The 24th Bay Area Discrete Math Day in honor of Bernd Sturmfels' 50th birthday (BERNDMath Day) will take place at the University of California, Berkeley in Evans Hall **Room 10** on Saturday, March 17, 2012 between 9AM and 5:30PM.

### Registration

To register, go here and fill out the form by **March 15**. If you are also attending the (subsidized) dinner (Cha-Am on Shattuck) after the talks, please RVSP by **March 15 **with an email to chillarmsri.org with the subject **MATHRSVP** and body containing your name and total number of people in your party attending the dinner. BADMath Day is
a free
event, so you can still attend if you have not registered.

### Speakers and schedule

9:00-9:30 | Coffee Break | |

9:30-10:15 | Jesus De Loera | Some polyhedral geometry problems from the early 1990's |

10:15-10:30 | Break | |

10:30-11:15 | Diane Maclagan | Tropical hypersurfaces of tropical linear spaces |

11:15-11:45 | Coffee Break | |

11:45-12:30 | Rekha Thomas | Algebra and optimization from computer vision |

12:30-2:30 | Lunch | |

2:30-3:15 | Nick Eriksson | The genetics of human traits |

3:15-3:30 | Break | |

3:30-4:15 | Laura Matusevich | A survey on hypergeometric ranks |

4:15-4:45 | Coffee Break | |

4:45-5:30 | Seth Sullivant | Positive margins and primary decomposition |

6:15PM- | Subsidized Dinner at Cha-Am (1543 Shattuck Ave at Cedar) |

### Transportation and parking

Directions to Berkeley can be found on the UC Berkeley website. If you are driving, we recommend that you park in Lower Hearst Garage or nearby on Euclid and Hearst.

### About BAD Math

BADMath Days are one-day meetings aimed at facilitating communication between researchers and graduate students of discrete mathematics around the San Francisco Bay Area. These days happen twice a year and strive to create an informal atmosphere to talk about discrete mathematics.

The term "discrete mathematics" is chosen to include at least the following topics: Algebraic and Enumerative Combinatorics, Discrete Geometry, Graph Theory, Coding and Design Theory, Combinatorial Aspects of Computational Algebra and Geometry, Combinatorial Optimization, Probabilistic Combinatorics, Combinatorial Aspects of Statistics, and Combinatorics in Mathematical Physics.

Organizing committee: Federico Ardila (SFSU), Andrew Berget (UCD), Tim Hsu (SJSU), Carol Meyers (Lawrence Livermore National Lab), Kelli Talaska (UCB), Rick Scott (Santa Clara U.), Ellen Veomett (Saint Mary's College of California).

Guest Committee Members: Chris Hillar (UCB), Serkan Hosten (SFSU), Lek-Heng Lim (U. Chicago), Seth Sullivant (NCSU), Lauren Williams (UCB).

Support for BAD Math Day is provided by

### Abstracts

(**De Loera**) I first met Bernd Sturmfels in Fall 1990, a mere 4 years since the publication of his first paper in 1986. 22 years and 205 papers later some of the polyhedral mathematics we discussed in Ithaca are still as fascinating as before. I will revisit two old mathematical themes and present a few fresh results and plenty of open questions for the new generation of polytope lovers: (A) Triangulations and subdivisions of polytopes and (B) Diameters of polytopes.

(**Eriksson**) Practically everything is claimed to "run in families". And in fact, over the last few years, there has been an explosion of research in human genetics, with thousands of genetic factors discovered affecting hundreds of diseases and traits. I'll explore two themes that occur in this research. First, how these genetic discoveries shed light on biology (the correlation graph of traits will play a role here). Second, how accurate are predictions of risk based on genetics? How accurate can they be? In honor of Bernd, I'll present some novel results about the genetics of morning-evening preference.

(**Maclagan**) Tropical geometry turns questions about algebraic varieties into polyhedral geometry questions. One of the best understood examples from the tropical perspective is when the variety is the solutions to linear equations. In this case the combinatorics of the associated matroid controls the tropical variety. I will discuss some generalizations of this, addressing what tropical hypersurfaces of a tropical linear space can occur. This is motivated by questions in birational geometry.

(**Thomas**) The triangulation problem in computer vision seeks to reconstruct the 3D-coordinates of a scene from noisy images in n cameras. The underlying mathematical problem is to find a point in the space of

images of the given cameras that is closest to a point outside this set. The first part of this problem involves understanding the equations for the space of images, and the second part involves solving a polynomial optimization problem. In this talk I will describe our results on both parts of this problem. The algebraic first piece is joint work with Chris Aholt and Bernd Sturmfels, and the second optimization piece is joint work with Chris Aholt and Sameer Agarwal.

(**Matusevich**) An A-hypergeometric system is a linear system of PDEs that is constructed from a point configuration A and a parameter vector beta. One fixes A and studies how the system behaves when beta varies. One of the earliest results about such systems concerns their holonomic rank, that is, the dimension of the holomorphic solution space about a nonsingular point. This result states that the holonomic rank of an A-hypergeometric system is bounded below by the normalized volume of the convex hull of A, and moreover, this bound is achieved for generic parameters beta. The fist example of an A-hypergeometric system for which this bound is strict was found by Bernd Sturmfels and Nobuki Takayama in an article that appeared in 1998. Saito, Sturmfels and Takayama have also provided an upper bound for the rank of an A-hypergeometric systems. I will survey what is currently known about hypergeometric ranks, with an emphasis on elementary constructions of examples where ranks are very high.

(**Sullivant**) Markov bases are special subsets of an integer lattice which are used to construct irreducible Markov chains over the lattice points in the family of all polytopes of the form P(b) = { x : Ax = b, x >= 0 }. The Markov basis depends on the matrix A, but not on the vector b, so provides the set of moves for an irreducible Markov chain for all b. While there has been much recent progress in terms of both developing algorithms and theory for computing Markov bases, for many situations it remains hopeless to determine these Markov bases for even moderately sized problems. Perhaps, instead of asking for a Markov basis, we should determine a subset of the Markov basis which works for "most" b? Indeed, for most problems of practical interest, there is a natural choices of "moves" to use, even when we cannot compute the Markov basis. Now the question becomes: which fibers are connected by the given set of moves?

Diaconis, Eisenbud, and Sturmfels proposed a solution to this problem using primary decomposition. I will report on recent work where we pursue this line of attack for the special case of graphical models, where our natural choice of moves comes from the global Markov statements implied by the graph. All the ideals we are able to compute are radical ideals, and, in many cases, we can use the structure of the decomposition to deduce that fibers are connected if all margins are positive. This is joint work with Thomas Kahle and Johannes Rauh.

original design: Andrew Berget