TATERS

Symplectic topology gives a midpoint between smooth topology and complex algebraic geometry. We will look at commonalities and divergences between symplectic topology and complex algebraic geometry in the setting of curves in the complex projective plane (to a topologist, surfaces in a 4-manifold).

The p-adic numbers are a particular completion of the rational numbers that has applications to number theory. Unlike the reals, they do not have a nice topology, but one can still do geometry over them. In this talk, we will discuss a notion of integration, due to Coleman, on algebraic curves over the p-adic numbers and some of its number-theoretic consequences. Time permitting, we will discuss extensions to bad reduction curves.

Abstract: Linear algebra is the foundation to methods for finding structure in matrix data. There are many challenges in extending this to the multi-linear setting of tensors. I will discuss the comparison of rank and symmetric rank for a tensor, including recent progress in finding tensors whose rank and symmetric rank differ (in joint work with Kexin Wang, building on work of Yaroslav Shitov). We will see connections to classical algebraic geometry, via cubic surfaces, and small open problems of a combinatorial nature.

Free modules are, in many respects, the simplest kinds of modules over a ring. A free resolution of a module M is an approximation of M in terms of the simple building blocks given by free modules. In this talk, I’ll discuss a geometric generalization of the notion of a free resolution, due to Berkesch-Erman-Smith, called a virtual resolution, and some open questions about them. 

Projective space, rational maps, and other notions from algebraic geometry appear naturally in the study of image formation and various applications of computer vision such as 3D reconstruction. In the algebraic study of vision, polynomial constraints encoded in the multiview ideal have so far been the stars of the show, owing in large part to their application to the task of triangulation. However, the analogous constraints associated with other tasks, namely resectioning and bundle adjustment, have not received such systematic attention. In joint work with Sameer Agarwal, Max Lieblich, and Rekha Thomas, we seek to remedy this situation by introducing an Atlas of the Pinhole Camera which catalogues a rich network of algebraic varieties and their vanishing ideals connected to these various tasks. To show the utility of this framework, we characterize various “generalized multiview ideals” associated with the bundle adjustment problem, which are interesting objects in their own right. Some nice previous results about multiview ideals also fall out as corollaries from our framework. For example, we give a new proof of a result by Aholt, Sturmfels, and Thomas that the multiview ideal has a universal Groebner basis consisting of k-focals (also known as k-linearities in the vision literature) for k in {2,3,4}. 

Tropical mathematics uses a min-plus algebra rather than the normal plus-times algebra. A strength of tropical geometry is that it allows us to look at a "linear" skeleton of a potentially complicated variety, reducing algebro-geometric questions to those of combinatorics. A strong trend in modern algebraic geometry is the study of moduli (parameter) spaces. Broadly, a moduli space parameterizes geometric objects, and we can define algebraic and tropical moduli spaces independently. My research investigates tropicalization questions involving moduli spaces of curves, that is, which algebraic moduli spaces "tropicalize" to their tropical counterparts.

In this talk, I will give a brief introduction to tropical mathematics and moduli spaces. I will also present a classification statement for moduli spaces of rational graphically stable curves that tropicalize "nicely". No prior knowledge of moduli spaces or tropical geometry is necessary. 

Border apolarity is a new and powerful tool that gives us much information about the border rank of a lot of tensors of interest. I will talk about two results about applications of it involving different notions of tensor rank and the border rank of 3 \times 3 permanent. The first result relates different notions of tensor rank to polynomials of vanishing Hessian. The second one computes the border rank of 3 \times 3 permanent, which is isomorphic to the second Kronecker power of a small Coppersmith-Winograd tensor. This tensor is of interest in the study of the exponent of matrix multiplication complexity. These are joint works with Emanuele Ventura, JM Landsberg, Austin Conner, and Mateusz Michalek.

Abstract:  The classical Beckman-Quarles theorem states that a transformation F of a Euclidean space of dimension two or higher which satisfies ||F(x)-F(y)||=1 whenever ||x-y||=1 is necessarily an isometry: ||F(x)-F(y)||=||x-y|| for all points x and y.  That is, a transformation preserving distance 1, preserves all distances. I'll discuss a conjectural generalization of this theorem for Riemannian manifolds, some supporting theorems, and will highlight the role of convex sets in the proofs.  The talk is based on collaborative work with Meera Mainkar (Central Michigan University).

Abstract:  A group is said to be bi-orderable if its elements enjoy a strict total ordering invariant under both left and right multiplication. Free groups and free abelian groups are examples of bi-orderable groups.  Via the Artin representation, an n-strand braid corresponds to an automorphism of the rank n free group $F_n$.  If there is a bi-ordering of $F_n$ preserved by the automorphism, we’ll say the braid is order-preserving.  We apply order-preserving braids to study pairs of minimal volume cusped hyperbolic 3-manifolds.  For example, for two cusps there are two inequivalent minimal examples: one has bi-orderable frundamental group while the other does not. A similar phenomenon happens to the two distinct one-cusped examples.  We also consider higher-cusped examples.  This is joint work with Eiko Kin.

The Springer fiber of a square matrix X is a kind of generalization of an eigenspace: instead of looking at lines and asking which are fixed by X, we look at lines contained in planes contained in 3-dimensional linear spaces contained in (etc.), and ask when each of those nested linear subspaces is fixed by X.  Springer fibers have beautiful geometry and also can be described very concretely in terms of the underlying linear algebra.  At the same time, they are connected to deep mathematics: one of the classical examples of geometric representation theory shows that the cohomology of Springer fibers admits a representation of the symmetric group (or the Weyl group, in general Lie type).  

Like better-known Schubert varieties, the geometry of Springer fibers is deeply entwined with combinatorics.  Unlike Schubert varieties, very little is known about even straightforward questions about this geometry.  In this talk, we study the combinatorics and geometry of a particular family of Springer fibers that arise in combinatorics, representation theory, and knot theory. We give some results about how to partition these Springer fibers into cells that are encoded by a kind of graph called a web.

The Jordan Curve theorem stands as a unique example of a theorem which seems so intuitive, it hardly needs proving. That being said, it's surprising to learn that few theorems have garnered so much controversy over their proofs. In this talk I will introduce a proof given in 1992 which relies only on graph theoretic methods and the unsolvability of the Three Utilities Problem. I will then use the introduced methods to sketch a proof of the Jordan-Schonflies theorem and talk a little bit about the extension of this theorem into higher dimensions.

Abstract:    This talk will review lots of background in the area of poset topology along the way towards discussing a new technique for studying the topological structure of simplicial complexes known as order complexes of partially ordered sets.  Our main results give a new twist on a popular tool of poset topology called lexicographic shellability.  When Bjoerner and Wachs introduced one of the main forms of lexicographic shellability, namely CL-shellability, they also introduced the notion of recursive atom ordering, and they proved that a finite bounded poset is CL-shellable if and only if it admits a recursive atom ordering.  We generalize the notion of recursive atom ordering, and we prove that any such generalized recursive atom ordering may be transformed via a reordering process into a recursive atom ordering.  We also prove that a finite bounded poset admits a generalized recursive atom ordering if and only if it is ``CC-shellable'' in a way that is self-consistent in a certain sense.  This allows us to conclude that CL-shellability is equivalent to self-consistent CC-shellability.  As an application, we prove that the uncrossing orders, namely the face posets for stratified spaces of planar electrical networks, are dual CL-shellable.  This is joint work with Grace Stadnyk.