Seminars archive

Following a suggestion of Erdos (1974), Alon and Yuster (2006) investigated the maximum number of orientations graphs of a given order admit if we forbid copies of a fixed tournament. We discuss the analogous problem in which certain restricted orientations of typical graphs of a given order and a given number of edges are considered.\\ \\This is joint work with M. Collares (Belo Horizonte), R. Morris (Rio de Janeiro) and G. O. Mota (Sao Paulo).

Tensors (aka multiway arrays) can be instrumental in revealing latent correlations residing in high dimensional spaces. Despite their applicability to a broad range of applications in machine learning, speech recognition, and imaging, inconsistencies between tensor and matrix algebra have been complicating their broader utility. Researchers seeking to overcome those discrepancies have introduced several different candidate extensions, each introducing unique advantages and challenges. In this talk, we review some of the common tensor definitions, discuss their limitations, and introduce our tensor product framework which permits the elegant extension of linear algebraic concepts and algorithms to tensors. Following introduction of fundamental tensor operations, we discuss in further depth tensor decompositions and in particular the tensor SVD (t-SVD) and its randomized variant, which can be computed efficiently in parallel. We present details of the t-SVD, theoretical results, and provide numerical results that show the promise of our approach for compression and analysis of operators and datasets, highlighting examples such as facial recognition and model reduction.

Elliptic modular forms are functions on the complex upper half plane that are invariant under a certain action of the special linear group with integer entries. Their history comprises close to two centuries of amazing discoveries and application: The proof of Fermat's Last Theorem is probably the most famous; The theory of theta functions is among its most frequently employed parts.\\ \\During the past decade it has been à la mode to study relaxed notions of modularity. Relevant keywords that we will discuss are mock modular forms and higher order modular forms. We have witnessed their application, equally stunning as surprising, to conformal field theory, string theory, combinatorics, and many more areas.\\ \\In this talk, we suggest a change of perspective on such generalizations. Most of the novel variants of modular forms (with one prominent exception) can be viewed as components of vector-valued modular forms. This unification draws its charm from the past and the future. On the one hand, we integrate results by Kuga and Shimura that hitherto seemed almost forgotten. On the other hand, we can point out connections, for example, between mock modular forms and so-called iterated integrals that have not yet been noticed. Experts will be pleased to have in the future a ``Petersson pairing'' for mixed mock modular forms at their disposal.\\ \\This is joint work with Michael Mertens

The Birkhoff--von Neumann decomposition expresses a doubly stochastic matrix as a convex combination of permutation matrices. This talk will be an introduction to this decomposition. We are going to see its use in solving sparse linear systems, and investigate some algorithmic and combinatorial problems associated with it. This talk contains results from joint work with Michele Benzi (Emory Univ., Atlanta), Fanny Dufosse (Inria, France), Kamer Kaya (Sabanci Univ, Turkey), and Ioannis Panagiotas (ENS Lyon, France).

The u-invariant of a field is the maximal dimension of a nonsingular anisotropic quadratic form over that field, whose order in the Witt group of the field is finite. By a classical theorem of Elman and Lam, the u-invariant of a linked field of characteristic different from 2 can be either 0,1,2,4 or 8. The analogous question in the case of characteristic 2 remained open for a long time.. We will discuss the proof of the equivalent statement in characteristic 2, recently obtained in a joint work by Andrew Dolphin and the speaker.

In order to control the spectrum of singularities of the generalized Riemann function, we need to find sharp bounds for certain kind of Gauss sums with frequencies on polynomials. This can be achieved by Weil bounds on completely irreducible algebraic curves, which lead us to prove some theorems on irreducibility of polynomials in two variables. We will prove a general theorem in this field. An example of the theorem is the absolute irreducibility of p(x)-p(y)+1, for any p(x) with integer coefficients.

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.