TATERS

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The notions of Yang-Baxter (YB) operator and braided categories have long been studied in algebra, and they have found important applications in geometric topology (invariants of knots and $3$-manifolds) and theoretical physics (topological quantum field theory). In this talk, I will survey some topics connecting YB operators, topology and physics. Then, I will present a (co)homology theory for associative algebras endowed with a Yang-Baxter operator satisfying some extra coherence axioms. I will discuss how the 2nd cohomology group classifies the infinitesimal deformations of these algebras and give diagrammatic interpretations of the differentials. I will give examples of IYI algebras arising from Hopf algebras, Frobenius algebras and multiple conjugation quandles. Lastly, I will discuss a topological motivation for the study of these objects, based on the theory of compact surfaces with boundary embedded in $3$-space.

One of the most important problems in the research area of cryptographic functions is the construction of permutations (bijections) that have a strong resistance against differential cryptanalysis. In this talk, we introduce the notion of permutation resemblance, which is a way to measure the distance of a given polynomial over a finite field from being a permutation. We introduce an integer programming approach to compute the permutation resemblance. Moreover, using the information of permutation resemblance and integer programming methods, we give an algorithm that constructs permutations with a good resistance against differential attack. 

Hurwitz's automorphism theorem says that the any Riemann surface of genus g ≥ 2 has at most 84(g-1) automorphisms. In this talk we will discuss the generalization of this result to higher dimensional algebraic varieties of general type.

We prove a new class of inequalities for submodular set functions, indexed by chordal graphs. Since entropy is a particularly useful example of a submodular function, we deduce some entropy inequalities. As a further corollary, we construct a novel family of determinant inequalities for sums of positive definite Hermitian matrices. As a special case we recover an inequality of Barrett, Johnson, and Lundquist (1989). 

This talk is based on joint work with Mokshay Madiman.

An n-manifold is a topological space where every point has the neighborhood of an open n-ball. We are interested in studying simple procedures for constructing examples of 3-manifolds with a variety of properties. Every 2-manifold can be obtained from a polygon via edge-pairing, and every 3-manifold can be obtained from a polyhedron via face-pairing. However the face pairing method in dimension 3 has a defect: the method yields a pseudo 3-manifold, but very rarely a 3-manifold. Twisting a given face-pairing however will always produce a 3-manifold. This was discovered by Cannon, Floyd, and Parry 20 years ago. In my talk I will discuss face-pairings and twisted face-pairings on polyhedra.

Decategorification collapses objects of categories to their isomorphism classes. Decategorification can yield familiar mathematical structures, and those structures’ respective categorifications can reveal interesting connections. Through the course of this talk we will ascend from the concrete to the abstract and descend back again to the concrete. First we will begin by looking at the natural numbers as the decategorification of the category of finite sets. This will lead us toward groupoids and some notions from homotopy and higher categories, which will in turn motivate a notion of “structure” which generalizes combinatorial species. Finally, we will see that decategorifying species and their operations yield certain power series, thus landing us back into the concrete in applications of generating functions to counting problems.

Primitive integral Apollonian circle packings are fractal arrangements of tangent circles with integer curvatures.  The curvatures form an orbit of a 'thin group,' a subgroup of an arithmetic group having infinite index in its Zariski closure.  The curvatures that appear must fall into one of six or eight residue classes modulo 24. The twenty-year old local-global conjecture states that every sufficiently large integer in one of these residue classes will appear as a curvature in the packing. We prove that this conjecture is false for many packings, by proving that certain quadratic and quartic families are missed. The new obstructions are a property of the thin Apollonian group (and not its Zariski closure), and are a result of quadratic and quartic reciprocity, reminiscent of a Brauer-Manin obstruction. Based on computational evidence, we formulate a new conjecture.  This is joint work with Summer Haag, Clyde Kertzer, and James Rickards.

Asymptotic dimension was introduced by Gromov as a basic large-scale invariant of metric spaces and groups. It is analogous to the classical covering dimension of topological spaces, but uses coverings with sets of large diameter. I will go over some basic theorems of asymptotic dimension theory and some examples, including Gromov's theorem that hyperbolic groups have finite asymptotic dimension. If there is time, I will outline the proof that mapping class groups have finite asymptotic dimension (my joint work with Bromberg and Fujiwara). I will end with some open questions.

The full shift over a finite set S is the collection of bi-infinite strings of symbols in S. A shift space is a subset of a full shift defined by a collection of “forbidden” blocks, i.e., finite strings which are not allowed to appear. Many shift spaces arise as the set of bi-infinite walks on a labeled graph, and many dynamical systems can be encoded as shift spaces where the dynamics is expressed as the left-shift map on the shift space. I will introduce shift spaces and their characteristics, then discuss the notion of conjugacy and describe some conjugacy invariants including entropy and zeta functions.

In 1966, Abraham Robinson used ideas from model theory to come up with nonstandard analysis, an approach to analysis allowing infinitesimals as actually existing objects. This talk is not about that. Instead, this talk is about a different area where nonstandard methods have been fruitful, namely combinatorics. With Timothy Trujillo, we were interested in whether nonstandard methods could be applied to understand work in topological Ramsey theory. After all, this area studies a generalization of the combinatorial-topological Ellentuck space on the integers, and nonstandard methods have enjoyed use in integer combinatorics.

In this talk I will give an introduction to the use of nonstandard methods and how they can be used to prove results like Ramsey's theorem. I'll discuss how these ideas can be used to prove the Nash-Williams separation theorem, and I'll gesture toward how to generalize this to the setting of abstract Ramsey spaces.

(Slides)

A fundamental problem at the confluence of algebraic geometry, commutative algebra and representation theory is to understand the structure and vanishing behavior of the cohomology of line bundles on flag varieties. Over fields of characteristic zero, this is the content of the Borel-Weil-Bott theorem and is well-understood, but in positive characteristic it remains wide open, despite important progress over the years. By embedding smaller flag varieties as Schubert subvarieties in larger ones, one can compare cohomology groups on different spaces and study their eventual asymptotic behavior. In this context I will describe a sharp stabilization result, and discuss some consequences and illustrative examples. Joint work with Keller VandeBogert.

Mapping class groups have been studied for over a century using a wide variety of tools. While many linear representations of these groups exist, they have almost no representation theory. In this talk, we will discuss an important class of representations, and some of the important questions, challenges, and advances in understanding them. No prior knowledge regarding these groups will be assumed.