Introductory Lectures

Lecture 1, by Tatyana Sorokina: "Approximation Theory and Numerical Analysis"

Lecture 2, by Hal Schenck: "Algebra"

Lecture 3, by Julianna Tymoczko: "Geometry and Topology"

Research Talks

Dave Anderson, "A q-enumeration of lattice points from toric arc schemes"

Abstract: We study a certain weighted enumeration of lattice points in a polytope, motivated by a construction from quantum K-theory of toric varieties. Applying an appropriate version of the localization theorem, we obtain a q-analogue of Brion’s theorem, expressing the weighted enumerator as a sum of certain q-series attached to each vertex of the polytope. A rescaled Fourier transform defines a type of Duistermaat-Heckman measure on the polytope. This is ongoing joint work with Aniket Shah.

Cesare Bracco, "C1 hierarchical splines on multipatch geometries for isogeometric methods"

Abstract: The rise of adaptive isogeometric methods for the solution of PDEs (see, e.g. [Bracco, Buffa, Giannelli, Vázquez, Discret. Contin. Dyn. S., 2019] for an overview) led to an even greater interest in the design of suitable spline spaces allowing local refinement. In particular, C1 spline spaces defined on multipatch geometries are a natural choice for isogeometric methods when solving high-order problems on complex domains. Works dealing with the construction of this kind of spaces in the tensor-product case, e.g. [Kapl, Sangalli, Takacs, Comput. Aided Design, 2019], have been presented in recent years. Our goal is to combine these techniques with the hierarchical framework: starting from the results obtained for the two-patch case [Bracco, Giannelli, Kapl, V`azquez, Comput. Math. Appl., 2020], and relaxing some constraints of the classical hierarchical construction, we show that we can obtain the desired hierarchical C1 spaces on multipatch geometries. The hierarchical construction with relaxed constraints holds in general, allowing the application of the hierarchical framework to a wider class of spaces. (Joint with Carlotta Giannelli, Mario Kapi, Rafael Vázquez).

Oleg Davydov, "Overlap splines and finite difference methods"

Abstract: Meshless finite difference (FD) methods offer the most flexibility for modern numerical simulations of complex fluid flows or solid structures in complex geometries. Yet, their numerical analysis is complicated by the fact that finite difference solutions are discrete and therefore do not fit into standard functional framework of the analysis of PDEs. Because of this, the analysis of FD methods has been restricted to either the grid based approaches, or to the cases where FD methods may be interpreted as discretizations of mesh based methods, in both cases losing the core advantages of a meshless method. I this talk I will relate the FD methods to collocation, least squares or Galerkin methods in a pseudo-functional setting, where the approximation tool are "overlap" splines consisting of local patches glued together pointwise instead of the standard definitions where they are connected across common boundaries of the elements of a partition. This approach has allowed to prove convergence and error bounds for a least squares version of the meshless finite difference method, and further applications are in the development. In certain cases overlap splines make use of polynomial least interpolation and seem to possess interesting algebraic properties.

Mike Dipasquale, "Homogeneous trivariate splines on the star of a vertex"

Abstract: In the mid-1990s, Alfeld, Neamtu, and Schumaker derived a dimension formula for trivariate homogeneous C^r splines of degree at least 3r+2 on a vertex star which is the direct analog for Schumaker's well-known lower bound for planar C^r splines. We call this the ANS formula. Unlike Schumaker's planar formula, the ANS formula is no longer a lower bound on the dimension of the homogeneous spline space on a vertex star for d<3r+2, however. This raises a natural (and practical) question - in what degrees is the ANS formula a lower bound on the dimension of the spline space?

We can get a good idea of when the ANS formula is a lower bound using a technique called apolarity (or Macaulay-Matlis duality). In joint work with Nelly Villamizar, we show that the ANS formula is a lower bound on the dimension of the spline space on a (generic) vertex star for degrees d\ge (3r+2)/2. Our method uses an invariant of points in projective space called the Waldschmidt constant. We also indicate that a result of Whiteley from the 1990s shows that the only trivariate homogeneous C^r splines of degree d\le (3r+1)/2 on a generic vertex star are global polynomials. Thus, on generic vertex stars, the lower bound on the spline space given by the dimension of the space of polynomials is optimal right up until the ANS formula can be used as a lower bound.

Salah Eddargani, "C^2 quartic splines defined on mixed macro-structures"

Abstract: This work deals with the construction of normalized B-splines of degree four and C^2 smooth everywhere on triangulations endowed with mixed splits. The main splits involved herein are Powell-Sabin (6-), and Modified Morgan-Scott (10-) splits. With the help of Marsden identity, a family of C^2 quartic quasi-interpolation splines of optimal orders has been provided.

Mike Floater, "Supersmoothness and the connection to dimensions of spline spaces on triangulations"

Abstract: TBA (joint work with Kaibo Hu).

Jan Grošelj, "On constructing non-negative edge basis functions for representation of splines over triangulations"

Abstract: Spaces of continuously differentiable splines over general triangulations are often defined in a way that they can be characterized by interpolation problems. Splines can then be represented by basis functions dual to the interpolation functionals, which are typically associated with vertices, edges, and triangles of the triangulation. In recent years, attempts have been made to improve such representation by modifying the basis functions to be non-negative and form a partition of unity. The key idea behind this research is based on a geometric consideration of degrees of freedom related to the vertices of the triangulation, which enables a B-spline-like representation of several types of splines, e.g., Powell-Sabin splines, reduced Clough-Tocher splines, and reduced Argyris splines. However, this method does not cover the degrees of freedom related to the edges.

In this talk we will discuss a new approach to capture the degrees of freedom of splines that are related to the edges of the triangulation. For a spline in the Bernstein-Bezier representation defined over two triangles with a common edge, we will show how to replace certain Bernstein basis polynomials by a suitable number of non-negative spline basis functions. This way the continuity and differentiability conditions are automatically incorporated into the representation

without any additional constraints on the spline coefficients. We will then use the presented construction in combination with the well-established construction of vertex basis functions to introduce normalized representations of higher degree Argyris and Clough-Tocher splines.

Kaibo Hu, "Finite element and spline diagram chasing"

Abstract: The Bernstein-Gelfand-Gelfand (BGG) complexes have their roots in representation theory and differential geometry and encode fundamental algebraic and differential structures in continuum mechanics, geometry and relativity. The BGG machinery also provides a constructive tool for deriving compatible spline and finite element spaces and sequences. In this talk, we first review the BGG construction and its applications. Then we derive splines and finite elements by diagram chasing. Examples include conforming and compatible discretizations of the Riemannian metric and linearized curvature. This cohomological approach may provide a new perspective for investigating dimension of spline spaces.

Bert Juettler, "On box meshes with algebraically complete systems of tensor-product B-splines"

Abstract: Spline spaces on adaptively refined box meshes, which are spanned by tensor-product B-splines (with local knot vectors), have attracted substantial interest, due to their numerous applications in geometric modeling and numerical simulation. These include T-splines, ``locally refined'' (LR) splines and hierarchical splines. The talk discusses instances of this construction that possess the property of algebraic completeness. In particular, we discuss the case of LR-splines, where both local linear independence and algebraic completeness can be achieved with the help of a suitable refinement procedure, and algebraically complete systems of hierarchical B-splines.

Kiumars Kaveh, "Intersection index/theory of subspaces of rational functions"

Abstract: This is, basically counting solutions of systems of algebraic equations. In more detail, if L_1, ..., L_n are finite dimensional vector subspaces of rational functions on an n-dimensional variety X, their intersection index [L_1, ..., L_n] is the number of solutions of a generic system f_1(x) = ... = f_n(x)=0, f_i \in L_i, where the solutions are counted away from a bad set Z. The main results are that this intersection index is well-defined and multi-linear (and of course birationally invariant). Moreover, the self-intersection index of a subspace is given by n! volume of a convex body (Newton-Okounkov body). Naturally these ideas get related to Grobner theory and computational algebra and give rise to the notion of a Khovanskii basis.

Tom Lyche, " Simplex spline bases for smooth splines on refined triangulations"

Abstract: Splines on triangulations have widespread applications in many areas, ranging from finite element analysis and physics/engineering applications to computer graphics and entertainment industry. High smoothness spline spaces are often preferred. When dealing with a general triangulation, to obtain splines of high smoothness in a stable manner sufficiently large degrees have to be considered. An alternative is to use lower-degree macro-elements that subdivide each triangle into a number of subtriangles (or more generally subdomains). Simplex splines are one of the most elegant generalizations of univariate B-splines to the multivariate setting. They can be interpreted as the density function of a simplex shadow. This geometric construction allows to easily derive properties such as smoothness and recursion, knot insertion and degree elevation formulas. In this talk, after reviewing the main properties of simplex splines, we consider a family of macro-elements of degree p and maximal smoothness p-1 on a triangular region and we discuss the construction of a suitable local representation for the related spline space using simplex splines. In particular, we detail the important cases of C^2 cubic and C^3 quartic macro-elements and we discuss several interesting properties, such as local support, linear independence, and nonnegative partition of unity of the provided simplex spline basis. The talk is based on joint work with Carla Manni and Hendrik Speleers.

Angelos Mantzaflaris, "Explicit geometrically continuous spline reconstruction on subdivision surfaces"

Abstract: We study explicit formulas, in the form of local masks over a polyhedral mesh, for the construction of geometrically continuous splines that approximate Catmull-Clark (CC) subdivision surfaces. The latter can be represented as bicubic Bézier patches away from extraordinary vertices (EVs). However, they are non-polynomial in the vicinity of EVs. We use biquintic patches around EVs and gluing data across patch boundaries to express and solve the tangent continuity constraints. The resulting local masks yield globally smooth surfaces that converge quadratically to the CC limit surface with respect to CC subdivision. We further investigate the dimension of the space, as well as the construction of local basis functions and their approximation power. The talk is based on joint work with M. Marsala and B. Mourrain.

Michael Neilan, "Exact finite element sequences on Alfeld and Worsey-Farin triangulations"

Abstract: We present several exact sequences, consisting of piecewise polynomials, on Alfeld and Worsey-Farin triangulations. Locally, these results lead to simple dimension counts for a variety of spline spaces (e.g., C^1-conforming and H^1(curl)-conforming spaces). Globally, the results lead to simple, structure-preserving finite element discretizations for the Stokes problem and the Maxwell eigenvalue problem. This is joint work with Guosheng Fu, Johnny Guzman, and Anna Lischke.

Leonardo Patimo, "Using GKM graphs to construct statistics in representation theory"

Abstract: The weight multiplicities of representations of reductive groups admit a q-refinement, called the Kostka-Foulkes polynomials. Finding combinatorial statistics which express these polynomials is a long-standing open problem in algebraic combinatorics, which has only been solved in type A. We propose a new approach to construct statistics based on the geometry of the GKM graph of the affine Grassmannian. This approach recovers the original statistics in type A of Lascoux and Schützenberger, and at the same time allows to find statistics for Kostka-Foulkes polynomials beyond type A.

Francesco Patrizi, "Isogeometric de Rham complex discretization in solid toroidal domains"

Abstract: We present an IGA discretization of the continuous de Rham complex by means of adequate spline spaces which assemble in a discrete complex sustaining the same cohomological structure, when the underlying physical domain is a toroidal solid. Discretizations preserving such homological invariant of the physical model are commonly exploited in electromagnetics to obtain numerical solutions satisfying important conservation laws at the discrete level. Thereby one avoids spurious behaviors and, on the contrary, improves accuracy and stability. The toroidal geometries are of particular interest, for example, in the context of magnetically confined plasma simulations. The singularity of the parametrization of such physical domains demands the construction of suitable restricted spline spaces, called polar spline spaces, ensuring an acceptable smoothness to set up the discrete complex.

Jenna Rajchgot, "Symmetric quivers and symmetric varieties"

Abstract: Since the 1980s, mathematicians have found connections between orbit closures in type A quiver representation varieties and Schubert varieties in type A flag varieties. For example, singularity types appearing in type A quiver orbit closures coincide with those appearing in Schubert varieties in type A flag varieties; combinatorics of type A quiver orbit closure containment is governed by Bruhat order on the symmetric group; and classes of type A quiver orbit closures in equivariant cohomology and K-theory can be expressed in terms of formulas involving Schubert polynomials, Grothendieck polynomials, and other objects from Schubert calculus.

After recalling some of this story, I will discuss the related setting of Derksen-Weyman's symmetric quivers and their representation varieties. I will show how one can adapt results from the ordinary type A setting to unify aspects of the equivariant geometry of type A symmetric quiver representation varieties with Borel orbit closures in corresponding symmetric varieties G/K (G = general linear group, K = orthogonal or symplectic group). This is joint work with Ryan Kinser and Martina Lanini.

Sara Remogna, "Approximation in Bernstein-Bézier form on type-1 triangulations"

Abstract: Quasi-interpolation is a general and powerful approximation approach for defining local approximants to a given function or a given set of data with low computational cost.

In this talk we present a method for the construction of quasi-interpolating splines where the spline is directly determined by setting its Bernstein-Bézier coefficients to appropriate combinations of the given data values instead of defining it as linear combinations of compactly supported bivariate splines. In particular we construct and analyse such a kind of quasi-interpolating operators in the space of C^1 quartic and cubic splines on a type-1 triangulation and in the space of C^1 quadratic splines over Powell-Sabin 6-splits of a type-1 triangulation. This is a joint work with Domingo Barrera, Catterina Dagnino, Salah Eddargani and María José Ibáñez.

Espen Sande, "Poincaré path integrals for elasticity"

Abstract: In this talk we describe a general strategy to derive null-homotopy operators for differential complexes based on the Bernstein-Gelfand-Gelfand (BGG) construction and properties of the de Rham complex. Focusing on the elasticity complex, we derive path integral operators p for elasticity satisfying dp + pd = I and p^2 = 0, where the differential operators d correspond to the linearized strain, the linearized curvature and the divergence, respectively. As a special case, this gives the classical Cesáro-Volterra path integral for strain tensors satisfying the Saint-Venant compatibility condition. The construction of Kozsul operators and possible applications to the construction of piecewise polynomial finite element spaces for elasticity will be mentioned.

Alessandra Sestini, "Control and motion design with quaternion algebra"

Abstract: The quaternion algebra is a powerful mathematical tool often adopted in the control con- text for representing spatial rotations and kinematic models of general rigid body motions. Besides that, within the Computer Aided Design community it is the main tool for path- planning based on spline extensions of spatial Pythagorean Hodograph (PH) curves and for defining piecewise rational motions of rigid bodies with orientation constraints. After introducing such algebra and recalling some main results about spatial PHs and relevant subfamilies of parametric curves, I will focus on real-time PH spline motion design from 3D data streams. A different application recently developed for defining a suitable guidance law for autonomous underwater vehicles will be also presented. Researches developed with: V. Calabrò, C. Giannelli, L. Sacco.

Hendrik Speleers, "Cubic B-spline representations with super-smoothness on Powell-Sabin triangulations"

Abstract: In this talk, we investigate subspaces of the full C^1 cubic Powell-Sabin spline space that are obtained by imposing additional smoothness conditions. The full space is well defined on general triangulations endowed with a Powell-Sabin 6-split and consists of splines that admit a B-spline representation constructed in a geometrically intuitive way. Such splines have a large number of degrees of freedom and we present some approaches for their reduction by prescribing C^2 smoothness conditions at particular vertices or across particular edges of the Powell-Sabin triangulation. Furthermore, we consider how to recombine the basis functions of the original B-spline representation in order to adjust them to the reduced subspaces. Finally, with the aim of narrowing the gap between globally C^1 and C^2 splines, we restrict ourselves to three-directional triangulations and show how additional geometric symmetries help in a further reduction of degrees of freedom. This is joint work with Jan Grošelj.

Deepesh Toshniwal, "Structure-preserving discretizations using smooth splines"

Abstract: Finite element exterior calculus (FEEC) is a framework for designing stable and accurate finite element discretizations for a wide variety of systems of PDEs. The involved finite element spaces are constructed using piecewise polynomial differential forms, and stability of the discrete problems is established by preserving at the discrete level the geometric, topological, algebraic and analytic structures that ensure well-posedness of the continuous problem. The framework achieves this using methods from differential geometry, algebraic topology, homological algebra and functional analysis. In this talk I will discuss the use of smooth splines within FEEC, motivated by the fact that smooth splines are the de facto standard for representing geometries of interest in engineering and because they offer superior accuracy in numerical simulations (per degree of freedom) compared to classical finite elements. In particular, I will present new results for smooth splines defined on unstructured meshes (i.e., non-Cartesian and/or locally-refined meshes). Joint work with: Kendrick Shepherd (BYU).

Nelly Villamizar, "A lower bound on the dimension of tetrahedral splines in large degree"

Abstract: We prove a lower bound on the dimension of trivariate spline spaces on tetrahedral partitions for degree at least 8r+1, where r is the order of smoothness of the spline functions. For this, we use Bernstein-Bézier methods to compute the size of the determining sets associated to the edges, triangles, and tetrahedra of the partition. A lower bound for the determining sets which are associated to vertices is computed by applying a new lower bound on the dimension of splines on vertex stars which was proved purely by algebraic methods. The use of Bernstein-Bézier methods allows us to generalize the lower bound proved by DiPasquale and Villamizar (2020) by being able to drop genericity conditions on the partition and giving an explicit bound on the smallest degree for which the formula starts being a lower bound on the dimension of the spline space. This result is part of a joint work with P. Alfeld, M. Sirvent, T. Sorokina, W. Whiteley, M. DiPasquale, and B. Yuan.

Beihui Yuan, "Homological methods on Geometric continuous spline spaces"

Abstract: Homological methods, especially the Billera-Schenck-Stillman spline complex, is a powerful tool on investigating spline properties. In particular, it works efficiently on dimension counting problem for spline spaces over planar domains. However, when the domain is 2-dimensional but not homeomorphic to a planar region, we must use multiple coordinate charts and consider geometric continuity for functions over such a domain. In the hope that homological methods can be generalized to working on these cases, we make some attempts to apply the spline complex and spline sheaves to study geometric continuous splines. This talk is based on an ongoing project with Angelos Mantzaflaris, Bernard Mourrain and Nelly Villamizar.

Kirill Zaynullin, "Coproduct on structure algebras of moment graphs"

Abstract: This is the joint project with Martina Lanini and Rui Xiong. We construct a coproduct on the structure algebra of a moment graph (the case of a continuous spline). We use it to determine the Hopf-algebra structure on the respective quotients and use it to compute the Hopf-structure on group cohomology. We also establish the top Leibniz rule which gives interesting combinatorial identities between double Schubert polynomials.