Rational Parking Functions and LLT Polynomials

We prove that the combinatorial side of the "Rational Shuffle Conjecture" provides a Schur-positive symmetric polynomial. Furthermore, we prove that the contribution of a given rational Dyck path can be computed as a certain skew LLT polynomial, thus generalizing the result of Haglund, Haiman, Loehr, Remmel and Ulyanov. The corresponding skew diagram is described explicitly in terms of a certain (m,n)-core.Comment: 14 pages, 8 figure

Infinitesimal change of stable basis

The purpose of this note is to study the Maulik–Okounkov K-theoretic stable basis for the Hilbert scheme of points on the plane, which depends on a “slope” m∈ R. When m=ab is rational, we study the change of stable matrix from slope m- ε to m+ ε for small ε> 0 , and conjecture that it is related to the Leclerc–Thibon conjugation in the q-Fock space for Uqgl^ b. This is part of a wide framework of connections involving derived categories of quantized Hilbert schemes, modules for rational Cherednik algebras and Hecke algebras at roots of unity

A combinatorial formula for LLT cumulants of melting lollipops in terms of spanning trees

We prove a combinatorial formula for LLT cumulants of melting lollipops as a positive combination of LLT polynomials indexed by spanning trees. The result gives an affirmative answer to a general positivity question for this class of unicellular LLT cumulants, and gives an independent proof of their Schur-positivity. In the special case of the complete graph, we also express the formula in terms of parking functions.Comment: An extended abstract of this work with fewer results and a different title is available at arXiv:2011.15080v

Rectangular analogues of the square paths conjecture and the univariate Delta conjecture

In this paper, we extend the rectangular side of the shuffle conjecture by stating a rectangular analogue of the square paths conjecture. In addition, we describe a set of combinatorial objects and one statistic that are a first step towards a rectangular extension of (the rise version of) the Delta conjecture, and of (the rise version of) the Delta square conjecture, corresponding to the case q=1 of an expected general statement. We also prove our new rectangular paths conjecture in the special case when the sides of the rectangle are coprime

Singularities

Singularity theory is a central part of contemporary mathematics. It is concerned with the local and global structure of maps and spaces that occur in algebraic, analytic or differential geometric context. For its study it uses methods from algebra, topology, algebraic geometry and complex analysis

kk-Schur functions and affine Schubert calculus

This book is an exposition of the current state of research of affine Schubert calculus and $k$-Schur functions. This text is based on a series of lectures given at a workshop titled "Affine Schubert Calculus" that took place in July 2010 at the Fields Institute in Toronto, Ontario. The story of this research is told in three parts: 1. Primer on $k$-Schur Functions 2. Stanley symmetric functions and Peterson algebras 3. Affine Schubert calculusComment: 213 pages; conference website: http://www.fields.utoronto.ca/programs/scientific/10-11/schubert/, updates and corrections since v1. This material is based upon work supported by the National Science Foundation under Grant No. DMS-065264

Buffer Overflow Vulnerability Diagnosis For Commodity Software

Buffer overflow attacks have been a computer security threat in software-based systems andapplications for decades. The existence of buffer overflow vulnerabilities makes the systemsusceptible to Internet worms and denial of service (DDoS) attacks which can cause hugesocial and financial impacts. Due to its importance, buffer overflow problem has been intensively studied. Researchershave proposed different techniques to defend against unknown buffer overflow attacks. Theyhave also investigated various solutions, including automatic signature generation, automatic patch generation, etc., to automatically protect computer systems with known vulnerabilities. The effectiveness and efficiency of the automatic signature generation approaches andthe automatic patch generation approaches are all based on the accurate understanding ofthe vulnerabilities, the buffer overflow vulnerability diagnosis (BOVD). Currently, the results of automatic signature generation and automatic patch generation are far from satisfaction due to the insufficient research results from the automatic BOVD. This thesis defines the automatic buffer overflow vulnerability diagnosis (BOVD) problemand provides solutions towards automatic BOVD for commodity software. It targets oncommodity software when source code and symbol table are not available. The solutionscombine both of the dynamic analysis techniques and static analysis techniques to achievethe goal. Based on the observation that buffer overflow attack happens when the size of the destination buffer is smaller than the total number of writes after the data copy process if the buffer overflow attack happens through a data copy procedure, the diagnosis results return the information of the size of destination buffer, the total number of writes of a data copy procedure and how the user inputs are related with them. They are achieved through bound analysis, loop analysis and input analysis respectively. We demonstrate the effectiveness of this thesis approach using real world vulnerable applications including the buffer overflow vulnerabilities attacked by the record-setting Slammer and Blaster worms. This thesis also does the complete case study for buffer overflow vulnerabilities which may have independent interests to researchers. Our buffer overflow case study results can help other researchers to design more effective defense systems and debugging tools against buffer overflow attacks

Application of Physics-Based Image Formation Models to Change Detection in The Context of Indoor Workplace Video Surveillance

The aim of this thesis is to investigate the application of physics-based image formation models to change detection in the context of indoor workplace video surveillance. First, video surveillance applications are reviewed. Based on this review, a new classification of video surveillance applications is proposed and indoor workplace surveillance is chosen as the target application. A new workplace surveillance modelis then introduced, which relates the needs of workplace surveillance applications, their implications and the capabilities of video surveillance techniques. Furthermore, a set of requirements for workplace surveillance applications are elicited and a videobased workplace system structure is proposed. Change detection is then reviewed, and the suitability of using physics-based image formation models to enhance change detection algorithms is investigated. Two physics-based change detection techniques are developed. The foundations of these techniques are advances of colour constancy techniques which extract physical features from the camera output; this approach is unlike other change detection algorithms which use the camera output directly without considering its physical meaning. The performance of the proposed techniques is measured and compared against the Horprasert algorithm, using objective and computational complexity evaluation methods, where the quality of the change detection is measured using recall and precision measures. The Horprasert algorithm was shown, in an independent study by other researchers, to have the best trade-off between segmentation quality and computational complexity among other state-of-the-art algorithms such as Cavallaro, McKenna and Shen under experimental conditions which covered different lightings and background structures