Schedule for: 23w5070 - Random Algebraic Geometry

A buffet dinner is served daily between 5:30pm and 7:30pm in Vistas Dining Room, top floor of the Sally Borden Building.

Breakfast is served daily between 7 and 9am in the Vistas Dining Room, the top floor of the Sally Borden Building.

A brief introduction to BIRS with important logistical information, technology instruction, and opportunity for participants to ask questions.

Seminal work by Khovanskii showed that the number of nondegenerate real zeros of a fewnomial system in $n$ variables is bounded by $n$ and its number $t$ of monomials. However, Khovanskii’s bound by is exponential in $t$, while many people believe that there should be an upper bound, which is polynomial in $t$. We show that ``generically'' this is indeed the case: more specifically, the expected number of real zeros of random fewnomial systems with prescribed set of exponent vectors can be neatly bounded (similarly as for Kushnirenko’s conjecture). When focusing on structured polynomials, even the univariate case is challenging. Koiran's real tau conjecture for the number of real zeros of a sum of sparse polynomials implies the separation of the complexity classes VP and VNP. Recently, we proved that random univariate polynomials typically have as few real zeros as predicted by the real tau conjecture. The randomness refers here to formulas with fixed combinatorial structure and independent standard Gaussian coefficients. The proofs rely on tools from probability (Kac-Rice formula) and integral geometry.

The essential variety is an algebraic subvariety of dimension 5 in real projective space which encodes the relative pose of two calibrated pinhole cameras. The 5-point algorithm in computer vision computes the real points in the intersection of the essential variety with a linear space of codimension 5. The degree of the essential variety is 10, so this intersection consists of 10 complex points in general. We will present the expected number of real intersection points with respect to different probability distributions, and some interesting geometry and computational work that arises from this problem. This is based on joint work with Paul Breiding, Pierpaola Santarsiero, and Elima Shehu.

Lunch is served daily between 11:30am and 1:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

Meet in the PDC front desk for a guided tour of The Banff Centre campus.

Meet in foyer of TCPL to participate in the BIRS group photo. The photograph will be taken outdoors, so dress appropriately for the weather. Please don't be late, or you might not be in the official group photo!

Convex regression is the problem of fitting a convex function to a collection of input-output pairs, and arises naturally in applications such as economics, engineering and computed tomography. We present a new approach to this problem called spectrahedral regression, in which we fit a spectrahedral function to the data, i.e. a function that is the maximum eigenvalue of an affine matrix expression of the input. This method generalizes polyhedral (also called max-affine) regression, in which a maximum of a fixed number of affine functions is fit to the data. We first provide bounds on how well spectrahedral functions can approximate arbitrary convex functions via statistical risk analysis. Second, we analyze an alternating minimization algorithm for the non-convex optimization problem of fitting a spectrahedral function to data. Finally, we demonstrate the utility of our approach with experiments on synthetic and real data sets. This talk is based on joint work with Venkat Chandrasekaran.

A buffet dinner is served daily between 5:30pm and 7:30pm in Vistas Dining Room, top floor of the Sally Borden Building.

Breakfast is served daily between 7 and 9am in the Vistas Dining Room, the top floor of the Sally Borden Building.

By looking closely to combinatorial properties of free resolutions of monomial ideals we show that random monomial ideals almost always have maximal projective dimension and that they are almost never Cohen-Macaulay. We also characterize when random monomial ideals are generic and when their Scarf complex gives a minimal free resolution. This is based on joint work with DeLoera, Krone, and Silverstein.

Many problems in computer vision, engineering, chemistry, and more can be formulated using a parameterized system of polynomials which must be solved for given instances of the parameters. Solutions and behaviour over the real numbers are those that provide meaningful information for these applications. Homotopy continuation within numerical algebraic geometry can be used to solve these parameterized polynomial systems and understand the “geography” of the corresponding parameter space. This talk will cover various methods that exist to compute real monodromy action, symmetries, the discriminant locus, and more—all with various applications. In addition, future directions and questions regarding the parameter space exploration will be mentioned.

Lunch is served daily between 11:30am and 1:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

Unlike complex polynomial systems, real polynomial systems don't have generically the same amount of real zeros. However, the disparities don't end here. In this talk, we will show how—unlike in the complex setting—the condition number of a real polynomial system controls the number of real zeros of a real polynomial system. Then, out of these condition-based bounds, we derive robust probabilistic bounds for all the moments of the number of real zeros of a random real polynomial system (both in the KSS and the Kac frameworks). This is joint work with Elias Tsigaridas.

The term ``probabilistic lens'' loosely refers to the indirect application of probabilistic methods to prove deterministic results. The terminology is especially used in combinatorics where the method is well-established. In this talk, I will discuss some of the basic principles (linearity of expectation, large deviation estimates, union bounds) and how they can be utilized in the study of polynomials to gain deterministic insights. As particular case studies, I will consider the zeros of complex harmonic polynomials (which gives rise to a bivariate real algebraic system) and a problem of Erdős on the area of polynomial (filled) lemniscates.

A buffet dinner is served daily between 5:30pm and 7:30pm in Vistas Dining Room, top floor of the Sally Borden Building.

Breakfast is served daily between 7 and 9am in the Vistas Dining Room, the top floor of the Sally Borden Building.

The expected number of real projective roots of orthogonally invariant random homogeneous real polynomial systems is known to be equal to the square root of the Bézout number. A similar result is known for random multi-homogeneous systems, invariant through a product of orthogonal groups. In this note, those results are generalized to certain families of sparse polynomial systems, with no orthogonal invariance assumed.

This talk will explain that Convolutional Neural Networks without activation parametrize semialgebraic sets of real homogeneous polynomials that admit a certain space factorization. We will investigate how the geometry of these semialgebraic sets (e.g., its singularities and relative boundary) changes with the network architecture. Moreover, we will start to explore how these geometric properties affect the optimization of a loss function for given training data. This talk is based on (ongoing) work with Guido Montúfar, Vahid Shahverdi, and Matthew Trager.

I’ll give an overview of several ways that randomness has been used to develop and analyze conjectures about syzygies, with a particular focus on models based on random monomial ideals. I’ll also outline some potential new areas where I think random techniques could provide useful insights.

Lunch is served daily between 11:30am and 1:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

A buffet dinner is served daily between 5:30pm and 7:30pm in Vistas Dining Room, top floor of the Sally Borden Building.

Breakfast is served daily between 7 and 9am in the Vistas Dining Room, the top floor of the Sally Borden Building.

I will discuss work studying systems of parameters over finite fields from a probabilistic perspective, and how such an approach relates to an effective version of Noether normalization over finite fields. The key idea is one may compute the asymptotic probability that random polynomials of a given degree over a finite field form a system of parameters by an adaptation of Poonen’s closed point sieve, where on sieves over higher dimensional subvarieties, and we express the desired probabilities via a zeta function-like power series that enumerates higher dimensional varieties instead of closed points. This is joint work with Daniel Erman.

Can one tell if an ideal is radical just by looking at the degrees of the generators? In general, this is hopeless. However, there are special collections of degrees in multigraded polynomial rings with the property that any multigraded ideal generated by elements of those degrees is radical. We call such a collection of degrees a radical support. In this talk, I give a combinatorial characterization of radical supports. I also discuss the relation between the notion of radical support and that of Cartwright-Sturmfels ideals, which are multigraded ideals with a radical generic initial ideal. The talk is based on a joint work with Aldo Conca and Emanuela De Negri.

Lunch is served daily between 11:30am and 1:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

In a joint work with Michele Stecconi, we study the zero set of a nice enough random smooth function. We show how to compute the expectation of the Riemannian volume of such zero sets. This computation involves a family of convex bodies (zonoids) in the exterior powers of the cotangent bundle. I will explain this construction and how this is linked to the recently introduced zonoid algebra (j.w. with Breiding Bürgisser and Lerario). I will show how the zonoids satisfy some elegant properties for pull backs and independent intersections. As an application, I will explain how this gives rise to Crofton formulae in Finsler geometry.

A fundamental challenge in statistics and machine learning is to learn a posterior distribution from its pointwise evaluations. Optimization-based approaches using variational inference have recently emerged for this learning especially when the support of the distributions have complex shapes such as multi-modality in high-dimensional settings. By viewing annealing as a homotopy in the space of probability distributions, this talk will describe how to apply techniques from homotopy continuation to variational inference. Many examples will be presented showing the computational efficiency of the approach including posterior estimation of parameters for dynamical systems and probability density approximation in multimodal and high-dimensional settings. This is joint work with Emma Cobian, Fang Liu, and Daniele Schiavazzi.

A buffet dinner is served daily between 5:30pm and 7:30pm in Vistas Dining Room, top floor of the Sally Borden Building.

Breakfast is served daily between 7 and 9am in the Vistas Dining Room, the top floor of the Sally Borden Building.

The first problem deals with estimating the volume of tubes around possibly singular real algebraic varieties. We resolve an open problem stated in the book ``Condition" by Bürgisser and Cucker. This is joint work with A. Lerario. The second problem is on obtaining bounds on certain oscillatory integrals of the form $\int_{[0,1]^d} \exp(i Q(\xi)) d\xi$ where $Q$ is a continuous semi-algebraic function. The bound obtained is then used to prove sharp convergence exponents of Tarry's problem in dimension two and to prove sharp $L^{\infty} \to L^p$ Fourier extension estimates for a family of monomial surfaces. This is joint work with Shaoming Guo, Ruixiang Zhang and Pavel Zorin-Kranich.

We give a new method to attempt to prove that, for a given $n$, there are finitely many equivalence classes of planar central configurations in the Newtonian $n$-body problem for generic masses. The human part of the proof relies on tropical geometry. The crux of our technique is in a computation that we have completed for $n\leq 5$, thus confirming the celebrated result of Albouy and Kaloshin.

5-day workshop participants are welcome to use BIRS facilities (TCPL ) until 3 pm on Friday, although participants are still required to checkout of the guest rooms by 11AM.