Two algebra and combinatorics seminars today

[Engström Alexander]

Dear all,

My guests Thomas Kahle from ETH Zürich and Maria Angelica Cueto from Columbia University will speak at todays seminar.

Best,

Alex

Speaker:

   Thomas Kahle, ETHZ

Time:

   Today, 13:15-15:00, in U510

Title:

   Positive margins and primary decompositions

Abstract:

   In algebraic statistics one uses Markov bases to assess the goodness of fit in graphical and other log-linear models.  Even if Markov bases are known, for many models they are too big to be practical.  We study the consequences of replacing a Markov basis of a graphical model by a simple set of moves coming from conditional independence statements.  While those moves do not connect all fibers, they may still be sufficient to connect all fibers satisfying certain linear constraints on the margins, for example strict positivity.  This is the positive margins property and it can be studied systematically using decompositions of binomial conditional independence ideals.  This method allows to clarify the positive margins property for graphical models of decomposable graphs, cycles, and some complete bipartite graphs.  (Joint work with Johannes Rauh and Seth Sullivant)

Speaker:

   Maria Angelica Cueto, Columbia

Time:

   Today, 15:15-17:00, in U510

Title:

   Mixed discriminants

Abstract:

   The mixed discriminant of Laurent polynomials in n variables with fixed support is the irreducible polynomial in the coefficients which vanishes whenever two of the roots coincide. It represents the variety of ill-posed systems. By means of the Cayley trick, we can express the mixed discriminant as an A-discriminant in the sense of Gelfand, Kapranov and Zelevinski. Our goal is to characterize its degree. I will discuss in detail the case of two plane curves where an explicit degree formula can be provided. In the case of two dense polynomials, this formula recovers the classical tact invariant of Salmon. Finally, inspired by the tropical approach to computing A-discriminants, I will show that the degree of the mixed discriminant is a piecewise linear function in the Plucker coordinates of a mixed Grassmannian. This is joint work with E. Cattani, A. Dickenstein, S. Di Rocco and B. Sturmfels (arXiv:1112.1012v1)