Events for 08/19/2024 from all calendars

Geometry Seminar

Time: 3:00PM - 4:00PM

Location: BLOC 302

Speaker: A. Conner, Harvard

Title: Geometry and the complexity of matrix multiplication

Abstract: Strassen's well known algorithm for multiplying matrices is at core a certain method to multiply 2 by 2 matrices using 7 multiplications. The data describing this method is equivalently an expression to write the structure tensor of the 2 by 2 matrix algebra as a sum of 7 decomposable tensors. Any such decomposition of an n by n matrix algebra yields a Strassen type algorithm, and Strassen showed that one essentially cannot do better than algorithms coming from such decompositions. Bini later showed all of the above remains true when we allow the decomposition to depend on a parameter and take limits. I discuss a technique for lower bounds for this decomposition problem, border apolarity. Two key ideas to this technique are (i) to not just look at the sequence of decompositions, but the sequence of ideals of the point sets determining the decompositions and (ii) to exploit the symmetry of the tensor of interest to insist that the limiting ideal has an extremely restrictive structure. I discuss its applications to the matrix multiplication tensor and other tensors potentially useful for obtaining upper bounds via Strassen's laser method.

Student/Postdoc Working Geometry Seminar

Time: 4:00PM - 5:00PM

Location: BLOC 302

Speaker: JM Landsberg, Texas A&M

Title: Tensors of minimal border rank: new components of the boundary

Abstract: The set of (m,m,m) tensors of minimal border rank is an orbit closure. What lies on the boundary? I'll discuss what is known and work in progress with F. Gesmundo