## Algebra/Algebraic Geometry

### Talks

#### B. Dayton: The Functorial Property of the Global Dual

posted Mar 29, 2012, 6:24 AM by Tony Shaska

 For an affine ring $R=\C[x_1,\dots,x_n]/I$ the global dual is the linear space $G(R)=\{d: R \rightarrow \C, | d \text{ is linear }\}$ where $\C$ is the field of complex numbers. If $\phi$ is a polynomial map $A^n \rightarrow A^m$ with $V(J)$ the closure of the image of $\phi(V(I))$ then there is a linear map $\phi_*:G(R) \rightarrow G(C[x_1,\dots,x_m]/J)$ making $\phi_*$ into a covariant functor from algebraic sets to $\C$-vector spaces. This can be implemented in terms of Sylvester and Macaulay matrices by a single matrix multiplication and easy extraction of an implicit description for $\overline{\phi(V(I))}$. Applications include implicitization of polynomial or rational parametric curves and surfaces, and affine or projective transformations of curves and surfaces. One can work numerically giving a nice method for computer graphing of space curves given implicitly.

#### R. Lewis: Dixon-EDF: A Premier Method for Parametric Polynomial Systems

posted Mar 18, 2012, 2:03 PM by Tony Shaska

 Using examples of interest from real problems, we will discuss the Dixon-EDF resultant as a method for symbolic solution of parametric polynomial systems. We will brieﬂy describe the method itself, then discuss problems arising in Nash equilibria, geometric computing, ﬂexibility of molecules, chemical reactions, global positioning systems, operations research, and others. We will compare Dixon-EDF toseveral implementations of Groebner bases algorithms on several systems. We ﬁnd that Dixon-EDF is greatly superior.

#### S. Basu: The Role of Plane Algebraic Curves in Bifurcations of Certain Discrete Dynamical Systems

posted Mar 17, 2012, 3:45 AM by Tony Shaska

 We analyze the global behavior of solutions to a class of nonlinear discrete dynamical systems with applications in the natural sciences. In particular, we show that the global dynamics of these solutions are determined by certain plane algebraic curves.

#### T. Sorokina: Computation of dimensions of multivariate spline spaces via Hilbert polynomials and Hilbert series.

posted Mar 17, 2012, 3:42 AM by Tony Shaska

 A computational method to obtain explicit formulae for the dimension of spline spaces of smoothness $r$ and degree $d$ over simplicial partitions is described. We show how to derive these formulae in the form of a linear combination of binomial coefficients using computed values of this dimension for a finite number of parameters $r$ and $d$ to interpolate the Hilbert polynomial. Then we apply Hilbert series to obtain explicit formulae. The method is applied to conjecture the dimension formulae for the Alfeld split of an $n$-simplex and for several other tetrahedral partitions.

#### M. Moreno Maza: On Fulton's Algorithm for Computing Intersection Multiplicities

posted Mar 16, 2012, 6:03 AM by Tony Shaska   [ updated Mar 17, 2012, 3:52 AM ]

 Remarkably, and as pointed out by Fulton in his Intersection Theory, the intersection multiplicities of the plane curves V (f) and V (g) satisfy a series of 7 properties which uniquely deﬁne I(p; f, g) at each point p ∈ V (f, g). Moreover, the proof of this remarkable fact is constructive, which leads to an algorithm, that we call Fulton’s Algorithm. This construction, however, does not generalize to n polynomials f1, . . . , fn (generating a zero-dimensional of k[x1, . . . , xn], for an arbitrary ﬁeld k) for n > 2. Another practical limitation, when targeting a computer implementation, is the fact that the coordinates of the point p must be in the base ﬁeld k. Approaches based on standard or Gr¨obner bases suﬀer from the same limitation.In this work, we adapt Fulton’s Algorithm such that it can work at any point of V (f, g), rational or not. In addition, and under genericity assumptions, we add an 8-th property to the 7 properties of Fulton, which ensures that these 8 properties uniquely and constructively deﬁne I(p; f1, . . . , fn) at any p ∈ V (f1, . . . , fn). The implementation of this 8-th property has lead us to a new approach for computing the tangent cones that do not involve standard or Gr¨obner bases. In fact, all our algorithms simply rely on the theory of regular chains and are implemented in the RegularChains library in Maple.Joint work with:Steﬀen Marcus,  Paul Vrbik

#### N. Gjini: Dual tilings and their connectdness

posted Mar 16, 2012, 4:27 AM by Tony Shaska   [ updated Mar 17, 2012, 5:19 AM ]

 The talk is devoted to the study of connectivity of fractal tilings related to well-known radix representations. We study graph-directed self-aﬃne tiles associated to β-expansions with respect to Pisot units. These tiles are examples of Rauzy fractals. The objective is the case of Pisot units of degree three and four. In the case of cubic Pisot units we prove that the associated Rauzy fractal is connected. In the quartic case we ﬁnd examples of disconnected Rauzy fractals. The existence of such examples is surprising, as the digits of quartic β-expansions are consecutive integers, which leads one to expect connected tiles. Furthermore we give a complete description of all connected as well as disconnected Rauzy fractals corresponding to quartic β-expansions. In the case of discontinuity, we prove that each tile contains inﬁnitely many connected components.In order to treat the case of β-expansions, a characterization of Pisot numbers in terms of the coeﬃcients of their minimal polynomials is needed.  So ﬁrst we present such characterization. The connectivity proof yields another interesting result: it provides the complete description of the representation of 1 for β-expansions with respect to quartic Pisot units.

#### C. Wampler: Cell Decomposition of Almost Smooth Real Algebraic Surfaces

posted Mar 15, 2012, 5:22 AM by Tony Shaska   [ updated Mar 29, 2012, 9:58 AM ]

 This talk will describe the application of numerical algebraic geometry to decompose an almost smooth real algebraic surface into a cell structure consisting of faces, edges, and vertices. The algorithm starts with a witness set for a complex surface and returns a decomposition of the real surface contained therein. Noncompact surfaces are treated by casting them into projective space. The method has applications in robot workspace studies and mechanism design. This is joint work with G.-M. Besana, S. Di Rocco, J. Hauenstein, and A. Sommese.

#### C. Vinzant: Hyperbolic polynomials and definite determinantal representations

posted Mar 15, 2012, 5:21 AM by Tony Shaska   [ updated Mar 18, 2012, 1:59 PM ]

 Hyperbolic polynomials are real polynomials whose corresponding hypersurfaces have a special topological property. These polynomials appear in many areas of mathematics, including optimization, combinatorics and differential equations. In an important breakthrough in 2007, Helton and Vinnikov showed that every hyperbolic polynomial in three variables can be written as the determinant of a definite symmetric matrix of linear forms. We'll talk about this theorem and possible methods for constructing such determinantal representations.

#### R. S. Kulkarni: Geometry of Ulrich bundles and represenations of Clifford algebras

posted Mar 15, 2012, 5:21 AM by Tony Shaska   [ updated May 6, 2012, 3:12 PM ]

 Abstract: Vector bundles on a smooth projective variety whose direct image under a general projection to the projective space of the same dimesnion are called Ulrich bundles. The existence of stable Ulrich bundles on projective varieties has been a subjct of much research in the past. We discuss geometry of these bundles on cubic and K3 surfaces. The existence of stable Ulrich bundles on some special hypersurfaces in same as existence of irreducible representations of Clifford algebras. We will discuss this connection as well.

#### W. Fulton: Degeneracy Loci and Equivariant Cohomology

posted Mar 15, 2012, 5:20 AM by Tony Shaska   [ updated Apr 10, 2012, 12:16 PM ]

 AbstractThe most basic degeneracy locus is a space of m by n matrices of rank at most r. This is defined by many more equations -- the vanishing of all minors of size r+1 -- than its codimension. Computing degrees of such loci in projective spaces, when the entries are general homogeneous polynomials of suitable degrees, was a challenge to 19th century mathematicians. The challenge increases if the matrix is symmetric or skew-symmetric.  In the 20th century, one considered loci where maps between vector bundles drop rank. Slowly we realized that this problem is essentially  equivalent to computing in equivariant cohomology rings of flag bundles, and that there should be loci and formulas for each element in the corresponding Weyl group. We will discuss recent progress in this problem, especially by Ikeda, Mihalcea, and Naruse, in the equivariant setting.

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