Date:

Mon, 03/06/201911:00-13:00

Location:

CS Rothberg bldg, room B-500, Safra campus

First talk:

Speaker: Madeleine Weinstein (Berkeley)

Title: Voronoi Cells of Varieties

Abstract:

Every real algebraic variety determines a Voronoi decomposition of its ambient Euclidean space. Each Voronoi cell is a convex semialgebraic set in the normal space of the variety at a point. We compute the algebraic boundaries of these Voronoi cells. Using intersection theory, we give a formula for the degrees of the algebraic boundaries of Voronoi cells of curves and surfaces. We discuss an application to low-rank matrix approximation. This is joint work with Diego Cifuentes, Kristian Ranestad, and Bernd Sturmfels.

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Second talk:

Speaker: Madeline Brandt (Berkeley)

Title: Symmetric powers of algebraic and tropical curves

Abstract:

Recently, tropical geometry has emerged as a tool for studying classical moduli spaces by associating to every variety a polyhedral complex which comes as its non-Archimedean skeleton. Classically, it is known that the d-th symmetric power of a smooth, projective algebraic curve X is again a smooth, projective algebraic variety which functions as the moduli space of effective divisors of degree d on X. In this talk, I will discuss two ways to tropicalize this statement. The first way is to take the d-th symmetric power of the tropicalization of X, and the second is to tropicalize the d-th symmetric power of X itself. In recent work with Martin Ulirsch, we show that in fact the two agree. I will present all necessary definitions for understanding the above statement and I will sketch the proof.

Speaker: Madeleine Weinstein (Berkeley)

Title: Voronoi Cells of Varieties

Abstract:

Every real algebraic variety determines a Voronoi decomposition of its ambient Euclidean space. Each Voronoi cell is a convex semialgebraic set in the normal space of the variety at a point. We compute the algebraic boundaries of these Voronoi cells. Using intersection theory, we give a formula for the degrees of the algebraic boundaries of Voronoi cells of curves and surfaces. We discuss an application to low-rank matrix approximation. This is joint work with Diego Cifuentes, Kristian Ranestad, and Bernd Sturmfels.

-----------------------

Second talk:

Speaker: Madeline Brandt (Berkeley)

Title: Symmetric powers of algebraic and tropical curves

Abstract:

Recently, tropical geometry has emerged as a tool for studying classical moduli spaces by associating to every variety a polyhedral complex which comes as its non-Archimedean skeleton. Classically, it is known that the d-th symmetric power of a smooth, projective algebraic curve X is again a smooth, projective algebraic variety which functions as the moduli space of effective divisors of degree d on X. In this talk, I will discuss two ways to tropicalize this statement. The first way is to take the d-th symmetric power of the tropicalization of X, and the second is to tropicalize the d-th symmetric power of X itself. In recent work with Martin Ulirsch, we show that in fact the two agree. I will present all necessary definitions for understanding the above statement and I will sketch the proof.