Abstracts collection

Title: Bernstein-Bézier techniques for differential operators on piecewise polynomial fields

Joint work with Shangyou Zhang and Peter Alfeld.

Title: Dimension of bi-degree splines on T-meshes

Abstract: Polynomial splines on triangulations and quadrangulations have myriad applications and are ubiquitous, especially, in the fields of computer aided geometric design and computational mechanics. Meaningful use of splines for these purposes requires the construction and analysis of a suitable set of basis functions for the spline spaces. The dimension of these spaces can depend on an interplay between geometry, topology and combinatorics, and a theoretical understanding of its computation (or estimation) can be a useful tool when assessing constructive approaches. I will discuss this problem for bivariate splines on T-meshes, (for a full abstract please click here).

Title: High-order mesh generation and morphing for a tight integration of geometry and finite element simulations

Abstract: The last two decades have seen a growing interest in high-order finite element techniques due to the possibility to reduce the computational cost required to achieve a desired, when compared to low order methods currently used in industry. The low dissipation low dispersion errors inherent to high-order approximations are particularly attractive for the high-fidelity simulation of transient phenomena, including wave propagation and fluid dynamics applications. To exploit the benefits of high-order methods, (for a full abstract please click here).

Title: Reconstruction and Understanding of 3D Geometry

Abstract: Obtaining a digital representation of real-world objects and scenes is an essential step in many downstream applications. The increasing availability of depth sensors makes 3D acquisition much more accessible to end users. However, due to sensor restrictions and unavoidable occlusion, multiple views are often required to produce output geometry of sufficient quality. In this talk, I will overview our recent work on this, as well as a related problem, namely semantic understanding of captured scenes. In fact, these two problems are highly related: high quality reconstruction makes semantic understanding easier, and semantic understanding can also provide strong clues for improving 3D reconstruction. To achieve this, both geometry processing and machine learning are exploited.

Title: Determinantal resultant matrices for multivariate systems of polynomials

Abstract: Multivariate resultant matrices characterize the roots of a polynomial system and reduce their computation to an eigenvalue problem. However, determinantal formulas (i.e. without extraneous factors) for the resultant do not exist for arbitrary systems. They have been constructed mostly for families of unmixed systems, that is, systems of polynomial equations with a common Newton polytope of special structure. In this talk we derive determinantal formulations for the multivariate resultant of structured systems with distinct supports per equation. Among the families that admit determinantal formulas we find multilinear polynomial systems, bivariate tensor-product systems as well as systems with scaled supports.

Title: Semi-Implicit Geometry Representation with B-Spline and 3D Interactive Segmentation

Abstract: Segmenting complex 3D geometry is a challenging task due to rich structural details and complex appearance variations of target object. Shape representation and foreground- background delineation are two of the core components of segmentation. Explicit shape models, such as mesh based representations, suffer from poor handling of topological changes. On the other hand, implicit shape models, such as level-set based representations, have limited capacity for interactive manipulation. Fully automatic segmentation for separating foreground objects from background generally utilizes non-interoperable machine learning methods, which heavily rely on the off-line training dataset and are limited to the discrimination power of the chosen model. To address these issues, we propose a novel semi-implicit representation method, namely Non-Uniform Implicit B-spline Surface (NU-IBS), which adaptively distributes parametrically blended patches according to geometrical complexity. Then, a two-stage cascade detector is introduced to carry out efficient foreground and background delineation, where a simplistic Naive-Bayesian model is trained for fast background elimination, followed by a stronger pseudo-3D Convolutional Neural Network (CNN) multi-scale detector to precisely identify the foreground objects. A localized interactive and adaptive segmentation scheme is incorporated to boost the delineation accuracy by utilizing the information iteratively gained from user intervention. The segmentation result is obtained via deforming an NU-IBS according to the probabilistic interpretation of delineated regions, which also imposes a homogeneity constrain for individual segments. We evaluate the proposed method by segmenting the aortic system from a 3D cardiovascular Computed Tomography Angiography (CTA) image dataset to segment the aortic system and ascending arch.

Title: Signal detection and spike sorting in noisy time series using higher criticism

Abstract: I will talk about a novel and robust method based on making use of higher criticism for detecting signals and sorting peaks in electrophysiological measurements of neuronal activities. The method relies solely on the intrinsic statistical properties of the data and avoids any preprocessing, which prevents the loss of any invaluable information. This is join work with E. Mitricheva, R. Kimura, N. K. Logothetis and H. R. Noori.

Title: Spline spaces on topological manifolds

Abstract: In this talk, we present algebraic techniques for analyzing the space of piecewise polynomial differentiable functions on a mesh of general topology and its dimension. We start by spline spaces on planar subdivisions and describe techniques to bound their dimension. We then focus on spaces of spline functions on topological manifolds, characterized by gluing data across the shared edges. We present some dimension formula, basis constructions and illustrative examples.

Title: The computation of multiple roots of a Bernstein basis polynomial

Abstract: The Bernstein basis is used extensively in geometric modelling because of its enhanced numerical properties and elegant geometric properties in the unit interval. The computation of the points of intersection of curves and surfaces is an important problem in geometric modelling, and it gives rise to a polynomial equation. Although there is an extensive literature on numerical methods for the solution of polynomial equations, they fail to address satisfactorily an important consideration of the polynomial equations that arise in geometric modelling. In particular, multiple roots are of particular interest because they define conditions of tangency, which are important for smooth intersections of curves and surfaces. There are, however, significant numerical problems associated with the computation of multiple roots of a polynomial because of their instability - a small perturbation in the coefficients of the polynomial is sufficient to cause the roots of the polynomial to break up into complex conjugate pairs, which is unsatisfactory. This presentation will show that multiple roots of a polynomial can be computed using an algorithm developed by Gauss. Although this algorithm is conceptually simple, its computational implementation is not trivial because it involves many greatest common divisor computations and polynomial divisions (deconvolutions), both of which are ill-posed operations. Furthermore, the combinatorial factors that arise in computations with Bernstein polynomials can cause numerical problems, which must also be considered, (for a full abstract please click here).

Title: Apolarity and trivariate piecewise polynomials

Abstract: Apolarity is a classical construction in algebraic geometry which makes it possible to answer certain questions about powers of linear forms by analyzing those polynomials which vanish to particular orders on the set of points dual to the linear forms. For instance, this is one of the main ways to approach Waring's problem for polynomials. In this talk I will explain what role apolarity plays for analyzing the dimension of the space of piecewise polynomial functions (called splines) on a three-dimensional simplicial or polyhedral complex.

Title: Introduction to Code-based Post-Quantum Cryptography

Abstract: Being able to communicate in a safe way has been essential in political, diplomatic, economic and military affairs. Nowadays, when the information is digital, cryptology plays an even more important role. In this talk I will give a brief introduction to cryptology, from the classical to the post-quantum one, and I will focus on code-based cryptography.

Title: Where can toric syzygies live?

Abstract: In this talk I will give an introduction to syzygies and will discuss some open questions on them. I will then talk about the toric case, where defining equations and higher syzygies have a natural grading by the character lattice of the torus, and give some results on the the regions in the character lattice in which these equations and syzygies can be supported.

Title: Geometry and topology of data.

Abstract: Last few years are often refereed to as data revolution; a time when ability to produce and process vast amounts of data called for new scalable algorithms to analyze them. As a consequence, many at hoc solutions that lack theoretical guarantees have been introduced. In this talk I will introduce a few basic methods of geometry and topology - computable, stable and provably general tools that can be applied to analyze data and have the potential to become key tools in explainable machine learning and artificial intelligence.

Title: Sparse nonnegative solutions to underdetermined Diophantine systems

Abstract: In this talk we will give an overview of the recent results on the size of support of the solutions to a system of linear Diophantine equations. Using Siegel’s Lemma one can show that an underdetermined system Ax=b, with integer mxn matrix A and integer vector b, always has a “sparse” nonnegative integer solution, with the size of support bounded from above by m+log_2(sqrt(det(AA^t))). We discuss the structural properties of sparse integer solutions and show that the general bound can be significantly improved in the case of one single equation. Our methods include geometry of numbers and algebraic techniques.

The talk is based on a series of joint works with G. Averkov, J. De Loera, F. Eisenbrand, T. Oertel, C. O’Neill and R. Weismantel.

Title: Implicitization of tensor product surfaces with basepoints via residual resultants

Abstract: In this talk I will introduce a residual resultant for P^1xP^1 following the work of Busé, Elkadi and Mourrain and show that it can be computed using virtual projective resolutions. Afterwards I will explain how to use the residual resultant in P^1xP^1 to compute implicit equations of tensor product surfaces with basepoints. Many examples and concrete computations will be presented.

Title: A surprising tool on the mobile linkage classification problem

Abstract: Linkages are rigid bodies assembled together by mechanical joints that allow for movement between the bodies when there is no physical constraint between them. These are arranged in 3-dimensional Euclidean space forming a closed loop. When the number of joints is not high enough, linkages are not mobile in general and are called overconstrained. However, mobile overconstrained linkages do exist and usually present very special geometric arrangements. A very recent, and surprising, combinatorial tool used in order to retrieve what these arrangements might be is called bond theory and it has been applied in the quest for understanding and classifying such linkages. In this talk we explain what the linkage classification problem is and how to tackle it from the perspective of bond theory.