A \(\mathbb{Z}\) -graded \(K\) -algebra \(R\) is Koszul if \(K\) has a linear free resolution over \(R\). It is known that quotients of a polynomials ring or exterior algebra or tensor algebra that are Koszul must have quadratic relations. It is sufficient to have a Groebner basis of quadratic relations – we call these algebras G-quadratic. Neither of these implications is reversible in general. There is an intermediate notion, called LG-quadratic, which describe algebras that are quotients of G-quadratic algebras by a regular sequence of linear forms. The goal of this talk will be to show that there are G-quadratic quotients of exterior algebras that are not LG-quadratic, and Koszul quotients that are not LG-quadratic. This is joint work with Zach Mere (former ISU ugrad, now Utah grad student).

Links: arXiv preprint, video of talk, lecture notes

We construct infinite-dimensional analogs of classical representations of the nonstandard quantized enveloping algebra \(U_q(\mathfrak{so}_n)\) by rationalizing the classical formulas from finite-dimensional representations. These also provide representations for the universal enveloping algebra \(U(\mathfrak{so}_n)\) as \(q\) approaches 1. We use these new representations to embed \(U_q(\mathfrak{so}_n)\) into a skew group algebra of shift operators.

Links: video of talk

We study under what conditions the canonical module of a Cohen-Macaulay Koszul algebra has a linear free resolution over its Koszul algebra, motivated by a question of Stillman asking whether this holds for all superlevel Koszul algebras. Using recent work of D’Alì and Venturello, we show that Stillman’s question has a negative answer in general. However, we also highlight a number of cases where the canonical module does have a linear resolution and give some consequences for resolutions of modules over certain rings of geometric interest. This talk is based on joint preliminary work with Paolo Mantero.

Links: video of talk

A graded k-algebra is said to be Koszul if the minimal R-free graded resolution of k is linear. In this talk we study the Koszul property of the homogeneous coordinate ring R of a set of m lines in complex projective space. Kempf proved that the coordinate ring of s points in general linear position is Koszul if s is less than or equal to 2n. Further, Conca, Trung and Valla showed that if the points are algebraically independent over Q, then the coordinate ring is Koszul if and only if s is less than or equal to 1+n+n^{2}/4. We expand on Kempfs Theorem with the exception we consider lines in projective space.

Links: video of talk

I will report on recent progress of our investigation into the diagonal reduction algebra for the orthosymplectic Lie superalgebra \(\mathfrak{osp}(1|2)\), including presentation, PBW basis and ghost center.

Joint work with Dwight Williams II.

Links: video of talk

Nichols algebras of diagonal type are generalizations of (positive parts) of small quantum groups that play a crucial role in the classification of finite-dimensional pointed Hopf algebras with abelian group of group-likes. Remarkably, many combinatorial tools of quantum groups were extended to this setting which led to the classification of such Nichols algebras. In this talk, we will introduce these tools through examples and show several recent developments in the theory of Nichols algebras.

Nichols algebras of diagonal type are braided Hopf algebras that play a crucial role in the classification of pointed Hopf algebras. In this talk, we will show that (positive parts) of small quantum groups are Nichols algebras. Then we will explain how both versions of big quantum groups at roots of unity can be recovered using the theory of Nichols algebras. Finally, we will recall many combinatorial and structural tools for quantum groups that have been generalized to Nichols algebras. Time permitting, we will discuss how these tools led to the classification of Nichols algebras of diagonal type and other important results on pointed Hopf algebras.

Algebraic geometry has furnished fruitful tools for studying matroids, which are combinatorial abstractions of hyperplane arrangements. We first survey some recent developments, pointing out how these developments remained partially disjoint. We then introduce certain vector bundles (K-classes) on permutohedral varieties, which we call "tautological bundles (classes)" of matroids, as a new framework that unifies, recovers, and extends these recent developments. Our framework leads to new questions that further probe the boundary between combinatorics and geometry. Joint work with Andrew Berget, Hunter Spink, and Dennis Tseng.

Matroids are combinatorial structures that arise naturally in diverse areas of mathematics. The simplest and most ubiquitous class of matroids are representable matroids (over a field), which are fully captured by a finite set of vectors. However, not every matroid is representable over a field, and finding a representation for a given matroid is a well-known difficult computational problem. Recently, Baker and Lorscheid have developed a general categorical framework for matroid representability, in terms of algebraic structures called pastures (which generalize fields). I will discuss how to utilize this framework to efficiently compute representations of matroids over finite fields. This is joint work with Tianyi Zhang.

Motivated by some problems from theoretical physics, we describe a toy model problem of counting isometric immersions of certain graphs in the flat torus. This is joint work with David Aulicino and Harry Richman. There will be lots of pictures, and hopefully, some discussion of the motivations.

The Chow ring of an algebraic variety is an algebro-geometric analog of the cohomology ring of a smooth manifold that encodes important information about the intersections between its subvarieties. Feichtner and Yuzvinsky computed a presentation for the Chow ring of a smooth toric variety associated to a matroid (and some other data) which is now called the Chow ring of the matroid. These rings have garnered significant attention in recent years thanks to their role in establishing long-standing conjectures on the combinatorics of matroids, including the resolution of the Heron-Rota-Welsh Conjecture by Adiprasito, Huh, and Katz and the resolution of the Top-Heavy Conjecture by Braden, Huh, Matherne, Proudfoot, and Wang.

From a commutative algebra standpoint, Chow rings of matroids are very nice graded Artinian Gorenstein rings defined by quadratic relations, and so, a natural conjecture posed by Dotsenko is that the Chow ring of a matroid is always Koszul. In this talk, we will discuss how the combinatorics of a matroid influences algebraic properties of its Chow ring, culminating in recent joint work with Jason McCullough giving an affirmative answer to Dotsenko’s conjecture.

We study the Ehrhart polynomials of certain slices of rectangular prisms. We give a combinatorial formula for all the Ehrhart coefficients in terms of the number of weighted permutations satisfying certain compatibility properties. This result proves that all these polytopes are Ehrhart positive i.e. the coefficients of the Ehrhart polynomial are positive. Additionally, providing an extension of a result by Early and Kim, we give a combinatorial interpretation for all the coefficients of the \(h^*\)–polynomial, and hence we solve the problem of providing an interpretation for the numerator of the Hilbert series, also known as the h-vector of all algebras of Veronese type. As consequences of our results, we obtain an expression for the volumes of these slices of prisms as weighted combinations of Eulerian numbers; we use this to provide some generalizations of Laplace’s result on the combinatorial interpretation of the volume of the hypersimplex.

Links: slides

This talk has three interrelated goals. On one hand it will introduce the basic framework of sub-Riemannian geometry, defining sub-Riemannian manifolds and structures, horizontal curves, sub-Riemannian geodesics, as well as the relation of sub-Riemannian problems to certain optimal control problems. Then it will illustrate how the concept of symmetry enters sub-Riemannian geometry and how it may be helpful in reducing the dimensionality of a problem and eventually reducing it to a Riemannian problem on a lower dimensional manifold. The third goal of the talk is to illustrate the above concepts for a structure on SL(2). Using symmetry reduction, the problem of finding length minimizing geodesics between any two points will be reduced to a Riemannian problem on the plane where the geodesics can be easily visualized. Using such an explicit graphical description a complete solution and a description of the sub-Riemannian geodesics between any two points on SL(2) will be given.

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