TATERS

We impose rank one constraints on marginalizations of a tensor given by a simplicial complex. If the tensor encodes a discrete probability distribution, the rank constraints correspond to mutual independences among the random variables. These statistical models become toric after a linear change of coordinates. We study their toric ideals with emphasis on random graph models and use the new coordinates to compute degrees for their parameter estimation problems.

Algebraic combinatorics, as the name suggests, is the mathematical discipline that marries together topics from combinatorics and algebra. Among other things, it considers the classification and enumeration of algebraic objects, such as counting p-groups, and it employs algebraic methods to study combinatorial structures, such as in algebraic graph theory. One well-studied and extremely important combinatorial structure is the partition of a set. Through the lens of algebraic combinatorics, it is natural then to ask that when a set G is algebraic, e.g. G is a group, whether a partition S of G is likewise algebraic. After all, when studying groups, not all subsets are generally studied; instead, the study of subgroups takes paramount, that is, those subsets of the group which are themselves groups. What analog is there for algebraic partitions? In the category of groups, the natural candidate is known as a Schur ring, first studied by Schur and Wielandt in the early 20th century. In this talk, we discuss the history and applications of Schur rings, as well provide examples. We will also discuss recent efforts to classify and enumerate Schur rings, particularly over cyclic groups. Through the lens of universal algebra, we will also discuss efforts to generalize the notion of a Schur ring to algebraic categories beyond groups, thus answering the original question about what makes a partition algebraic.

Tanglegrams are formed by taking two rooted binary trees with the same number of leaves and uniquely matching their leaves. They can be represented by certain drawings in the plane called layouts. In a layout, the trees are drawn planarly with their leaves facing each other, and the matching between leaves is drawn with straight lines. One problem of interest is the Tanglegram Layout Problem, which is to efficiently find a layout that minimizes the number of crossings. This parallels the Graph Drawing Problem, which attempts to draw a graph in the plane with the fewest number of crossings possible, where one common approach to approximating a solution is to insert edges into a planar subgraph. In this talk, we present a characterization of all planar layouts of a planar tanglegram, which are tanglegrams that have a layout with no crossings. We then discuss the tanglegram analogue of edge insertion for graphs.

I'll introduce some of the history of thin groups and discuss a proof that there are Zariski dense surface groups in SL(2k+1,Z).

Many combinatorial objects, such as matroids, graphs, and posets, can be realized as generalized permutahedra - a beautiful family of polytopes. This realization respects the natural multiplication of these objects as well as natural "breaking" operations. Surprisingly many of the important invariants of these objects, when viewed as functions on polytopes, satisfy an inclusion-exclusion formula with respect to subdivisions. Functions that satisfy this formula are known as valuations. In this talk, I will discuss work with Federico Ardila that completely describes the relationship between the algebraic structure on generalized permutahedra and valuations. Our main contribution is a new easy-to-apply method that converts simple valuations into more complicated ones.

We examine the Waring rank for certain types of symmetric polynomials. The first type of polynomials we refer to as a Power of a Fermat type polynomial, or a PFT polynomial. This is a Fermat type (or power sum) polynomial over n variables with degree p taken to some power k. We establish there exist no degree k annihilators, and find the degree k+1 annihilator ideal for general PFT polynomial when p>k and k>2 using the method of partial derivatives. We also examine Schur polynomials, and find the Waring rank of one general type, and a lower bound for another.

In a brief but important 1974 paper, L. R. Welch gave a family of bounds on the maximum modulus of inner products between distinct vectors in a set of m unit vectors in an n-dimensional space. It was noted that these bounds have implications in the design of sequences having desirable correlation properties for multichannel communication systems. Over the years, motivated by several applications, sets that attain the Welch bound have been sought after. On the other hand, mutually unbiased bases (MUBs) are a primitive used in quantum information processing. From a mathematical standpoint, the existence and construction of maximal sets of MUBs in all dimensions pose important open questions.  Maximal sets of MUBs turn out to be the same objects as complex projective 2-designs. This talk will discuss these various mathematical objects in connection to sets that attain the Welch bounds. 

Current advances in machine learning and computer vision have allowed for more sophisticated methods in automatic license plate recognition (ALPR) to be commercialized. Most existing methodologies, however, only achieve state of the art (SOTA) accuracy on datasets containing approximately frontal images of vehicles with license plates (LPs) from a specific region (e.g. the US, or China). In a recent internship, I was tasked with combining multiple SOTA deep learning architectures in order to build a highly accurate ALPR system that works in unconstrained scenarios. That is, situations where the LP may be distorted due to oblique camera views. The system was also to be fast, with a minimum processing speed of 5 FPS. I combined 3 Convolutional Neural Networks (CNNs) to perform vehicle recognition, LP detection, and optical character recognition (OCR). My proposed system achieved 96% accuracy on an unseen dataset which consisted of 200 vehicles with U.K. license plates.

Each point $x$ in $Gr(r,n)$ corresponds to an $r \times n$ matrix $A_x$ which gives rise to a matroid $M_x$ on its columns. Gel'fand, Goresky, MacPherson, and Serganova showed that the sets $\{ y \in Gr(r,n) \mid M_y = M_x \}$ form a stratification of $Gr(r,n)$ with many beautiful properties. However, results of Mnëv and Sturmfels show that these strata can be quite complicated, and in particular may have arbitrary singularities. We study the ideals $I_x$ of matroid varieties, the Zariski closures of these strata. We construct several classes of examples based on theorems from projective geometry and describe how the Grassman-Cayley algebra may be used to derive non-trivial elements of $I_x$ geometrically when the combinatorics of the matroid is sufficiently rich.

Phylogenetic tree and network reconstructions in evolutionary biology lead not only to a better understanding of our natural world, but also have applications in other fields, such as conservation and epidemiology. For example, taking into account phylogenetic diversity in the restoration of natural vegetation can lead to restorations that establish quicker and are heartier, while understanding the phylogenetics of pathogens can aid in back-tracing the spread of a disease and guiding epidemiological interventions. While trees are a natural choice for representing evolution combinatorially, by restricting to the class of trees, it is possible to miss more complicated events such as hybridization and horizontal gene transfer. For more complete descriptions, phylogenetic networks, directed acyclic graphs, are increasingly becoming more common in evolutionary biology. In this talk, we discuss Markov models for phylogenetic networks and illustrate how we can view these models as algebraic varieties. This algebraic and geometric framework can lead us to interesting results. In particular, assuming specific group-based models of evolution, using tools from computational algebraic geometry, we show that the semi-directed network topology of level-one networks is generically identifiable and discuss how these results can be used for reconstruction.

Hyperbolic (and relatively hyperbolic) groups generalize the fundamental groups of compact (and finite volume) hyperbolic manifolds.  For an example of a hyperbolic group, consider the group of symmetries of one of Escher's "Circle Limit" pictures.   An example of a relatively hyperbolic group is PSL(2,Z).   I'll talk about why I think hyperbolic groups are interesting, and about some recent constructions of Italiano-Martelli-Migliorini answering a twenty-year old question in the field.

In a first course on Galois theory, one often learns about the normal basis theorem for field extensions. I shall discuss various aspects of what happens if we replace fields by rings of integers. This will be a colloquium-style talk, and no previous knowledge of this topic will be assumed.

Consider these basic questions: What can we say about finite sums of powers of consecutive whole numbers? What can we say about whole number solutions to polynomial equations? What about factorizations into primes? What about values of the Riemann zeta function? In interesting families of examples — elementary and sophisticated, ancient and modern — "Bernoulli numbers" unify these seemingly unrelated questions. After an introduction to the Bernoulli numbers, we will explore related developments for these intertwined problems, which lead to central challenges in number theory and beyond.  This talk will be appropriate for undergraduates, graduate students, and faculty in all areas of math.