TATERS

 A real algebraic variety is a set of points in real Euclidean space that satisfy a system of polynomial equations. Metric algebraic geometry is the study of properties of real algebraic varieties that depend on a metric. In this talk, we introduce metric algebraic geometry through a discussion of Voronoi cells, bottlenecks, and the reach of an algebraic variety. We describe applications to the study of the geometry of data with nonlinear models and to low-rank matrix approximation.

For a multivariable polynomial that factors as a product of "random" linear factors, I will describe the vector space spanned by its derivatives, and the collection of differential equations (with constant coefficients) satisfied by the polynomial. At the end I will pose some questions.

Banach asked whether a normed space all of whose k-dimensional linear subspaces are isometric to each other, for some fixed 2 ≤ k < dim(V), must necessarily be Euclidean. At present, an affirmative answer is known for k = 2 (Auerbach-Mazur-Ulam, 1935), all even k (Gromov, 1967), all k = 1 (mod 4) but k = 133 (Bor-Hernandez Lamoneda-Jimenez Desantiago-Montejano Peimbert, 2021), and k = 3 (Ivanov-Mamaev-Nordskova, 2023). These developments, except perhaps the recent resolution of the k = 3 case, can be considered spiritual successors to the original argument of Auerbach-Mazur-Ulam for k = 2 which is based on a topological obstruction. In this talk, I will present a stable version of their result: if all 2-dimensional linear subspaces are approximately isometric to each other, then the normed space is approximately Euclidean. This talk is based on joint work with Dmitry Faifman.

 An ideal in a polynomial ring with several variables is called symmetric if it remains invariant as a set under the action of the symmetric group, which permutes the variables of the polynomials. In mathematics, it is a recurring principle that a "general" or "random" member of a family often exhibits desirable properties. Motivated by this idea, we undertake the task of identifying an appropriate notion of a general symmetric ideal within the family of symmetric ideals with a fixed number of generators, considered up to symmetry. This framework enables us to uncover remarkable homological properties inherent to these ideals. This talk is based on joint work with Megumi Harada and Liana Şega and separately with Liana Şega.

Metric tensor learning serves as an approach for uncovering hidden structures within high-dimensional spaces. By acquiring a suitable distance metric, algorithms reliant on distance measurements can more effectively capture the inherent structure of data points, resulting in enhanced performance. In contrast to single-metric learning methods, the effectiveness of multi-metric and geometric metric learning becomes evident when handling intricate data distributions and diverse data characteristics. These alternative approaches offer heightened flexibility and interpretability, making them especially valuable for representation learning in complex, non-linear, multi-modal datasets. In this context, a concise introduction to the concepts of distance metric learning is given, followed by an overview of methods that extend its applicability to high-dimensional spaces, graphs, and manifolds.

Classical combinatorics has extensively studied ordered and unordered sequences of positive integers summing to n; i.e., the compositions and partitions of n.  But at least in these venerable branches of pure mathematics, it is still possible to make publication-worthy contributions (with a little help from the pros) with elementary methods; e.g., single variable generating functions.   Emphasis is on the initial phases of various discoveries, including formulas for: k-step Fibonacci numbers, both positively-indexed and negatively-indexed; the number of times k is the m th largest part in the partitions of n; the number of parity palindromes of n; the number of separable and inseparable compositions of n; the number of compositions of n with some parts frozen.  Mathematics is indeed kind to its amateurs!

OUTLINE OF TOPICS

Emphasis is on the initial phases of discovery throughout.

Describing two-toned tiling functions

Using two-toned tiling functions to find formulas for positively-indexed and negatively-indexed k-step Fibonacci numbers

Discovering the formula for the number of times k is the mth largest part within the partitions of n, starting with a well-known recursion formula

Coming up with the idea of parity palindromes and, more generally, modulo k palindromes

Coming up with the idea of latent properties of partitions and, as a result, finding the formula for separable and inseparable partitions

Hybrid compositions/partitions

 Kreck and Schafer produced the first examples of stably diffeomorphic closed smooth 4-manifolds which are not homotopy equivalent. They were constructed by applying the doubling construction to 2-complexes over certain finite abelian groups of odd order. By extending their methods, we formulate a new homotopy invariant on the class of 4-manifolds arising as doubles of 2-complexes with finite fundamental group. As an application we show that, for any k≥2, there exist a family of k closed smooth 4-manifolds which are all stably diffeomorphic but are pairwise not homotopy equivalent. This is joint work with Ian Hambleton.