Although Vietoris–Rips (VR) complexes are frequently used in topological data analysis to approximate the “shape” of a dataset, their theoretical properties are not fully understood. In the case of the circle, these complexes show an interesting progression of homotopy types as the scale increases, moving from the circle S^1 to S^3 to S^5, and so on. However, much less is known about VR complexes of higher-dimensional spheres.

I will present work exploring the VR complexes of the n-sphere S^n and show how the appearance of nontrivial homotopy groups of these complexes can be controlled by the covering properties of S^n and real projective space RP^n. Specifically, if the first nontrivial homotopy group of a VR complex of S^n at scale π - t occurs in dimension k, then S^n can be covered by 2k + 2 balls of radius t, but there is no covering of RP^n by k balls of radius t/2. This is joint work with Henry Adams and Žiga Virk.

The study of fragile economic systems is important in identifying systems that are vulnerable to a dramatic collapse. For instance, complex systems like supply chains are at risk of being fragile because they require many parts to work well simultaneously. Even when each individual firm has a small susceptibility to a shock, the global system may still be at great risk. A recent survey by Matthew Elliot and Ben Golub review fragile economic systems from the point of view of networks. In a network, the reliability that the final product (e.g., a car, computer, or lifesaving medication) is made by a firm is determined the probabilities of shocks being in the system. Thus, reliability transitions from being zero to a positive probability depending on the chances of a shock --- characterizing these phase transitions is an important problem in the theory of economic fragility. In our work, we view these phase transitions through the algebraic geometry lens by using resultants. As a result, we bring new tools to econometrics to analyze multi-parameter models, and we fully describe the reliability of many new network models using computational algebraic geometry. Our most significant application is a surprising case study on a mixture of two multi-parameter supply chain models. This is joint work with Jiayi Li (UCLA).

We will finish the proof of Quillen-Suslin. Please note the non-standard meeting time for the seminar this week.

Many problems in computer vision are represented using a parametrized overdetermined system of polynomials which must be solved quickly and efficiently. Classical methods for solving these systems involve specialized solvers based on Groebner basis techniques or utilize randomization in order to create well-constrained systems for numerical techniques. We propose new methods in numerical linear algebra for solving such overdetermined polynomial systems and provide a complete error analysis showing that the numerical approach is stable. Examples will be provided to show the efficacy of the method and how the error in the data affects the error in the solution.

This is the first part of the talk on the Quillen-Suslin theorem, formerly known as Serre's Conjecture. The assertion is that a finitely generated projective module over a polynomial ring over a field is always free. In 1976, Quillen and Suslin independently provided a complete proof of this theorem. Quillen was awarded a fields medal in part because of his proof this conjecture. In this talk we'll discuss a simplification of Suslin's proof given by Vaserstein. 

In the first part, we'll develop the required machinery, starting with the concepts such as stably free modules and unimodular column extension property (UCEP). Based on these, we'll see what stably free modules of a commutative ring having UCEP look like. Also, we'll see the proof of Horrock's theorem and will draw some observations required to prove the main theorem.

When is it possible to project two sets of labeled points lying in a pair of projective planes to the same points on a projective line? Here one answer: such projections exist if and only if the two 2D point sets are themselves images of a common point set in 3D projective space. Furthermore, when the two sets of points are in general position, it is possible to give a complete description of the loci of pairs of projection centers. I will describe the roles of classical invariant theory, Cremona transformations, and geometric computer vision in this description.

Let \(R\) be a Noetherian local ring  of dimension \(d\).   We define an intersection product on schemes \(Y\) which are birational and projective over \(Spec(R)\)  which allows us to interpret multiplicity in \(R\) as an intersection product, generalizing a theorem of Ramanujam and others. We use this to give new geometric formulations and proofs of some classical theorems about multiplicity. In particular, we give a new geometric proof of a celebrated theorem of Rees about degree functions. This is joint work with Jonathan Montaño.