FB 6 Mathematik/Informatik

Institut für Mathematik


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SS 2016

12.04.2016 um 16:15 Uhr in 69/125:

Mateusz Michalek (Freie Universität Berlin)

From topology to algebraic geometry and back again

Secant varieties are known to play an important role in complexity theory, representation theory and algebraic geometry, relating to ranks of tensors. In my talk I would like to present applications of secants in topology through k-regular embeddings. An embedding of a variety in an affine space is called k-regular if any k points are mapped to linearly independent points. Numeric conditions for the existence of such maps are an object of intensive studies of algebraic topologists dating back to the problem posed by Borsuk in the fifties. Recent new results were obtained by Pavle Blagojevic, Wolfgang Lueck and Guenter Ziegler. Our results relate k-regular maps to punctual versions of secant varieties. This allows us to prove existence of such maps in special cases. The main new ingredient is providing relations to the geometry of the punctual Hilbert scheme and its Gorenstein locus. The talk is based on two joint works: with Jarosław Buczynski, Tadeusz Januszkiewicz and Joachim Jelisiejew and with Christopher Miller: arXiv:1511.05707 and arXiv:1512.00609.

19.04.2016 um 16:15 Uhr in 69/125:

Holger Brenner (Universität Osnabrück)

Symmetric Signature

We prove that for invariant rings of a small finite group G the differential symmetric signature is 1/|G|. This is based on joint work with Alessio Caminata.

03.05.2016 um 14:15 Uhr in 69/E15:

Peter Symonds (University of Manchester)

Groups of power series under substitution and automorphisms of curves

It is known that the group formed by formal power series over a field of characteristic p with no constant term under substitution contains every finite p-group, but it is very hard to realise these finite subgroups explicitly. We discuss this problem and show how it is related to group actions on curves. Using this connection, we can construct an explicit element of order 4; no such construction was known before.

17.05.2016 um 16:15 Uhr in 69/125:

Richard Sieg (Universität Osnabrück)

Representations of Hilbert Series in Normaliz

One of the major computation goals in Normaliz is the Hilbert Series of an affine monoid, that is, the generating function for counting elements in each degree. We will discuss the current calculation and representation of this series in Normaliz. Next we will present a new form which is compact and allows for a combinatorial interpretation. This form is attained by calculating the degrees of a homogeneous system of parameters for the monoid algebra and we will present and discuss an algorithm to compute them.

09.06.2016 um 16:15 Uhr in 69/125:

Davide Bolognini (University of Genova)

Betti splitting of monomial ideals and simplicial complexes

Splitting a topological space in order to describe its invariants from the invariants of its pieces is a very old idea, due to Mayer and Vietoris. Nevertheless, the algebraic version of this approach for ideals and simplicial complexes, the so-called Betti splitting theory, is relatively new. We say that a homogeneous ideal I admits a Betti splitting I=J+K if the graded Betti numbers of I can be determined in terms of the graded Betti numbers of the ideals J, K and JnK. We present some new results on this topic, generalizing several previous results in the literature and showing that they are suitable for recursive procedures in a combinatorial context. Moreover we get information on the Alexander dual ideals of simplicial complexes, in particular we relate some nice splitting properties of triangulated manifolds with orientability and Discrete Morse Theory.

14.06.2016 um 16:15 Uhr in 69/125:

Ulrich von der Ohe (Universität Osnabrück)

On Prony's method

Motivated by a problem from physics, in 1795 de Prony proposed a method to reconstruct the parameters of an exponential sum, i.e. a linear combination of exponential functions, given a finite set of samples of sufficient cardinality. By his approach the original interpolation problem is translated into the problem of solving a single univariate polynomial equation. Several variants and generalizations of Prony's method have been studied recently. In particular, the multivariate case has been studied. This talk is about a generalization of Prony's method to the case of multivariate exponential sums that is based on systems of multivariate polynomial equations. In particular, we consider some of its algebraic properties. This talk is based on joint work with Stefan Kunis, H. Michael Möller, Thomas Peter, and Tim Römer.

28.06.2016 um 16:15 Uhr in 69/125:

Rainer Sinn (Georgia Institute of Technology)

Positive semidefinite matrix completion, sums of squares, and free resolutions

We will translate the positive semidefinite matrix completion problem into an algebraic geometric setup and use the convex geometry of sums of squares to relate its geometry to combinatorial commutative algebra. This is joint work with Greg Blekherman and Mauricio Velasco.

30.06.2016 um 16:15 Uhr in 69/125:

Francesco Strazzanti (Università di Pisa)

Rigidity properties of local cohomology modules

Given a polynomial ring R and a homogeneous ideal I of R, several authors showed some '``rigi'' behaviours of the Betti numbers of I with respect to those of its lexicographic ideal Ilex. For instance Conca, Herzog, and Hibi proved that if βi(R/I) = βi(R/Ilex) for some i, then the same equality holds for all k ≥ i. We quickly survey some of these properties and talk about similar results on the Hilbert function of local cohomology modules. Among others, in this context we present an analogous statement of the theorem above. This is a joint work with Enrico Sbarra.

21.07.2016 um 18:15 Uhr in 69/125:

Matthias Köppe (University of California, Davis)

Refined asymptotics of lattice point counting functions with LattE integrale

We study sums S(P(b,h) of polynomials h over the lattice points in multi-parameter polytopes P(b)={Ax<=b}, where A is rational and b∈RN is a real parameter vector. On every chamber, S(P(b),h) is a generalized quasi-polynomial function of b, called the (weighted, real, multi-parameter) Ehrhart quasi-polynomial. It is a polynomial in variables bi whose coefficients are periodic functions of the real parameters b_i. We give an algorithm for computing them in a certain closed form ("rational step-polynomials"). This extends to the case of intermediate sums, which interpolate between integrals and discrete sums. Extending work by Barvinok (2006), we show that certain linear combinations of the intermediate Ehrhart quasi-polynomials give an approximation of the Ehrhart quasi-polynomial. This gives an algorithm for computing the coefficients of the terms of the highest k degrees in parameters bi. In various interesting settings for varying dimension but fixed k, it runs in polynomial time. The algorithm is a tool for computing refined asymptotics of counting functions when exact counting is out of reach. If time permits, I will also say a few words about a very fast specialized algorithm for computing refined asymptotics in the case of knapsacks. The talk is based on several papers and the software implementation LattE integrale, joint with Velleda Baldoni, Nicole Berline, Jesús De Loera, Brandon Dutra, and Michèle Vergne.

30.08.2016 um 16:15 Uhr in 69/125:

Satoshi Murai (Osaka University, Japan)

Ring isomorphisms between cohomology rings of certain toric manifolds

Recently, people in toric topology are interested in ring isomorphisms between cohomology rings of toric manifolds. In this talk, I will introduce some problems on this topic and show some partial affirmative answers to them for certain special toric manifolds called Bott manifolds. This is a joint work with Suyoung Choi and Mikiya Masuda.

06.09.2016 um 16:15 Uhr in 69/125:

Anna Bigatti (Università di Genova, Italy)

Implicitization of Hypersurfaces

We present new, practical algorithms for the hypersurface implicitization problem: namely, given a parametric description (in terms of polynomials or rational functions) of the hypersurface, find its implicit equation.
Two of them are for polynomial parametrizations: one algorithm, ``ElimTH'', has as main step the computation of an elimination ideal via a truncated, homogeneous Gröbner basis. The other algorithm, ``Direct'', computes the implicitization directly using an approach inspired by the generalized Buchberger-Möller algorithm.  Either may be used inside the third algorithm, ``RatPar'', to deal with parametrizations by rational functions.
Finally we show how these algorithms can be used in a modular approach, algorithm ``ModImplicit'', for avoiding the high costs of arithmetic with rational numbers. We exhibit experimental timings to show the practical efficiency of our new algorithms.

27.09.2016 um 16:15 Uhr in 69/125:

Dumitur Stamate (University of Bucharest, Romania)

On the Koszul property for numerical semigroup rings

Let H be a numerical semigroup. We give effective bounds for its multiplicity e(H) such that  gr(K[H]) is Koszul.
We conjecture that not all the values in the range are possible, and this correlates to a series of conjectures of Eisenbud, Green and Harris on a Generalized Cayley-Bacharach statement.
We  describe the Koszul property for several classes of numerical semigroups and we study the relationship with the Cohen-Macaulay property of the gr(K[H]).
Joint work with Juergen Herzog.