The aim of the workshop "Combinatorial Methods in Algebraic Geometry and Commutative Algebra", organized by the MPI MiS and the University of Leipzig, is to give an overview of the recent developments and challenging questions in this research area, as well as to bring together scientists who are working on this subject.

The central curve of a linear program is an algebraic curve specified by the associated hyperplane arrangement and cost vector. This curve is the union of the various central paths for minimizing or maximizing the cost function over any region in this hyperplane arrangement. Here we will discuss the algebraic properties of this curve and its beautiful global geometry, both of which are controlled by the corresponding matroid of the hyperplane arrangement.

Spectrahedra are the feasible sets of semidefinite programming. An important step in classifying spectrahedra consists of writing polynomials as determinants of symmetric linear matrix polynomials. It turns out that there is an interesting relationship between such determinantal representations, and sums of squares decompositions of the Hermite matrix of the polynomial. I will explain this fact, which is recent work with Daniel Plaumann and Andreas Thom.

A celebrated result of Kodiyalam and Cutkosky, Herzog and Trung asserts that the regularity (in the sense of Castelnuovo-Mumford) of the powers of an ideal in a standard graded ring over a field is eventually a linear function of the power. In this talk, I will explain results with Amir Bagheri and Tai Ha showing that the shifts in the minimal free resolutions of powers also have such a type of linear behavior. Also that this holds in a pretty general context.

Let $K$ be a field and let $A \subseteq B\subseteq \mathbb N^d$ be affine semigroups such that the corresponding cones are equal, \mbox{i.\,e.}, $C(A)=C(B)$. We will discuss how $K[B]$ can be decomposed into a direct sum of monomial ideals in $K[A]$; this is a generalization of an idea by Hoa and Stückrad. In case that $B$ is homogeneous, we can use this to compute the Castelnuovo-Mumford regularity of $K[B]$. Since the decomposition can be computed very effectively by our Macaulay2 package MonomialAlgebras.m2, the regularity computation with our package in high codimension is much faster than the usual computation by using its free resolution. This enables us to test the Eisenbud-Goto conjecture for affine semigroup rings with high codimension. This is a joint work with Janko Böhm and David Eisenbud.Moreover, I will explain how this decomposition has led to positive answers for the E-G conjecture in case that $B$ is simplicial, including a new combinatorial proof for arbitrary monomial curves, the seminormal case, and the case of at most two elements.

The boundary of the convex hull of a compact algebraic curve in real 3-space defines a real algebraic surface. For general curves, that boundary surface consists of tritangent planes and a scroll of stationary bisecants. We express the degree of this surface in terms of the degree, genus and ingularities of the curve, we show how to compute its defining polynomials. We also discuss extensions to higher dimensions. This is based on work with Kristian Ranestad.

The coefficients of a degree $d$ homogeneous polynomial may be arranged in a $d$-dimensional matrix. Analogous to the determinant of a matrix, Cayley introduced the notion of the hyperdeterminant of a multi-dimensional matrix, and we consider this hyperdeterminant applied to a polynomial. In this talk we will describe some of the beautiful symmetry, geometry and combinatorics of this symmetrized hyperdeterminant. In particular we will give a geometric description via dual varieties, and we will use this to interpret the combinatorial formula for the degree of the hyperdeterminant.

We report on joint work with Chr.~Krattenthaler and J. Uliczka.Stanley decompositions of multigraded modules $M$ over polynomials rings have been discussed intensively in recent years. There is a natural notion of depth that goes with a Stanley decomposition, called the Stanley depth. Stanley conjectured that the Stanley depth of a module $M$ is always at least the (classical) depth of $M$. In this paper we introduce a weaker type of decomposition, which we call Hilbert decomposition, since it only depends on the Hilbert function of $M$, and an analogous notion of depth, called Hilbert depth. Since Stanley decompositions are Hilbert decompositions, the latter set upper bounds to the existence of Stanley decompositions. The advantage of Hilbert decompositions is that they are easier to find. We test our new notion on the syzygy modules of the residue class field of $K[X_1,\dots,X_n]$ (as usual identified with $K$). Writing $M(n,k)$ for the $k$-th syzygy module, we how that the Hilbert depth of $M(n,1)$ is $\lfloor(n+1) / 2\rfloor$. Furthermore, we show that, for $n > k \ge \lfloor n / 2\rfloor$, the Hilbert depth of $M(n,k)$ is equal to $n-1$. We conjecture that the same holds for the Stanley depth. For the range $n / 2 > k > 1$, it seems impossible to come up with a compact formula for the Hilbert depth. Instead, we provide very precise asymptotic results as $n$ becomes large.

Let $S$ be a polynomial ring over a field $K$ and let $R=S / I$ be a standard graded $K$-Algebra where $I$ is a graded ideal of $S$. The Multiplicity Conjecture of Herzog, Huneke, and Srinivasan which was recently proved using the Boij-Söderberg theory states that the multiplicity of $R$ is bounded above by a function of the maximal shifts in the minimal graded free resolution of $R$ over $S$ as well as bounded below by a function of the minimal shifts if $R$ is Cohen-Macaulay. We present results to the related problem that the total Betti-numbers of $R$ are also bounded above by a function of the shifts in the minimal graded free resolution of $R$ as well as bounded below by another function of the shifts if $R$ is Cohen-Macaulay.

The Jukes Cantor binary model associates to a trivalent tree a projective toric variety, or equivalently a lattice polytope. This phylogenetic model has been studied by Sturmfels, Sullivant, Buczyńska, Wiśniewski, Xu. Buczyńska generalized this construction to trivalent graphs, in which case a graded affine semigroup is associated to a trivalent graph. In this talk we discuss how the first Betti number of a trivalent graph is related to the degree of minimal generation of the associated semigroup. Specifically, the semigroup of a trivalent graph with the first Betti number equal to $g$ is minimally generated in degree less or equal to $g+1$. Furthermore, there are many graphs for which the bound is effective. The caterpillar graph with $g$ cycles is generated in degree $g+1$, for $g$ even, and in degree $g$, for $g$ odd.This talk is based on the joint work with Weronika Buczyńska, Jarosław Buczyński, Mateusz Michalek.

A multivariate real polynomial $p$ is nonnegative if $p(x)\geq0$ for all $x\in\mathbb R^n$. I will review the history and motivation behind the problem of representing nonnegative polynomials as sums of squares. Such representations are of interest for both theoretical and practical computational reasons. I will explain that the difference between nonnegative polynomials and sums of squares comes from Gorenstein ideals with special structure, which I will call positive Gorenstein ideals. Although the problem of nonnegative polynomials and sums of squares retains real flavor, this makes it amenable to the more familiar tools from complex algebraic geometry and commutative algebra. I will present some applications of these ideas.

It is well known that the classical discrete logarithm problem (the problem to compute indices modulo prime numbers) can be solved in subexponential expected time.\\In contrast, it is not known whether the discrete logarithm problem in the groups of rational points of elliptic curves over finite fields (the elliptic curve discrete logarithm problem) can be solved in subexponential expected time. Indeed, it was the lack of an obvious algorithm for this computational problem which was faster than "generic" algorithms which lead Miller and Koblitz to suggest the use of the problem for cryptographic applications.\\In 2004 Gaudry gave a randomized algorithm with which one can - under some heuristic assumptions - solve the elliptic curve discrete logarithm problem over all finite fields with a fixed extension degree at least 3 faster than with generic algorithms. By using a similar algorithm, I have shown that there exists a sequence of finite fields (of strictly increasing cardinality) over which the elliptic curve discrete logarithm problem can be solved in subexponential time.\\Recently, I have been able to extend the result in such a way that now for more families of finite fields the elliptic curve disctrete logarithm problem can be solved in subexponential expexted time. A corresponding algorithm and its analysis will be outlined in the talk.

Let $R$ be a commutative ring of characteristic $p>0$ and $M$ an $R$-module. A $p^e$-linear map $\varphi_e:M\to M$ is an additive map that satisfies $\varphi_e(rm) = r^{p^e}\varphi_e(m)$ for all $r\in R, m\in M$. The set of all $p^e$-linear maps $\mathcal{F}^e (M)$ is an abelian group. Since composing a $p^e$-linear map and a $p^{e'}$-linear map is a $p^{e+e'}$-linear map, this lead G. Lyubeznik and K. Smith to define in \textit{On the commutation of the test ideal with localization and completion}, Trans. Amer. Math. Soc. 353 0$, let $E_R$ be the injective envelope of its residue field. Then, we may give a precise description of $\mathcal{F}(E_R)$ that shows that this algebra can only be principally generated or infinitely generated depending on the minimal primary decomposition of $I$. Examples will be given showing that one may find both non Cohen-Macaulay ideals with principally generated Frobenius algebra and Cohen-Macaulay ideals with infinitely generated Frobenius algebra. As an application, we will see that independently of the finite or non finite character of the Frobenius algebra $\mathcal{F}(E_R)$, its Matlis dual algebra $\mathcal{C}(R)$ of Cartier operators is always gauge bounded, a notion introduced by M. Bilckle in Test ideals via algebras of $p^{-e}$-linear maps, arXiv/0912.2255, which implies that the set of $F$-jumping numbers of the corresponding generalized test ideals is always a discrete set.This is a joint work with Josep Àlvarez Montaner and Alberto Fernàndez Boix.

The talk is devoted to the families of lattice polytopes with a fixed number of interior lattice points. It is well-known that such families are important for algebraic geometry. I will present recent results motivated by the work of Hensley, Lagarias, Ziegler, Lawrence and Pikhurko. In particular, I will present new upper bounds on the size of lattice simplices with a single interior lattice point.

Every binomial ideal in a monoid algebra induces a congruence on the monoid. Decomposing the induced congruence is a fair approximation of a primary decomposition of a binomial ideal. We will present a decomposition theory of congruences that remedies many of the deficits of primary decomposition of commutative monoid congruences. Lifting to the monoid algebra produces a decomposition theory of binomial ideals that works over non-algebraically closed fields.

There is a purely combinatorial description of toric DM stack, which is very similar to the description in the case of varities by a fan. I will give a short review of the construction as given in an article by L. Borisov, L. Chen, and G. Smith. In doing so, I will focus on a simple example: the moduli space of elliptic curves.

Participants

Gennadiy Averkov

University of Magdeburg, Germany

Greg Blekherman

University of California, USA

Janko Böhm

Universität des Saarlandes, Germany

Winfried Bruns

Universität Osnabrück, Germany

Marc Chardin

CNRS, France

Mohammad Dashtpeyma

Universität Regensburg, Germany

Claus Diem

Universität Leipzig, Germany

Majid Eghbali

Martin-Luther University Halle-Wittenberg, Germany

Hans-Gert Gräbe

Univ Leipzig, Germany

Andreas Hochenegger

FU Berlin, Germany

Danijela Horak

Max-Planck-Institut für Mathematik in den Naturwissenschaften, Germany

Bobo Hua

Max-Planck-Institut für Mathematik in den Naturwissenschaften, Germany

Jürgen Jost

Max-Planck-Institut für Mathematik in den Naturwissenschaften, Germany

Michael Joswig

TU Darmstadt, Germany

Thomas Kahle

Institut Mittag-Leffler, Sweden

Franz Király

TU Berlin, Germany

Kaie Kubjas

FU Berlin, Germany

Paul Larsen

Humboldt-Universität zu Berlin, Germany

Michael Le Barbier Grünewald

HCM Bonn, Germany

Matthias Lenz

Technische Universität Berlin, Germany

Benjamin Lorenz

Goethe-Universität Frankfurt am Main, Germany

Antonio Macchia

Università degli Studi di Bari, Italy

Madhusudan Manjunath

Saarland University, Germany

Guido Montúfar

Max-Planck-Institut für Mathematik in den Naturwissenschaften, Germany

Julio José Moyano-Fernández

Universität Osnabrück, Germany

Tim Netzer

Universität Leipzig, Germany

Marco Neumann

Universität Leipzig, Germany

Max Nitsche

Max-Planck-Institut für Mathematik in den Naturwissenschaften, Germany

Luke Oeding

Università degli Studi di Firenze, Italy

Philipp-Jens Ostermeier

Max-Planck-Institut für Mathematik in den Naturwissenschaften, Germany

Andreas Paffenholz

TU Darmstadt, Germany

Markus Perling

Ruhr-Universität Bochum, Germany

Daniel Plaumann

Universität Konstanz, Germany

Jing Qin

Max-Planck-Institut für Mathematik in den Naturwissenschaften, Germany

Johannes Rauh

Max-Planck-Institut für Mathematik in den Naturwissenschaften, Germany

Jose Rodriguez

University of California at Berkeley, USA

Tim Römer

Universität Osnabrück, Germany

Camilo Sarmiento

Max-Planck-Institut für Mathematik in den Naturwissenschaften, Germany

Peter Schenzel

Martin-Luther University Halle-Wittenberg, Germany

Moritz Schmitt

FU Berlin, Germany

Konrad Schmüdgen

Universität Leipzig, Germany

Martina Scolamiero

Politecnico di Torino, Italy

Abel Stolz

Universität Leipzig, Germany

Jürgen Stückrad

Universität Leipzig, Germany

Bernd Sturmfels

University of California, USA

Lorenzo Taggi

Max-Planck-Institut für Mathematik in den Naturwissenschaften, Germany

Naoki Terai

Saga University, Japan

Danny Tobisch

Universität Leipzig, Germany

Tat Dat Tran

Max-Planck-Institut für Mathematik in den Naturwissenschaften, Germany

Cynthia Vinzant

University of Michigan, USA

Matthias Waack

Universität Leipzig, Germany

Santiago Zarzuela

Universitat de Barcelona, Spain

Piotr Zwiernik

Mittag-Leffler Institute, Sweden

Scientific Organizers

Jürgen Jost

Max-Planck-Institut für Mathematik in den Naturwissenschaften

Jürgen Stückrad

Universität Leipzig

Administrative Contact

Max Nitsche

Max-Planck-Institut für Mathematik in den Naturwissenschaften
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Antje Vandenberg

Max-Planck-Institut für Mathematik in den Naturwissenschaften
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