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# Geometric Methods in Representation Theory Seminar – Joseph Landsberg (TAMU)

## August 25, 2023 @ 4:00 pm - 5:00 pm

**Linear spaces of matrices of bounded rank**

**Abstract**. A classical problem in linear algebra is to classify linear spaces of matrices such that no element of the space has full rank. Work of Eisenbud and Harris showed that the problem may be rephrased in terms of classifying sheaves on projective space with certain properties. 40 years ago spaces of bounded rank at most three were classified and there have been interesting, isolated examples of spaces discovered that are related to well-studied objects in algebraic geometry such as instanton bundles, but there had been no progress on the classification problem. Motivated by questions in theoretical computer science and quantum information theory, H. Huang and myself revisited this problem. Using methods from algebraic geometry and commutative algebra, we classified spaces of bounded rank four.