decision making. The skyline of a set of multi-dimensional data points consists of the points for which no other point exists that is better in at least one dimension and at least as good in every other dimension. Although skyline computation and queries have been extensively studied in both computational geometry and database communities, there are still many challenges need to be fixed, especially in this big data ear. In this dissertation, I present several efficient and adaptive skyline computation algorithms. First, I show a faster output-sensitive skyline computation algorithm which is the state-of-the-art algorithm from the theoretical aspect. Second, traditional skyline computation is inadequate to answer queries that need to analyze not only individual points but also groups of points. To address this gap, I adapt the original skyline definition to the novel group-based skyline (G-Skyline), which represents Pareto optimal groups that are not dominated by other groups. Third, to facilitate skyline queries, I propose a novel concept Skyline Diagram, which given a set of points, partitions the plane into a set of regions, referred to as skyline polyominos. Similar to kth-order Voronoi diagram commonly used to facilitate k nearest neighbor (kNN) queries, any query points in the same skyline polyomino have the same skyline query results.

The talk is motivated by a practical issue: walk-based centrality measures regard all walks of the same length as being equally important, whereas it is intuitively reasonable to rule out certain classes of walk. We focus here on non-backtracking walks. The theory of zeta functions provides an expression for the generating function of non-backtracking walk counts on a directed network. This expression can be used to produce a centrality measure that eliminates backtracking walks at no cost. The new centrality measure may be interpreted as standard Katz on a modified network, where self loops are added, and where non-reciprocated edges are augmented with negative weights. We also give a multilayer interpretation of the new centrality measure, where (negatively) weighted walks between layers compensate for backtracking walks on the only non-empty layer. We further show that the radius of convergence of the generating function is determined by the spectrum of a three-by-three block matrix involving the original adjacency matrix. This gives a means to choose appropriate values of the attenuation parameter and, in particular, we show that we obtain a larger range of choices for the attenuation parameter than that obtained for standard Katz. By studying the effect of pruning operations on the network (i.e., removing nodes), we show that there is potential for the non-backtracking centrality to be computed more cheaply than Katz for appropriate network structures. Studying the limit as the attenuation parameter approaches its upper bound allows us to propose an eigenvector-based non-backtracking centrality measure in this directed network setting. We illustrate the centrality measure on a synthetic network, where it is shown to eliminate a localization effect present in standard Katz centrality. We also give results for real networks. Finally, we discuss some preliminary results on the non-backtracking version of the total communicability and of some alternating walk-based centrality measures.\\ \\This talk is based on joint work with Prof. Peter Grindrod (University of Oxford, UK), Prof. Desmond J. Higham (University of Strathclyde, UK), and Dr. Vanni Noferini (University of Essex, UK).

The excedance algebra, given by the noncommutative quotient Z<a,b>/(ba-a-b-ab), is motivated by a recurrence for a permutation statistic. We examine this algebra and a related matrix construction known as the excedance matrix. We then consider properties of the box polynomials, which arise from applying the finite difference operator to monomials and whose coefficients are the entries of the rightmost column of the excedance matrix. Evaluating these polynomials yields a variety of identities involving set partition enumeration. We extend these identities using restricted growth words and a new operator called the fast Fourier operator. This talk is based on joint work with Richard Ehrenborg and Dustin Hedmark.

An old open problem in Euclidean Ramsey theory asks if the points in $E^3$ are coloured in red and blue, does there exist either a red pair of points at unit distance or six collinear blue points separated by unit distances? After a short survey of related problems, I will give a positive answer to this problem and outline the proof. This talk is based on a joint work with Sergei Tsaturian.

mall, low-cost, low-power sensing devices are transforming science and society, making it possible to transform the deployment of new applications and allow the collection and analysis of data far beyond the scale of what was previously possible. A commercially available embedded platform, such as Arduino, helps developers to prototype embedded systems easier, faster, and in a fun way. One of Arduino's fun projects is autonomous car, where a racing car must be able to move without a remote control. The car should be able to move forward, avoid any obstacles, stop and reverse if necessary, without any user input apart from turning it on. In this talk, we are going to learn how to control the electronic speed control and the steering servo of the car using an Arduino.

Rachel Pries (4pm) \\ Title: Galois action on homology of Fermat curves \\ Abstract: We prove a result about the Galois module structure of the Fermat curve using commutative algebra, number theory, and algebraic topology. Specifically, we extend work of Anderson about the action of the absolute Galois group of a cyclotomic field on a relative homology group of the Fermat curve. By finding explicit formulae for this action, we determine the maps between several Galois cohomology groups which arise in connection with obstructions for rational points on the generalized Jacobian. Heisenberg extensions play a key role in the result. This is joint work with R. Davis, V. Stojanoska, and K. Wickelgren. \\ Gopal Prasad (5:15pm) \\ Title: Weakly commensurable Zariski-dense subgroups of semi-simple groups and isospectral locally symmetric spaces \\ Abstract. I will discuss the notion of weak commensurability of Zariski-dense subgroups of semi-simple groups. This notion was introduced in my joint work with Andrei Rapinchuk (Publ. Math. IHES 109(2009), 113-184), where we determined when two Zariski-dense S-arithmetic subgroups of absolutely almost simple algebraic groups over a field of characteristic zero can be weakly commensurable. These results enabled us to prove that in many situations isospectral locally symmetric spaces of simple real algebraic groups are necessarily commensurable. This settled the famous question "Can one hear the shape of a drum?" of Mark Kac for these spaces. The arguments use algebraic and transcendental number theory.