TATERS

I'll present two theorems on partitioning the positive integers into two complementary sets. First, we can pick two positive irrational numbers and consider the round-downs of their integer multiples. Beatty's theorem characterizes when this gives a partition of the positive integers, with no gaps or collisions. Second, a frequency-counting construction similar to transposing tableau gives all the partitions into two sets.

The reconstruction from indirect measurements of a shade function defined in Euclidean space reveals that only black and white pictures can be identified from finitely many data. Working with polynomials as test function one isolates principal, real semi-algebraic sets as the only sets allowing an exact recovery. The mathematical foundations of the constructive aspects of this specific inverse problem go back to A. Markov. For a century and a half, his concepts have evolved with quite unexpected turns. We will survey this chapter of classical and modern analysis.

We extend the theory of graphical designs, which are quadrature rules for graphs, to positively weighted graphs. Through Gale duality for polytopes, we show that there is a bijection between graphical designs and the faces of eigenpolytopes associated to the graph. This bijection proves the existence of graphical designs with positive quadrature weights, and upper bounds the size of a graphical design. We further show that any combinatorial polytope appears as the eigenpolytope of a positively weighted graph. Through this universality, we establish two complexity results for graphical designs: it is strongly NP-complete to determine if there is a graphical design smaller than the mentioned upper bound, and it is #P-complete to count the number of minimal graphical designs.

One way to show two plane polygonal regions have the same area is to decompose them into finite unions of congruent pieces. If this is possible, the regions are called equidecomposable, and we have a strong explicit certificate of the equality of their areas. Conversely, it turns out that any two plane polygonal regions of equal area are always equidecomposable. Hilbert raised the question: Are three-dimensional polyhedra of equal volume always decomposable? Max Dehn answered this question by introducing the Dehn invariant. This talk will give an introduction to these ideas.

In computational mathematics a tensor is an array of numbers.  It can have more than two indices, and thus generalizes a matrix.  Operations with higher-order tensors, e.g. low-rank decompositions, enjoy stronger uniqueness properties than matrix factorizations in linear algebra do thanks to results in algebraic geometry. However, often they are intractable in theory (due to being NP-hard) and also in practice (due to their high dimensionality).  In this talk, I’ll present ideas that address some of these challenges for tensors arising as moments of multivariate datasets.  I will describe tensor-based methods for fitting mixture models to data drawn from Gaussian mixtures and a class of other mixtures, which in some cases perform better than other leading statistical estimation approaches.  Time permitting, I will mention related algebraic geometry and a scientific application.  Based on joint works with João Pereira and Tamara Kolda; Yifan Zhang; and Yulia Alexandr and Bernd Sturmfels.  

Abstract: Restricted secant varieties of Grassmannians are constructed from sums of points corresponding to k-planes with the restriction that their intersection has a prescribed dimension.  

We study dimensions of restricted secant of Grassmannians and relate them to the analogous question for secants of Grassmannians via an incidence variety construction. We define a notion of expected dimension and give a formula for the dimension of all restricted secant varieties of Grassmannians that holds if the BDdG conjecture on non-defectivity of Grassmannians is true. We also demonstrate example calculations in Macaulay2, and point out ways to make these calculations more efficient. We also show a potential application to coding theory.

Fix natural numbers $m$ and $n$ and let $\mathcal{P}_0^m(\mathbb{R}^n)$ be the ring of (real) $m$-jets in $n$ variables that vanish at the origin, i.e., the ring of at most $m$th degree Taylor polynomials in $n$ variables with a zero constant term. For any closed subset $E\subset\mathbb{R}^n$ that contains the origin we define $I^m(E)$ -- the ideal in $\mathcal{P}_0^m(\mathbb{R}^n)$ that consists of all $m$-jets ($m$th degree Taylor approximations at the origin) of $C^m$ functions that vanish on $E$. (1) Which ideals in $\mathcal{P}_0^m(\mathbb{R}^n)$  arise as $I^m(E)$ for some $E$? (2) Is it true that for any $E$ there exists a semi-algebraic set $E'$ such that $I^m(E)=I^m(E')$? These are the main questions this talk will address. We will also discuss the motivation for these problems.

In particular, we will present the notion of an ideal being closed and show that any ideal of the form $I^m(E)$ is closed (more precisely explain why this is true; no formal proofs will be presented). We do not know whether in general the converse also holds, namely whether any closed ideal is of the form $I^m(E)$ for some $E$. However, we will exhibit many rings in which this is indeed the case. For instance, if $m+n\leq 5$ then any closed ideal in $\mathcal{P}_0^m(\mathbb{R}^n)$ is of the form $I^m(E)$ for some $E$. We will also see that in these rings the answer to question (2) is positive.

No advance prior knowledge in analysis will be assumed and many elementary fun examples will be presented. This talk is based on a joint work with Charles Fefferman.

Transfer systems are certain discrete objects that have a rich combinatorial structure defined in terms of the subgroup lattice of  a group G. They encode information about commutative operations on spaces with a G-action. In this talk, we will define transfer systems, and briefly explain their importance in algebraic topology. We will show that for an abelian group, the lattice of transfer systems is self-dual, generalizing the work of Balchin-Bearup-Pech-Roitzheim for cyclic groups of squarefree order. Time permitting, we will discuss the relationship between transfer systems and model structures.

This is joint work with Kyle Ormsby, and Reed undergraduate students Evan Franchere, Usman Hafeez, Peter Marcus, Weihang Qin, and Riley Waugh.

We determine p-colorability of the paradromic rings. These rings arise by generalizing the experiment of bisecting a Möbius strip. Instead of joining the ends with a single half twist, use m half twists, and, rather than bisecting (n = 2), cut the strip into n sections. Replacing each thin strip with its midline results in the m, n paradromic link. Using the notion of p-colorability from knot theory, we determine, for each m and n, which primes p can be used to color the link. Amazingly, almost all admit 0, 1, or an infinite number of prime colorings! This is reminiscent of solutions sets in linear algebra. Indeed, the problem quickly turns into a study of the eigenvalues of a large, nearly diagonal matrix.  (This is joint work with Godzik, Ho, Jones, and Sours.)

Many classical families of complex hyperplane arrangements share two interesting properties: the characteristic polynomial of the intersection lattice factors completely over the integers, and the complement is aspherical. Neither condition implies the other. The first condition often arises from algebraic considerations, most generally in the class of free arrangements introduced in the 1980’s. Paul Edelman and Vic Reiner found an example of a rank-three free arrangement with real defining equations whose (complex) complement is not aspherical, using an ad hoc argument to prove the latter property. For supersolvable arrangements the factorization and asphericity both follow from a special fiber-bundle structure on the complement.

In an attempt to characterize freeness, combinatorial conditions for factorization were introduced, in particular the notion of ``factored arrangement” by Michel Jambu. In the early 1990’s, using a weight test for asphericity due to the speaker, Luis Paris showed that factored arrangements of rank three with real defining equations have aspherical complements. It has long been thought that the result might generalize to non-real or high rank arrangements. The difficulty is in proving asphericity: existing techniques are mostly restricted to real arrangements in three variables. Recently Gerhard Roehrle and Torsten Hoge showed that the non-real  arrangement defined by the linear factors of the polynomial Q(x,y,z)=x(x^3-y^3)(x^3-z^3)(y^3-z^3)(x+y+z) is factored. I'll sketch an argument showing this arrangement does not have aspherical complement, involving a generalization and formalization of the existing ad hoc methods. This is a preliminary report on joint work with Roehrle. 

The Hurwitz problem asks for necessary and sufficient conditions for constructing, from abstract branch data, a branched covering between Riemann surfaces. Hurwitz solved the problem in the 1900's, in terms of some permutations determined by the branch data. These conditions are not easy to check by hand.

S.M. Gersten in 1987 found another, simpler solution for the Hurwitz problem in the sphere case. He found a method which does not use the symmetric group, but instead uses the free group. He shows that abstract branch data can be realized as a branched covering if and only if the data gives rise to a cancellation diagram over the free group for the given data, along with a few other key criteria. These cancellation diagrams can be constructed by computer. Gersten's method boils down to simple graph theory, but is not as general as Hurwitz' results.

In my talk I will give a brief history of the Hurwitz problem and will work through a simple example that showcases both solution methods. 

We present a new explicit formula for the determinant that contains significantly fewer terms than the usual Leibniz formula. As an immediate corollary of our formula, we show that the rank of the n-by-n determinant tensor is no larger than the n-th Bell number, which is much smaller than the previously best-known upper bounds. Over fields of non-zero characteristic we obtain even tighter bounds, and in fields of characteristic 2 we obtain a formula for the permanent that has fewer terms than Ryser’s formula.