# Mathematics Colloquia and Seminars

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### Ranks of tropical matrices

**Algebra & Discrete Mathematics**

Speaker: | Melody Chan, UC Berkeley |

Location: | 2112 MSB |

Start time: | Fri, Apr 9 2010, 11:00AM |

What is the rank of a tropical matrix? There are many different sensible definitions of rank in the tropical setting. We will begin by examining two of them, Kapranov and tropical rank, and prove that they agree for 5 x n matrices. In doing so, we compute the moduli space of n labeled coplanar points in tropical projective 4-space. We also discuss several natural notions of Barvinok rank on symmetric matrices. Our study is essentially one of tropical secant varieties. Our techniques are drawn from graph and hypergraph theory, yet they yield some decidedly non-combinatorial results. For example, we give a tropical computation of the dimensions of secant varieties of the Grassmannian Gr(2,n). Joint work with Anders Jensen and Elena Rubei, and with Dustin Cartwright. PLEASE NOTE CHANGED ROOM (2112) THIS WEEK.