Georgia Institute of Technology College of Sciences

At its core, numerical algebraic geometry is the business of solving zero-dimensional polynomial systems over the complex numbers. Thanks to incredibly fast state-of-the-art software implementations, the bottleneck in these algorithms has shifted from computation time to memory usage.

To address this, recent work has introduced iterator datatypes for solution sets. An iterator represents a list by storing a single element and providing a mechanism to obtain the next one, thereby reducing memory overhead.

In this talk, we present our design of 'homotopy iterators' and 'monodromy coordinates', two iterator datatypes based on the most widely used numerical methods for solving polynomial systems. We highlight the substantial benefits of this low-memory perspective through several iterator-friendly adaptations of existing algorithms, including parameter space searches, data compression, and certification.

This talk features joint work with subsets of Paul Breiding, Hannah Friedman, and David K. Johnson.

We will start with a presentation by Griffin Edwards and continue with a free discussion.

The Heisenberg projection problem asks whether there is an analogue of the Marstrand projection theorem in the first Heisenberg group, namely whether Hausdorff dimension of sets generically decreases under projection, for a natural family of projections arising from the group structure. This problem is still open, but I will discuss a recent improvement to the known bound obtained through a variable coefficient local smoothing inequality. 

Rather than going through the proof in detail, I will spend most of the talk introducing the problem and explaining the connection to averaging operators over curves, and explaining why these operators are Fourier integral operators satisfying Sogge's cinematic curvature condition. This condition was originally introduced by Sogge to generalise Bourgain's circular maximal theorem, but it turns out to have useful applications to projection theory. 

How hard is it to estimate a sequence of length N, satisfying some *unknown* linear recurrence relation of order S and observed in additive Gaussian noise? The class of all such sequences is extremely rich: it is formed by arbitrary (complex) exponential polynomials with total degree S. This includes the case of stationary sequences, a.k.a. harmonic oscillations, a.k.a. sequences with discrete​ Fourier spectra supported on S *arbitrary* frequencies. Strikingly, it turns out that one can estimate such sequences with almost the same statistical error as if the recurrence relation was known (and a simple least-squares estimator could be used). In particular, stationary sequences can be estimated with mean-squared error of order O(S/N) up to a polylogarithmic factor, without any assumption of spectral separation—despite what one might learn in a high-dimensional statistics class. Moreover, these methods are computationally tractable. 

In this talk, I will highlight some mathematics underlying this result, putting emphasis on analytical, rather than statistical, side of things. In particular, I will show how to invert a polynomial while ensuring that the result is a polynomial—rather than a reciprocal of a polynomial—and what this has to do with reproducing kernels. Then, I will pitch some accessible open problems in this area.