Daniel Erman (University of Wisconsin):

Title:  Syzygies and monomial ideals

Abstract:
These lectures will provide an introduction to the study of syzygies and free resolutions.  We will lay some foundational results about free resolutions, and discuss how syzygies connect to topics in algebraic geometry and combinatorics.  There will be a particular emphasis on the free resolutions of monomial ideals, and the corresponding connections with simplicial complexes.  We will introduce Stanely-Reisner theory, which provides a combinatorial/algebraic dictionary between the simplicial complexes and monomial free resolutions.

We will also provide a glimpse of the many open questions related to syzygies including questions about: the structure of free resolutions, asymptotic behaviors, toric syzygies, and connections with combinatorics.  We will use monomial resolutions to provide a perspective on how to think about some of these overarching questions.


Fatemeh Mohammadi (University of Bristol, UK):

Title: Combinatorics of binomial and toric ideals

Abstract: A toric variety is a certain algebraic variety modeled on a convex polyhedron. Due to combinatorial tools, toric varieties are very well-studied and they play an important role in commutative algebra, algebraic geometry and combinatorics. Their corresponding ideals (toric ideals) are binomial and prime. These lectures will provide an introduction to toric varieties. We will focus more on examples and methods to generate such varieties. We will introduce the Gröbner fan of ideals as a natural way to produce toric ideals as initial ideals (of an arbitrary ideal) with respect to weight vectors.  The basic results on Gröbner degenerations and relations to triangulations of polytopes will be presented.  We also introduce several class of binomial ideals associated to graphs, posets and lattices and study their algebraic and homological properties. The main tools to study these objects are Gröbner basis theory, combinatorics and graph theory.

Throughout the lectures we will point out various open problems and conjectures related to binomial ideals and will try some examples and explicit computations in Macaulay2.

Trygve Johnsen (Norges Arktiske Universitet):

Title: Algebraic-geometric codes.

Abstract: The lecture series will treat various ways in which error-correcting codes can be produced or studied using techniques from algebra, combinatorics and algebraic geometry. We will treat the relations between matroids and linear codes. We will also introduce and study the larger class of almost affine codes and show how some of its properties are determined by matroids. In particular we will show how Stanley-Reisner rings of simplicial complexes, and resolutions of them, various kinds of Betti numbers, determine important properties of the codes. We will also introduce and briefly treat some aspects of algebraic-geometric codes, produced both from curves (Goppa codes) and higher dimensional varieties.


Johan Hansen (Aarhus University, Denmark):

Title: Toric Codes.

Abstract: Toric codes are algebraic geometry codes defined over toric varieties. Thanks to combinatorial properties of such varieties one is able to estimate some of the parameters of the resulting codes, such as its minimum distance. In the course we will highlight some properties of toric varieties that are used in the estimation of the minimum distance, e.g. intersection theory. Moreover, we plan to connect some of the results in toric codes with recent research problems such as Quantum Codes and Secret Sharing Schemes with a large number of players.


Sonja Petrovic (Illinois Institute of Technology):

Title: The role of algebraic statistics in estimation and modeling of random graphs and networks.

Abstract: The ubiquity of network data in the world around us does not imply that the statistical modeling and fitting techniques have been able to catch up with the demand. This series of lectures will discuss some of the basic modeling questions that every statistician knows are fundamental, some of the recent advances toward answering them, and the challenges that remain. The specific focus of the lectures will be on goodness of fit testing for random graph models. More broadly, we will summarize a few lines of research that are intimately connected to discrete mathematics and computer science, where sampling algorithms, hypergraph degree sequences, and polytopes play a crucial role in the general family of statistical models for networks called exponential random graph models.


Seth Sullivant (North Carolina State University):

Title: Algebraic Statistics and Applications to Biology.

Abstract: Tools from algebraic statistics have been successfully applied to address many problems including construction of Markov bases, identifiability problems for graphical models, and theoretical study of phylogenetic mixture models. The focus of this series of lectures will be to develop the basic background on statistical models to illustrate the core concepts of algebraic statistics. A crucial role is played by the family of graphical statistical models. We will then illustrate these ideas with examples of statistical models that are used in computational biology, including phylogenetic models. Beyond the concrete applications, the lectures will also emphasize how the study of problems in statistics and computational biology leads to challenging problems in algebra, and will highlight some open problems.