Conference Agenda
Overview and details of the sessions of this conference. Please select a date or location to show only sessions at that day or location. Please select a single session for detailed view (with abstracts and downloads if available).
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Session Overview | |
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Location: Unitobler, F-107 30 seats, 56m^2 |
| Date: Tuesday, 09/Jul/2019 | |
| 10:00am - 12:00pm | MS140, part 1: Multivariate spline approximation and algebraic geometry |
| Unitobler, F-107 | |
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10:00am - 12:00pm
Multivariate spline approximation and algebraic geometry The focus of the proposed minisymposium is on problems in approximation theory that may be studied using techniques from commutative algebra and algebraic geometry. Research interests of the participants relevant to the minisymposium fall broadly under multivariate spline theory, interpolation, and geometric modeling. For instance, a main problem of interest is to study the dimension of the vector space of splines of a bounded degree on a simplicial complex; recently there have been several advances on this front using notions from algebraic geometry. Nevertheless this problem remains elusive in low degree; the dimension of the space of piecewise cubics on a planar triangulation (especially relevant for applications) is still unknown in general. (25 minutes for each presentation, including questions, followed by a 5-minute break; in case of x<4 talks, the first x slots are used unless indicated otherwise) Algebraic Approaches to Spline Theory In this talk we will give a brief overview of some of the algebraic methods which are used in spline theory. We will give particular attention to the pioneering work of Billera, in which homological methods were introduced for the calculation of dimension formulas. These methods have proved very fruitful for splines on all types of subdivisions; we will attempt to give a flavor for the various results that have been obtained this way, the questions that remain open, and the connections to algebraic geometry that result from these methods. Computing dimension formulas is only a beginning in spline theory - time permitting we will address how the gluing data for splines may be solved in some cases to give basis functions. Polynomial splines of non-uniform degree: Combinatorial bounds on the dimension Polynomial splines on triangulations and quadrangulations have myriad applications and are ubiquitous, especially in the fields of computer aided design, computer graphics and computational analysis. Meaningful use of splines for these purposes requires the construction and analysis of a suitable set of basis functions for the spline spaces. In turn, the computation or estimation of their dimensions is useful which, following the definition of smooth splines, depends on an interplay between algebra and geometry. We consider the general case of splines with polynomial pieces of differing degrees. The flexibility of such splines would allow design of complex shapes with fewer control points, i.e., cleaner and simpler designs; while the same would also lead to more efficient engineering analysis. Using homological techniques, introduced by Billera (1988), we analyze the dimension of splines on triangulations and T-meshes. Specifically, we generalize the frameworks presented in Mourrain and Villamizar (2013) and Mourrain (2014) to the setting of both mixed polynomial degrees and mixed smoothness. Combinatorial bounds on the dimension are presented. Several examples are provided to illustrate application of the theory developed. Approximation power of C1-smooth isogeometric functions on trivariate two-patch domains Bases and dimensions of trivariate spline functions possessing first order geometric continuity on two-patch domains were studied in (Birner, Jüttler, Mantzaflaris, Graph. Mod. 2018). It was shown that the properties of the spline space depend strongly on the type of the gluing data that is used to specify the relation between the partial derivatives along the interface between the patches. Locally supported bases were shown to exist for trilinear geometric gluing data (that corresponds to piecewise trilinear domain parameterizations) and sufficiently high degree. In this talk we discuss the approximation properties of these spline functions. In particular, we perform numerical experiments with L2 projection in order to explore the approximation power. Despite the existence of locally supported bases, we observe a reduction of the approximation order for low degrees, and we provide a theoretical explanation for this phenomenon. This is joint work with Bert Jüttler and Angelos Mantzaflaris. Splines, representations, and the Stanley-Stembridge conjecture Splines can be used to construct the (equivariant) cohomology of certain algebraic varieties. We describe one such construction, and how the action of certain permutations on the splines relates to a longstanding open question in combinatorics called the Stanley-Stembridge conjecture. We also discuss certain steps towards resolving the conjecture in special cases. |
| 3:00pm - 5:00pm | MS168, part 1: Riemann Surfaces |
| Unitobler, F-107 | |
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3:00pm - 5:00pm
Riemann Surfaces In the past decades, the central role played by Riemann surfaces in pure mathematics has been strengthened with their surprising appearance in string theory, cryptography and material science. This minisymposium is intended for the curve theorists and the avant-garde applied mathematician. Our emphasis will be on the computational aspects of Riemann surfaces that are prominent in pure mathematics but are not yet part of the canon of applied mathematics. Some of the subjects that will be touched upon by our speakers are integrable systems, Teichmüller curves, Arakelov geometry, tropical geometry, arithmetic geometry and cryptography of curves. (25 minutes for each presentation, including questions, followed by a 5-minute break; in case of x<4 talks, the first x slots are used unless indicated otherwise) Real soliton lattices of KP-II equation and desingularization of spectral curves Planar bicolored (plabic) networks in the disk were originally introduced by A.Postnikov to parametrize positroid cells in totally nonnegative Grassmannians and used by Y. Kodama and L.Williams to explain the asymptotic behavior of real regular multiline soliton solutions (rrss) of Kadomtsev-Petviashvili II (KP) equation. In this talk based on recent papers in collaboration with P.G. Grinevich (arXiv:1801.00208, arXiv:1803.10968, arXiv:1805.05641) we explain a different relation of plabic networks with KP theory based on the spectral theory for degenerate finite-gap solutions on reducible curves by Krichever where the rrss play the role of potentials. In our construction the plabic graph is dual to a reducible curve which is the rational degeneration of a smooth M-curve of genus equal to the number of faces of the graph diminished by one. The boundary of the disk corresponds to the rational curve associated to the soliton data in the direct spectral problem and each internal vertex is a rational component. Edges are the double points where two such components are glued. We then introduce and characterize systems of edge vectors on plabic networks and use them to uniquely associate to generic soliton data a Krichever divisor satisfying the reality and regularity conditions of Dubrovin-Natanzon. Our approach is constructive and may be used to effectively desingularize curves. The case of soliton data in the positive part of Gr(2,4) is shown in detail. Conformal patterns on closed surfaces via discrete conformal maps and holomorphic differentials Using uniformization of discrete Riemann surfaces we construct conformal patterns on closed surfaces without cuts and overlapping. Arakelov invariants in the tropical limit In this talk we are interested in semistable degenerations of compact Riemann surfaces. We consider the asymptotic behavior, under such degenerations, of certain canonical metrics, Green's functions and related invariants as studied in Arakelov theory and string perturbation theory. We obtain precise expressions for the asymptotics under consideration in terms of potential theory on metric graphs, aka tropical curves. In particular, non-archimedean and tropical geometry appear naturally when studying degenerations of Riemann surfaces. Siegel modular forms and classical invariants For abelian varieties of dimension 2 and 3, Siegel modular forms for the full symplectic group can be reinterpreted as classical invariants for the action of GL2 or GL3. There are many applications and consequences of this dictionary. We will show in particular that one can decrease the number of generators for the ring of Siegel modular forms in dimension 3 obtained by Tsuyumine (1986). Joint work with Reynald Lercier. |
| Date: Wednesday, 10/Jul/2019 | |
| 10:00am - 12:00pm | MS140, part 2: Multivariate spline approximation and algebraic geometry |
| Unitobler, F-107 | |
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10:00am - 12:00pm
Multivariate spline approximation and algebraic geometry The focus of the proposed minisymposium is on problems in approximation theory that may be studied using techniques from commutative algebra and algebraic geometry. Research interests of the participants relevant to the minisymposium fall broadly under multivariate spline theory, interpolation, and geometric modeling. For instance, a main problem of interest is to study the dimension of the vector space of splines of a bounded degree on a simplicial complex; recently there have been several advances on this front using notions from algebraic geometry. Nevertheless this problem remains elusive in low degree; the dimension of the space of piecewise cubics on a planar triangulation (especially relevant for applications) is still unknown in general. (25 minutes for each presentation, including questions, followed by a 5-minute break; in case of x<4 talks, the first x slots are used unless indicated otherwise) Bounds on the dimension of spline spaces on polyhedral cells We study the space of spline functions defined on polyhedral cells. These cells are the union of 3-dimensional polytopes sharing a common vertex, so that the intersection of any two of the polytopes is a face of both. In the talk, we will present new bounds on the dimension of this spline space. We provide a bound on the contribution of the homology term to the dimension count, and prove upper and lower bounds on the ideal of the interior vertex which depend only on combinatorial (or matroidal) information of the cell. We use inverse systems to convert the problem of finding the dimension of ideals generated by powers of linear forms to a computation of dimensions of so-called fat point ideals. The fat point schemes that comes from dualizing polyhedral cells is particularly well-suited and leads to the exact dimension in many cases of interest that will also be presented in the talk. On the gradient conjecture for homogeneous polynomials The following conjecture was proposed by myself and Tom McKinley: Let p and f be homogeneous polynomials in n variables such that p(gradf)=0. Then p(grad)f=0. This intriguing conjecture is closely related to the work of Gordan and Noether on polynomials with vanishing Hessians and with some density problems proposed by Pinkus and Wainryb. In my talk I will indicate some particular cases when the conjecture holds true. In particular when the number of variables is at most 5 and when the deg(p)=2. Ambient Spline Approximation of Functions on Submanifolds Recently, a novel approach to approximation of functions on submanifolds has been made: It is based on extending the function constantly along the normals and approximating this extension by functions that exist in the ambient space with tensor product splines as a prominent example. For those, the concept is able to essentially reproduce the convergence orders on the submanifold one can expect in the ambient space. In the talk, we will give an introduction into the basic concept along with some theoretical results and numerical experiments. Watertight Trimmed NURBS Surfaces Trimmed NURBS are the standard for industrial surface modeling, and all common data exchange formats, like IGES or STEP, are based on them. Typically, trimming curves have so high degree and so complex knot structure that it seems to be impossible to match them properly to neighboring geometry. Thus, surfaces built from several trimmed NURBS patches are known to reveal gaps along inner boundaries, and it is a cumbersome and sometimes nontrivial task for designers to keep the magnitude of these gaps below an acceptable tolerance. In this talk, we present a novel methodology to construct trimmed NURBS surfaces with prescribed low order boundary curves, facilitating the representation of watertight surface models within the functionality of standard CAD systems. |
| 3:00pm - 5:00pm | MS168, part 2: Riemann Surfaces |
| Unitobler, F-107 | |
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3:00pm - 5:00pm
Riemann Surfaces In the past decades, the central role played by Riemann surfaces in pure mathematics has been strengthened with their surprising appearance in string theory, cryptography and material science. This minisymposium is intended for the curve theorists and the avant-garde applied mathematician. Our emphasis will be on the computational aspects of Riemann surfaces that are prominent in pure mathematics but are not yet part of the canon of applied mathematics. Some of the subjects that will be touched upon by our speakers are integrable systems, Teichmüller curves, Arakelov geometry, tropical geometry, arithmetic geometry and cryptography of curves. (25 minutes for each presentation, including questions, followed by a 5-minute break; in case of x<4 talks, the first x slots are used unless indicated otherwise) Computing endomorphism rings of Jacobians Let C be a curve over a number field, with Jacobian J, and let End(J) be the endomorphism ring of J. The ring End(J) is typically isomorphic to ZZ, but the cases where it is larger are interesting for many reasons, most of all because the corresponding curves can then often be matched with relatively simple modular forms. We give a provably correct algorithm to verify the existence of additional endomorphisms on a Jacobian, which to our knowledge is the first such algorithm. Conversely, we also describe how to get upper bounds on the rank of End(J). Together, these methods make it possible to completely and explicitly determine the endomorphism ring End(J) starting from an equation for C, with acceptable running time when the genus of C is small. This is joint work with Edgar Costa, Nicolas Mascot, and John Voight. Inverse Jacobian problem for cyclic plane quintic curves We consider the problem of computing the equation of a curve with given analytic Jacobian, that is, with a certain period matrix. In the case of genus one, this can be done by using the classical Weierstrass function, and it is a key step if one wants to write down equations of elliptic curves with complex multiplication (CM). Also in higher genus, the theory of CM gives us all period matrices of principally polarized abelian varieties with CM, among which the periods of the curves whose Jacobian has CM, and computing curve equations is the hardest part. Beyond the classical case of elliptic curves, efficient solutions to this problem are now known for both genus~2 and genus~3. In this talk I will give a method that deals with the case y5 = a5x5 + ... + a1x + a0, inspired by some of the ideas present in the method for the genus-3 family of Picard curves y3 = x(x-1)(x-λ)(x-μ). Teichmüller curves, Kobayashi geodesics and Hilbert modular forms Teichmüller curves are totally geodesic curves inside the moduli space of Riemann surfaces. By results of Möller, they can always be seen as Kobayashi geodesics inside a Hilbert modular variety parametrising abelian varieties with real multiplication. Our main objective is to cut Teichmüller curves out as the vanishing locus of a Hilbert modular form in order to calculate their Euler characteristics. The building blocks of these modular forms turn out to be certain theta functions and their derivatives, which can be made very precise. Counting special points on teichmüller curves A flat surface is a Riemann surface together with the choice of a non-zero holomorphic differential. The moduli space of flat surfaces admits a natural SL2(R) action and the closed orbits are Teichmüller curves in the moduli space of Riemann surfaces. While a lot of the original motivation stems from dynamical systems, the known examples of families of such Teichmüller curves carry a surprising amount of arithmetic information. This permits explicit formulas for the genus, the number of cusps and the number and types of orbifold points as well as, in many cases, precise asymptotic behavior of these numbers. |
| Date: Thursday, 11/Jul/2019 | |
| 10:00am - 12:00pm | MS140, part 3: Multivariate spline approximation and algebraic geometry |
| Unitobler, F-107 | |
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10:00am - 12:00pm
Multivariate spline approximation and algebraic geometry The focus of the proposed minisymposium is on problems in approximation theory that may be studied using techniques from commutative algebra and algebraic geometry. Research interests of the participants relevant to the minisymposium fall broadly under multivariate spline theory, interpolation, and geometric modeling. For instance, a main problem of interest is to study the dimension of the vector space of splines of a bounded degree on a simplicial complex; recently there have been several advances on this front using notions from algebraic geometry. Nevertheless this problem remains elusive in low degree; the dimension of the space of piecewise cubics on a planar triangulation (especially relevant for applications) is still unknown in general. (25 minutes for each presentation, including questions, followed by a 5-minute break; in case of x<4 talks, the first x slots are used unless indicated otherwise) Bivariate Semialgebraic Splines We consider bivariate splines over partitions defined by arcs of irreducible algebraic curves. We compute the dimension of the space of semialgebraic splines in two extreme cases. If the forms defining the edges span a three-dimensional space of forms of degree $n$, then we show that the dimensions can be reduced to the linear case. If the partition is sufficiently generic, we give a formula for the dimension of the spline space in large degree and bound how large the degree must be for the formula to be correct. We also study the dimension of the spline space in some examples where the curves do not satisfy either extreme. The results are derived using commutative and homological algebra. This is joint work with Michael DiPasquale. Geometrically smooth spline bases for geometric modeling Given a topological complex M with glueing data along edges shared by adjacent faces, we study the associated space of geometrically smooth splines that satisfy differentiability properties across shared edges. We present new constructions of basis functions of the space of $G^1$- spline functions on quadrangular meshes, which are tensor product bspline functions on each quadrangle and with b-spline transition maps across the shared edges. By analysing the syzygy equation induced by the $G^1$ constraints over a single edge, we show that the separability of the space of $G^1$ splines across an edge allows to determine the dimension and a bases of the space of $G^1$ splines on M. This leads to new explicit construction of basis functions attached to the vertices, edges and faces of M. The construction of smooth basis functions attached to a topological structure has important applications in geometric modeling. We illustrate it on the fitting of point clouds by $G^1$ splines on quadrangular meshes of complex topology. The ingredients are detailled and experimentation results showing the behavior of the method are presented. Splines, Stable Bundles, and PDE’s We will explain a number of connections between certain local and global problems in approximation theory related to spaces of splines and certain stable or semi-stable vector bundles/reflexive sheaves on complex projective spaces. These connections lead to an interesting relationship between the spaces of solutions of certain systems of constant coefficient partial differential equations and the first cohomology group of those vector bundles/reflexive sheaves. Using results of Grothendieck and Shatz, the case of two variables and the projective plane is analyzed. We will also discuss extensions to vector bundles on higher-dimensional projective spaces as they relate to splines and PDE’s in three or more variables. Computing the dimension of spline spaces using homological techniques Homological techniques have been successfully employed for computing the dimensions of piecewise polynomial spaces on triangulations and quad meshes in the plane. Examples of applications to other spline spaces will be presented. |
| 3:00pm - 5:00pm | MS136, part 1: Syzygies and applications to geometry |
| Unitobler, F-107 | |
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3:00pm - 5:00pm
Syzygies and applications to geometry In this minisymposium, titled "Syzygies and applications to geometry”, we will focus on the striking results and applications that the study of syzygies provides in algebraic geometry, in a wide sense. Topics should include but are not limited to the study of rational and birational maps, singularities, residual intersections and the defining equations of blow-up algebras. We plan to focus on recent progress in this area that result in explicit and effective computations to detect certain geometrical property or invariant. Applications to geometric modeling are very welcome. (25 minutes for each presentation, including questions, followed by a 5-minute break; in case of x<4 talks, the first x slots are used unless indicated otherwise) Fibers of multi-graded rational maps and orthogonal projection onto rational surfaces I will present a new algebraic approach for computing the orthogonal projection of a point onto a rational algebraic surface embedded in the three dimensional projective space, which is a joint work with Nicolás Botbol, Laurent Busé and Marc Chardin. Our approach amounts to turn this problem into the computation of the finite fibers of a generically finite trivariate rational map whose source space is either bi-graded or tri-graded and which has one dimensional base locus: the congruence of normal lines to the rational surface. This latter problem is solved by using certain syzygies associated to this rational map for building matrices that depend linearly in the variables of the three dimensional ambient space. In fact, these matrices have the property that their cokernels at a given point p in three dimensional space are related to the pre-images of the p via the rational map. Thus, they are also related to the orthogonal projections of p onto the rational surface. Then, the orthogonal projections of a point are approximately computed by means of eigenvalues and eigenvectors numerical computations.
Complete intersection points in product of projective spaces I will report on ongoing project with Marc Chardin. we study the bigraded Hilbert function of complete intersection sets of points in $mathbb{P}^n times mathbb{P}^m$. We give a sharp lower bound for the stabilization of the bigraded Hilbert function. In addition, we show that, in a specific and pretty large region, the bigraded Hilbert Function only depends upon the degree of the forms defining the points. Finally, we consider the case where the forms defining the points are chosen generically. In this case we show that the natural projections to $mathbb{P}^n$ and $mathbb{P}^m$ are one-to-one. Fibers of rational maps and Jacobian matrices A rational map $varphi$ from a projective space of dimension m to another is defined by homogeneous polynomials of a common degree d. We establish a linear bound in terms of d for the number of (m − 1)-dimensional fibers of $varphi$, by using ideals of minors of the Jacobian matrix. This is joint work with S. Dale Cutkosky and Tran Quang Hoa. Syzygies and the geometry of rational maps (introductory talk) During the past years, the analysis of the syzygies (i.e. the algebraic relations of first order) between the equations defining a geometric object leaded to important advances on many problems lying at the interface of commutative algebra and algebraic geometry, motivated in large part by computer aided assisted computations. In this introductory talk, I will provide an overview of a range of methods and results on the study of the geometric properties of rational maps by means of the syzygies of their defining polynomials, in particular on the understanding of their image and fibers. The computational aspects and the relevance of these results in the field of geometric modeling will also be discussed. |
| Date: Friday, 12/Jul/2019 | |
| 10:00am - 12:00pm | MS125: Efficient algorithms for geometric invariant theory |
| Unitobler, F-107 | |
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10:00am - 12:00pm
Efficient algorithms for geometric invariant theory Recently, motivated by the polynomial identity testing problem from computer science, and by questions arising in quantum information theory, efficient numerical algorithms for solving the null cone problem from geometric invariant theory have been proposed. The goal of the minisymposium is to review this progress and to report on recent advances. (25 minutes for each presentation, including questions, followed by a 5-minute break; in case of x<4 talks, the first x slots are used unless indicated otherwise) Algorithms for the separation of orbits of matrices We consider two group actions on the set of m-tuples of n by n matrices, namely the simultaneous conjugation action of GL(n) and the left-right action of SL(n) x SL(n). An invariant can separate two orbits of m-tuples if and only if the closures of these orbits are disjoint. In both cases we present a polynomial time algorithm that decides whether the orbit closures of two m-tuples intersect. This is joint work with Visu Makam. Analytic algorithms for the null cone problem Null cone is a fundamental object in invariant theory. It is the variety defined by all the homogeneous invariant polynomials for a particular group action. This talk will be focused on the computational complexity of testing membership in the null cone. From a purely algebraic point of view, this problem seems hard, since except in a few cases, one does not have any nice description of the invariant polynomials. However, we will see that there is a good reason to believe that analytic methods can provide provably efficient algorithms for the null cone and we will discuss several examples where this has already been achieved, the most notable being bipartite matching, linear programming, non-commutative rank etc. The analytic approach goes via the Kempf-Ness criterion and connects to the exciting area of geodesically convex optimization. Non-commutative rank of linear matrices, related structures and applications The non-commutative rank of a matrix with homogeneous linear entries is the rank when we consider the variables as elements of the appropriate free division algebra. This is the same as the maximum size of a square sub-matrix such that the tuple of coefficient matrices is not in the null-cone of the polynomial invariants for the left-right action of the suitable special linear group. There is a "dual" characterization in terms of large common rectangular zero blocks (after appropriate changes of bases).We will outline a deterministic polynomial time algorithm for computing the non-commutative rank in a two-fold constructive way. Firstly, it computes a polynomial invariant for an appropriate sub-matrix together with a non-vanishing substitution of values from a division algebra of relatively small dimension (actually, even from a matrix algebra) into the variables. Then, in the non-full rank case, common zero blocks with matching parameters are revealed. We will also discuss related common "echelon forms" for the coefficient matrices and some applications. The algorithm for finding the large zero blocks works along certain flags of subspaces that further "echelonize" the coefficient matrices. These flags are analogous to the alternating forests in algorithms for finding maximum matchings in bipartite graphs. Interestingly, their behavior explain certain special cases when the non-commutative rank coincide with or approximates the usual, "commutative" one (i.e., with the rank when the variables are considered as elements of a commutative field). Some of these special cases will be discussed as well. Analytic algorithms for the moment polytope Moment polytopes are convex bodies associated to certain group actions on manifolds. When the manifold is a projective variety invariant under the action of a reductive Lie group, it is known that the moment polytope encodes asymptotic information about the irreducible representations of the group occurring in the coordinate ring of the variety. This talk concerns the computational complexity of deciding moment polytope membership. Existing methods include enumerating the (potentially exponentially many) facets and evaluating highest weight polynomials (of potentially exponential degree), but we will discuss analytic algorithms that do not seem to face the same hurdles. These analytic algorithms are also notable for their simplicity and their ability to compute preimages under the moment map, a problem of practical interest. We will discuss how alternating minimization, one of the simplest approaches in optimization, leads to analytic algorithms for Horn's problem and the quantum marginal problem. Among the conceptual tools in the analysis of these algorithms are "reductions" to the null cone problem and lower bounds for Kempf-Ness functions via representation theory. |
| 3:00pm - 5:00pm | MS136, part 2: Syzygies and applications to geometry |
| Unitobler, F-107 | |
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3:00pm - 5:00pm
Syzygies and applications to geometry In this minisymposium, titled "Syzygies and applications to geometry”, we will focus on the striking results and applications that the study of syzygies provides in algebraic geometry, in a wide sense. Topics should include but are not limited to the study of rational and birational maps, singularities, residual intersections and the defining equations of blow-up algebras. We plan to focus on recent progress in this area that result in explicit and effective computations to detect certain geometrical property or invariant. Applications to geometric modeling are very welcome. (25 minutes for each presentation, including questions, followed by a 5-minute break; in case of x<4 talks, the first x slots are used unless indicated otherwise) Implicitization of Tensor Product Surfaces via Virtual Projective Resolutions (Part I) In this talk, we address the implicitization problem for tensor product surfaces. A tensor product surface is defined by a parametrization (rational map) $mathbb{P}^1times mathbb{P}^1to mathbb{P}^3$. We consider the problem of computing the defining equation for the tensor product surface based on the polynomials which give its parametrization. Towards this end, we use the residual resultants developed by Busé-Elkadi-Mourrain. Our perspective is informed by the new development of virtual resolutions, which afford the derivation of the implicit equation from a smaller, more manageable algebraic construction than the more standard projective resolutions. Part I of this talk will discuss the algebraic underpinnings of our method. Implicitization of Tensor Product Surfaces via Virtual Projective Resolutions (Part II) In this talk, we address the implicitization problem for tensor product surfaces. A tensor product surface is defined by a parametrization (rational map) $mathbb{P}^1times mathbb{P}^1to mathbb{P}^3$. We consider the problem of computing the defining equation for the tensor product surface based on the polynomials which give its parametrization. Towards this end, we use the residual resultants developed by Busé-Elkadi-Mourrain. Our perspective is informed by the new development of virtual resolutions, which afford the derivation of the implicit equation from a smaller, more manageable algebraic construction than the more standard projective resolutions. Part II of this talk will discuss computational considerations and the practical implementation of our method. The Hilbert quasipolynomial of a polynomial ring and generating functions related the Frobenius complexity for various classes of singularities The talk will present some open questions on the Hilbert quasipolynomial associated to a polynomial ring over a field in finitely many indeterminates, with nonstandard grading. These questions originate in investigations regarding the Frobenius complexity for finitely generated algebras over the integers. The investigation approaches this notion of complexity by analogy to the Hilbert-Samuel and Hilbert-Kunz functions. This is joint work with Yongwei Yao. Generalized Stanley-Reisner rings Given a simplicial complex, we study the structure of the subrings of the Stanley-Reisner ring associated to the simplicial complex. These subrings can be seen as the space of spline functions defined on the simplicial complex with higher order continuity conditions accross the faces of the partition. Their ring structure becomes particularly interesting by identifying the ring of continuous splines with the equivariant cohomology ring of a space with a torus action. In the talk, we will explore this identification as well as the the geometric realizations of genealized Stanley-Reisner rings via the description of certain syzygy modules thar encode the smoothness conditions on the splines.
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| Date: Saturday, 13/Jul/2019 | |
| 10:00am - 12:00pm | MS159: Intersections in practice |
| Unitobler, F-107 | |
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10:00am - 12:00pm
Intersections in practice This mini-symposium will focus on practical computational methods in intersection theory and their applications. At its most basic, intersection theory gives a means to study the geometric and enumerative properties of intersections of two varieties within another. These questions are fundamental to both algebraic geometry and its applications. Fulton-MacPherson intersection theory provides a powerful tool-set with which to study these intersections; however, many mathematical objects which are needed in this framework have long been computationally inaccessible. This barrier has limited the use of these ideas in computations and applications. In recent years several new and computable expressions for Segre classes, Polar classes, Euler characteristics, Euler obstructions, and other fundamental objects in intersection theory have been developed. This has led to a variety of computationally effective symbolic and numeric algorithms and opened the way for ideas from intersection theory to be applied to solve both mathematical and scientific problems. Some of this recent work will be highlighted in this mini-symposium. The first talk in the session will be an introductory talk, which will demonstrate the natural relations between intersection theory and numerical algebraic geometry and will highlight how intersection theory can be applied to solve classical problems such as testing ideal membership (without computing a Groebner basis). Subsequent talks will explore computational aspects of intersection theory in more detail and will highlight their practical applications. (25 minutes for each presentation, including questions, followed by a 5-minute break; in case of x<4 talks, the first x slots are used unless indicated otherwise) Segre-driven ideal membership testing In this talk we discuss new effective methods to test pairwise containment of arbitrary (possibly singular) subvarieties of any smooth projective toric variety and to determine algebraic multiplicity without working in local rings. These methods may be implemented without using Gröbner bases; in particular any algorithm to compute the number of solutions of a zero-dimensional polynomial system may be used. The methods arise from techniques developed to compute the Segre class s(X,Y) of X in Y for X and Y arbitrary subschemes of some smooth projective toric variety T. In particular, this work also gives an explicit method to compute these Segre classes and other associated objects such as the Fulton-MacPherson intersection product of projective varieties. These algorithms are implemented in Macaulay2 and have been found to be effective on a variety of examples. This talk is based on joint work with Corey Harris (University of Oslo). The bottleneck degree of a variety The talk is within the area of Algebraic Geometry of Data. Bottlenecks are pairs of points on a variety joined by a line which is normal to the variety at both points. These points play a special role in determining the appropriate density of a point-sample of the variety. Under suitable genericity assumptions the number of bottlenecks of an affine variety is finite and we call it the bottleneck degree. We show that it is determined by invariants of the variety, such as polar classes and Chern classes. The talk is based on joint work with D. Eklund and M. Weinstein. Symbolic Computation of Invariants of Local Rings For a local ring (A, m) and an ideal I such that A/I has finite length, the Hilbert-Samuel polynomial P(n) of I is a polynomial such that P(n)=length(A/I^n) for large n. The leading coefficient and degree of this polynomial are important invariants of the ideal and encode information about the singularities of its coordinate ring. We present methods for computing this polynomial for arbitrary ideals and give various geometric examples involving hypersurfaces, determinantal ideals, etc. Moreover, we will share progress on local computation of other invariants such as the Bernstein-Sato polynomial and multiplier ideals, which measure singularities of varieties. |
| 3:00pm - 5:00pm | MS136, part 3: Syzygies and applications to geometry |
| Unitobler, F-107 | |
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3:00pm - 5:00pm
Syzygies and applications to geometry In this minisymposium, titled "Syzygies and applications to geometry”, we will focus on the striking results and applications that the study of syzygies provides in algebraic geometry, in a wide sense. Topics should include but are not limited to the study of rational and birational maps, singularities, residual intersections and the defining equations of blow-up algebras. We plan to focus on recent progress in this area that result in explicit and effective computations to detect certain geometrical property or invariant. Applications to geometric modeling are very welcome. (25 minutes for each presentation, including questions, followed by a 5-minute break; in case of x<4 talks, the first x slots are used unless indicated otherwise) Inversion of polynomial systems and polar maps Given F=V(f) a reduced projective hypersurface defined by a homogeneous polynomial f in several variables, the gradient of f defines a rational map P_f between two projective spaces of the same dimension, called polar map of F. In general, it is a problem to describe all the homaloidal hypersurfaces, that is the hypersurfaces F=V(f) such that P_f is birational (i.e. P_f is an isomorphism between two Zariski opens) which aims to distinguish specific singular locus of the projective hypersurfaces. The classification of reduced homaloidal complex curves (i.e. when the base field is the complex field) was established by I.V.Dolgachev. It is formed by the smooth conics, the unions of three general lines and the unions of a smooth conics with one of its tangent. When the base field has characteristic p>2, the three curves in Dolgachev's classification are still homaloidal and a problem becomes to establish if they are the only ones. In this talk, I will explain how this question can be related to an analysis of the syzygies of the jacobian ideal of the hypersurfaces and I will show an explicit example of a homaloidal curve of degree 5 if p=3. This can be viewed algebraically as a study of the difference between the Rees and the symmetric algebra of the jacobian ideal or, equivalently, as a study of the variations of the Milnor number of the curves with respect to reduction modulo p. Singularities and radical initial ideals What kind of reduced monomial schemes can be obtained as a Gröbner degeneration of a smooth projective variety? Emanuela De Negri, Matteo Varbaro and myself conjecture that the answer is: Only Stanley-Reisner schemes associated to acyclic Cohen-Macaulay simplicial complexes. This would imply in particular, that only curves of genus zero have such a degeneration. We proved this conjecture for degrevlex orders, for elliptic curves over real number fields, for boundaries of cross-polytopes, and for leafless graphs. Consequences for rational and F-rational singularities of algebras with straightening laws will also be discussed. Syzygies and gluing for semigroup rings Two numerical semigroups can be glued to obtain another numerical semigroup in higher embedding dimension. This concept was originally introduced to classify numerical semigroups that are complete intersections and it was later generalized to arbitrary numerical semigroups and semigroups in higher dimension. In this talk, we will construct the syzygies of the semigroup ring k[C] of a semigroup C obtained by gluing two semigroups A and B in terms of the syzygies of k[A] and k[B]. This will provide formulas for several invariants like Betti numbers, projective dimension and Hilbert series. We will use our construction to show that gluing two semigroups in higher dimension is not as easy as in the numerical case. This is a joint work with Hema Srinivasan (Missouri University, USA). Specialization of rational maps We consider the behavior of the degree of a rational map under specialization of the coefficients of the defining linear system. The method rests on the classical idea of Kronecker as applied to the context of projective schemes and their specializations. For the theory to work one is led to develop the details of rational maps and their graphs when the ground ring of coefficients is a Noetherian integral domain. We will show specific applications to certain classes of rational maps. This is joint work with Aron Simis. |
