Pizza + Posters
Gaussoids are combinatorial structures generalizing conditional independence inference rules among Gaussian random variables, similar to how matroids generalize linear independence. Since their introduction in 2007 by Lněnička and Matúš, combinatorial results about gaussoids have been conﬁned to dimension ≤ 5.
Using an embedding of N-gaussoids into the N-dimensional cube, we show constructively that the logarithm of the number of gaussoids is asymptotically between n2n and n22n. This is considerably larger than the number of “statistically relevant” gaussoids — those realizable by a Gaussian distribution. Their number is known to the same degree of accuracy: the logarithm is asymptotically between n2 and n3.
It has been shown by Soprunov that the lattice mixed volume
of an n-tuple of n-dimensional lattice polytopes (minus one)
is a lower bound for the number of interior lattice points in the Minkowski sum of the polytopes. Tuples of mixed degree one are exactly those n-tuples for which this upper bound is attained. A generic class of examples is given by n-tuples admitting a projection sending all polytopes onto the same unimodular (n-1)-simplex. We show that, for each dimension n greater than three, all but finitely many n-tuples of mixed degree one are of this generic type. Furthermore we classify n-tuples of mixed degree one for n=3, showing the existence of infinitely many non-generic examples. This is joint work with Gabriele Balletti.
The signature of a path is a sequence of tensors, encoding its geometric properties. The signatures of a given class of paths form a subvariety of the space of tensors, whose geometry reflects the features of the paths. We will focus on the class of rough path, very important in stochastics. Their signature variety shows remarkable similarities to the Veronese variety.
We apply tools from real algebraic geometry to the problem of multistationarity of chemical reaction networks whose positive steady states admit a monomial parametrization. In particular, we show that in the space of total concentrations there is multistationarity for some value of the total concentrations if and only if there is multistationarity on the entire ray containing this value (possibly for different rate constants). Moreover, for these networks it is possible to decide about multistationarity by formulating semialgebraic conditions that involve only total concentrations. Hence quantifier elimination may give new insights into multistationarity regions in the space of total concentrations. To demonstrate this, we show that for the distributive phosphorylation of a protein at two binding
sites multistationarity is only possible if the total
concentration of the substrate is larger than either the
concentration of the kinase or the phosphatase. This result is enabled by the chamber decomposition from polyhedral geometry. This is a joint work with Carsten Conradi and Thomas Kahle.
References: C. Conradi, A. Iosif, T. Kahle, Multistationarity in the space of total concentrations
for systems that admit a monomial parametrization arXiv:1810.08152.
A simplicial arrangement is a ﬁnite set of hyperplanes in ℝℓ which cuts simplicial cones out of the ambient space. For example all Coxeter arrangements, i.e. the reﬂection arrangements of the ﬁnite real reﬂection groups are simplicial. At least in rank 3 there is a list of irreducible simplicial arrangements conjectured to be complete. But a classiﬁcation of these natural geometric objects remains an open problem.
A further important class in the theory of hyperplane arrangements are the supersolvable arrangements which possess nice algebraic, geometric and combinatorial properties.
We give a classiﬁcation of all supersolvable simplicial arrangements using tools inspired by M. Cuntz and I. Heckenberger’s recent work on crystallographic arrangements.
This is joint work with Michael Cuntz.
We will see that functions that count substructures generate algebras if we are given a combinatorial presheaf. These have been studied for permutations and graphs, and here we will observe that the algebra obtained by marked permutations is in fact free.
The components of the irreducible monomial primary decomposition of an edge ideal of a simple hypergraph are completely determined by the minimal covers of the hypergraph. In this talk, we will introduce the concept of edge ideal of a weighted oriented graph. We will study the irreducible monomial primary decomposition of these ideals. For get it, we introduce the concept of strong cover and we prove that these determine the components of the primary decomposition. We will study also when these edge ideals are unmixed and ﬁnally we will study the Cohen-Macaulay property for these ideals.
A fewnomial system in n variables with t terms is a polynomial system f1(x) = 0,…,fn(x) = 0 of equations where each polynomial fi in the system has a prescribed set of monomials described by a set A ⊆ ℕn of cardinality t. In 1980, Khovanskii proved that such a system with real coeﬃcients has at most
nondegenerate positive real solutions. It is important to note that this bound does not depend on the degrees of the polynomials. It is conjectured that this bound can be improved to be polynomial in t. However, all existing improvements are still exponential in t.
In this talk, we consider a probabilistic version of this conjecture. We show that the expected number of nondegenerate positive real solutions of a random fewnomial system in n variables with t terms is bounded by
which is polynomial in t.
This is joint work with Peter Bürgisser and Alperen A. Ergür.
(This is a joint talk with Alexandros Grosdos Koutsoumpelias.)