Algebra meets combinatorics in Neuchatel
15 - 17 July, 2019, Neuchatel
Algebra meets combinatorics in Neuchatel
15 - 17 July, 2019, Neuchatel
Speakers:
Bruno Benedetti (U of Miami)
Daniel Bernstein (M.I.T.)
Mats Boji (KTH, Stockholm)
Alessio Borzi' (U of Catania)
Alexandru Constantinescu (Freie U Berlin)
Alessio D'Alì (U Warwick)
Carlos D'Andrea (U Barcelona)
Emanuela De Negri (U Genova)
Emanuele Delucchi (U of Fribourg)
Ragnar Freij (Aalto, Helsinki)
Ana Garcia (U Graz)
Hiram Lopez (Cleveland State U)
Martina Juhnke-Kubitzke (U of Osnabrück)
Ralph Morrison (Williams College)
Alberto Ravagnani (U College Dublin)
Monday, July 15, 2019
9:00 - Martina Juhnke-Kubitzke
10:00 - Coffee
10:30 - Emanuele Delucchi
11:30 - Bruno Benedetti
12:30 - Lunch
14:00 - Alexandru Constantinescu
15:00 - Emanuela De Negri
Tuesday, July 16, 2019
9:00 - Hiram Lopez
10:00 - Coffee
10:30 - Alberto Ravagnani
11:30 - Ragnar Freij
12:30 - Lunch
14:00 - Carlos D'Andrea
15:00 - Ralph Morrison
Wednesday, July 17, 2019
9:00 - Mats Boji
10:00 - Coffee
10:30 - Alessio Borzì
11:30 - Alessio D'Alì
12:30 - Lunch
14:00 - Daniel Bernstein
15:00 - Ana Garcia
Where. All talks will be in room B103 at Unimail, the math department.
This is at the following address:
Rue Emile Argand 11, 2000 Neuchatel;
You find right below the google maps link:
Monday, July 15, 2019
9:00 - Martina Juhnke-Kubitzke
Graded Betti numbers of balanced simplicial complexes
I discuss upper bounds for the graded Betti numbers of Stanley-Reisner rings of balanced simplicial complexes in various settings, including general balanced simplicial complexes, balanced Cohen-Macaulay complexes such as balanced normal pseudomanifolds.
10:00 - Coffee
10:30 - Emanuele Delucchi
Fundamental polytopes of metric spaces via parallel connection of matroids
Motivated by applications in phylogenetics, we tackle the problem of a combinatorial classification of finite metric spaces via their fundamental polytopes, as suggested by Vershik in 2010. We consider a hyperplane arrangement associated to every split pseudometric and, for tree-like metrics, we study the combinatorics of its underlying matroid. We give explicit formulas for the face numbers of fundamental polytopes and Lipschitz polytopes of all tree-like metrics, and we characterize the metric trees for which the fundamental polytope is simplicial. This is joint work with Linard Hoessly.
11:30 - Bruno Benedetti
Graphs of polytopes
The graph of a polytope is its 1-dimensional frame, made of its vertices and edges. Diameter and connectivity of such graphs are important for applications. We’ll sketch an algebraic approach coming from an ongoing project (with Matteo Varbaro, Michela Di Marca, Barbara Bolognese) that has given some promising results.
12:30 - Lunch
14:00 - Alexandru Constantinescu
Deformations of toric singularities
The possibilities of splitting a lattice polytope with primitive edges into a Minkowski sum of lattice polytopes reflect the deformation theory of the induced toric Gorenstein singularity. This was proved by Altmann in 1996. Our goal is to reveal a similar correspondence for arbitrary toric singularities. To this aim we prove the existence of an initial object in the category of extensions of semigroups, and define a lattice structure inside the cone of Minkowski summands.
15:00 - Emanuela De Negri
Cartwright-Sturmfel Ideals of graphs and linear spaces
In order to study ideals of minors of multigraded matrices of linear forms, and inspired by a work of Cartwright and Sturmfels, we introduced two classes of ideals in a multigraded polynomial ring, named CS and CS*. It turns out that the ideals of these classes are radical, have nice universal Gr\"obner bases and good homological properties.
The aim of this talk is to present other classes of CS and CS* ideals, in particular binomial edge ideals, multi-graded homogenizations of linear spaces, and multiview ideals. This approach allows us to recover and generalize recent results of various authors.
These results have been obtained jointly with Aldo Conca and Elisa Gorla.
Tuesday, July 16, 2019
9:00 - Hiram Lopez
Basic commutative algebra and combinatorics tools for coding theory
In this talk we will give a basic introduction to coding theory. Then, we will show how some basic tools from commutative algebra, like the vanishing ideal or Hilbert function, and basic concepts from combinatorics, like convex polytopes and Minkowski length, can be useful on coding theory.
10:00 - Coffee
10:30 - Alberto Ravagnani
Network Coding and the Combinatorics of Error-Correcting Codes
In the context of network coding, one or multiple sources of information transmit messages to several terminals through a network of intermediate nodes (multicast). In order to maximize the throughput, the nodes are allowed to recombine the received packets before forwarding them towards the sinks.
In this talk, I will first give an introduction to network coding. In particular, I will introduce rank-metric codes as a solution to the issue of error amplification in network transmissions.
The second part of the talk is devoted to mathematical aspects of the theory of error-correcting codes. I will compare rank-metric codes with classical Hamming-metric codes, showing strong divergences in the behaviour of these families with respect to density properties.
In the last part of the talk I will discuss new lines of research intersecting coding theory, combinatorics, and number theory.
11:30 - Ragnar Freij
Privacy, capacity, and lifted codes
In private information retrieval (PIR), a user tries to download an item from a storage system, without revealing any information about the identity of the item he is downloading. this talk, I will give a mathematical introduction to PIR from coded data, and discuss some related qustions that arise in information theory and linear algebra and in the interface between those two fields.
In the first part of the talk, we will prove lower and upper bounds on the capacity of a coded PIR model with collusion, and show that these converge very quickly as the number of items grows. The proof uses combinations of conditional independence statements between vector spaces and probability spaces.
In the second part of the talk, we define the lift of a linear code over an abstract simplicial complex on the same ground set, to be the smallest code whose projection to any simplex agrees with that of the original code. We show that this is not a matroid invariant, and provide some support for the conjecture that it is matroid invariant for generic codes. Finally, time permitting, we will discuss how to maximize the difference between a code and its lift over a given simplicial complex, in various metrics.
12:30 - Lunch
14:00 - Carlos D'Andrea
Sparse resultants: combinatorial properties and Macaulay style formulas
Recently, a geometric definition of the sparse resultant was proposed, which differ from the classical one in an exponent. We study basic properties of this resultant like initial forms, homogeneities, factorization after the vanishing of some of the coefficients of the input system. We present also formula "a la Macaulay" for computing this resultant as the quotient of two determinants of a Sylvester-type matrix, conjectured by Canny and Emiris in the nineties. This is joint work with Gabriela Jeronimo and Martin Sombra.
15:00 - Ralph Morrison
The moduli space of tropical curves with fixed Newton polygon
In tropical geometry, a polynomial in two variables defines a curve in the plane, just like in algebraic geometry. Unlike a classical curve, however, a tropical curve is a piecewise-linear object that embeds into the plane in a balanced, polyhedral way. In both worlds, we can choose a lattice polygon and look at a moduli space: the moduli space of non-degenerate algebraic curves with that Newton polygon (as studied by Castryck and Voigt), and the moduli space of graphs arising in tropical curves with that Newton polygon. We'll establish a strong connection between these two moduli spaces, in particular that they have the same dimension. This talk includes joint work with Desmond Coles, Neelav Dutta, Sifan Jiang, and Andrew Scharf.
Wednesday, July 17, 2019
9:00 - Mats Boji
Waring rank and SLP for annihilators of symmetric forms
In a joint work with Migliore, Miró-Roig and Nagel, we study Gorenstein algebras defined by the annihilators of symmetric forms. In particular we obtain a complete picture of what happens for symmetric cubics in terms of the Strong Lefschetz Property (SLP) and the Waring rank. We also prove that complete symmetric forms of any degree give algebras satisfying the SLP and we provide a surprisingly short power sum decomposition for them.
10:00 - Coffee
10:30 - Alessio Borzì
Face poset and Grothendieck-Tutte polynomial of matroids over a domain
We present some algebraic and combinatorial invariants of matroids over a domain, in the sense of Fink and Moci. For instance, we define a face poset for these matroids, that results to be a disjoint union of isomorphic simplicial posets. We define a Grothendieck-Tutte polynomial that partly generalizes the definition of Fink and Moci. This polynomial satisfy the classical deletion-contraction property. Further, when the associated simplicial poset is finite, following Stanley, we can consider its face ring, then we show that the Hilbert series of this face ring is a specialization of the Grothendieck-Tutte polynomial.
This is a joint work with Ivan Martino.
11:30 - Alessio D'Alì
Symmetric edge polytopes in combinatorics and beyond
Symmetric edge polytopes are certain polytopes defined from the data of a finite simple graph. In the present talk we introduce some of the pleasant combinatorial properties of these objects and explore some surprising connections to the Kuramoto synchronization model in physics and to the theory of finite metric spaces. This is joint ongoing work with E. Delucchi and M. Michalek.
12:30 - Lunch
14:00 - Daniel Bernstein
The tropical Cayley-Menger variety
Varieties that arise in the algebraic study of matrix completion come embedded in a vector space whose coordinates are indexed by some graph. A recurring problem is to find combinatorial descriptions, in terms of these graphs, of the algebraic matroids underlying these varieties. Tropical geometry can be used to solve such problems. In this talk, I will show that the tropicalization of the Cayley-Menger variety of points in the plane has a simplicial complex structure that can be described in terms of rooted trees. Then, I will show how one can use this to obtain a new proof of Laman's theorem, a celebrated theorem from rigidity theory giving a combinatorial description of the algebraic matroid underlying the Cayley-Menger variety of points in the plane.
15:00 - Ana Garcia
Frieze mutations
Frieze is a lattice of positive integers satisfying certain rules. Friezes of type A were first studied by Conway and Coxeter in 1970’s, but they gained fresh interest in the last decade in relation to cluster algebras, that are generated via a combinatorial rule called mutation. Moreover, the categorification of cluster algebras developed in 2006 yields a new realization of friezes in terms of representation theory of Jacobian algebras. Thus, a frieze is an array of positive integers on the Auslander-Reiten quiver of a Jacobian algebra such that entires on a mesh satisfy a unimodular rule. In this talk, we will discuss friezes of type D and their mutations. This is joint work with K. Serhiyenko (UC Berkeley).
How to get to the seminar room from the train station.
You may want to walk:
You cal also take the bus 107 from the stop of the train station and exit at "Portes rouges".
From Bern to Neuchatel.
You may want to walk :) I am kidding :)
You need to take the train. There is a 38 minutes connection every hour.
Remember to validate your ticket even if you buy it on line on the app.
Here the link to the app.