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.

A frame (x_j) for a Hilbert space H allows for a linear and stable reconstruction of any vector x in H from the linear measurements (<x,x_j>).  However, there are many situations where some information of the frame coefficients is lost.   In applications such as signal processing and electrical engineering one often uses sensors with a limited effective range and any measurement above that range is registered as the maximum.  Depending on the context, recovering a vector from such measurements is called either declipping or saturation recovery.  We will discuss a frame theoretic approach to this problem in a similar way to what Balan, Casazza, and Edidin did for phase retrieval.  The talk is based on joint work with W. Alharbi,  D. Ghoreishi, B. Johnson, and N. Randrianarivony.

In this talk, we explore adaptations of semidefinite programming relaxations for solving PDE problems. Our approach transforms a high-dimensional PDE problem into a convex optimization problem, setting it apart from traditional non-convex methods that rely on nonlinear re-parameterizations of the solution. In the context of statistical mechanics, we demonstrate how a mean-field type solution for an interacting particle Fokker-Planck equation can be provably recovered without resorting to non-convex optimization. 

In this talk, we will show that a geometrical condition on \(2 \times 2\) systems of conservation laws leads to nonuniqueness in the class of 1D continuous functions. This demonstrates that the Liu Entropy Condition alone—which ensures the uniqueness of small BV solutions—is insufficient to guarantee uniqueness in the continuous setting, even within the mono-dimensional frame. We provide examples of systems where this pathology holds, even if they verify stability and uniqueness for small BV solutions. Our proof is based on the convex integration process. Notably, this result represents the first application of convex integration to construct non-unique continuous solutions in one spatial dimension. This is a joint work with Alexis Vasseur and Cheng Yu.
 

We prove exact leading-order asymptotic behaviour at the origin for nontrivial solutions of two families of nonlocal equations. The equations investigated include those satisfied by the cusped highest steady waves for both the uni- and bidirectional Whitham equations. The problem is therefore analogous to that of capturing the 120∘ interior angle at the crests of classical Stokes’ waves of greatest height. In particular, our results partially settle conjectures for such extreme waves posed in the series of recent papers by Ehrnström, Johnson, and Claassen (2019), Ehrnström and Wahlén (2019), and Truong, Wahlén, and Wheeler (2022). Our methods may be generalised to solutions of other nonlocal equations, and can moreover be used to determine asymptotic behaviour of their derivatives to any order.

We continue the discussion on projective equivalence of filtrations. We further look at the discrete valued filtrations and show that they have particularly nice properties . We generalize some of the theory of Rees valuations of ideals to these filtrations.