AWM at JMM 2016

Special Session Abstracts

Contents

Friday, January 8, 2016, 8:00 a.m. − 10:50 a.m. (Room 603, Convention Center)

AMS-AWM Special Session on Commutative Algebra, I

8:00 a.m. Enumeration of Double Cosets in Symmetric Groups and Beyond

Sara C. Billey*, University of Washington; Matjaz Konvalinka, University of Ljubljana; Frederick Matsen, Fred Hutchinson Cancer Research Center (1116-05-2035)

Let G be a group with subgroups H and K. The collection of double cosets H\G/K = {HgK|g ∈ G} partition G. The double cosets are generally more complicated than the one-sided cosets. For example, different double cosets can have different sizes. If G is finite, the size of H\G/K is given by the inner product on the character of the two trivial representations on H and K respectively induced up to G.

We will present recent results enumerating all distinct double cosets for certain types of subgroups of the symmetric groups. The first case was inspired by a problem in mathematical biology related to tanglegrams. This is joint work with Konvalinka and Matsen (see arXiv:1507.04976). The second case is inspired by the geometry/topology of generalized flag varieties related to parabolic subgroups. Some of our results extend to all Coxeter groups. This is joint work in progress with Konvalinka, Petersen, Slofstra and Tenner.

8:30 a.m. Constructing ideals with high Castelnuovo-Mumford regularity

Brooke S. Ullery*, University of Utah (1116-13-704)

The Castelnuovo-Mumford regularity of a module is a homological invariant that roughly measures complexity. Though straight-forward to define, it is difficult to find ideals in polynomial rings with high Castelnuovo-Mumford regularity. I will demonstrate a method that takes as input well-understood modules and outputs ideals which cut out schemes supported on linear spaces with high Castelnuovo-Mumford regularity and other desirable homological properties.

9:00 a.m. Conjectures on Symbolic Powers

Louiza Fouli*, New Mexico State University; Paolo Mantero, University of Arkansas; Yu Xie, Penn State, Altoona (1116-13-1581)

Given a finite set of points X in the projective space ℙkN , for some N, it is natural to ask what is the least degree, (αm, of
a hypersurface F≠0 passing through all the points with a given multiplicity m. Chudnovksy conjectured in 1981 that αm ⁄ m ≥ (α(X)+N−1)⁄N, where α(X) is the minimum degree of a hypersurface passing through every point in X. He established his conjecture in the case N = 2, but the conjecture is still open in full generality. We will discuss known results and some further progress towards this conjecture. This is joint work with Paolo Mantero and Yu Xie.

9:30 a.m. Dimensions of Formal Fiber Rings

Sarah Fleming, Williams College; Lena Ji, Columbia University; Susan Loepp*, Williams College; Peter McDonald, Williams College; Nina Pande, Williams College; David Schwein, Brown University (1116-13-1023)

Let R be a local ring with maximal ideal M and let R^? denote the M-adic completion of R. It has long been known that the formal fiber rings of R encode important information about R. If P is a prime ideal of R, the formal fiber ring of R at P is defined to be R^? ⊗R k(P ) where k(P ) = RP ⁄PRP . In this talk, we will discuss the dimensions of these formal fiber rings. We will give a history of known results as well as present new results.

10:00 a.m. A ring without a Boij-Soederberg theory

Courtney R. Gibbons*, Hamilton College; Luchezar Avramov, University of Nebraska-Lincoln; Roger Wiegand, University of Nebraska-Lincoln (1116-13-1967)

A graded short Gorenstein ring R can be thought of as a ring with Hilbert series 1 + es + s2, where e is the multiplicity of the ring. In joint work with Avramov and Wiegand, we show that when e ≥ 3, there are Betti diagrams of modules over R that cannot be realized as rational sums of diagrams that lie along extremal rays in the cone of Betti diagrams over R.

10:30 a.m. Polynomials in rigidity theory: special positions of frameworks

Ruimin Cai, Seattle, WA; James Farre, University of Utah; Jessica Sidman*, Mount Holyoke College; Audrey St. John, Mount Holyoke College; Louis Theran, Aalto University; Xilin Yu, Mount Holyoke College

In rigidity theory, a framework is specified by giving n full-dimensional rigid bodies in Rd and a set of geometric constraints among them. The fundamental question is to determine if the framework is rigid or if it admits relative motions between the bodies. Such a framework has an associated multigraph G encoding the combinatorics of the constraints, a rigidity matrix describing the conditions imposed on infinitesimal motions, and a bracket polynomial PG that lives in the homogeneous coordinate ring of a certain Grassmannian. The polynomial PG is the determinant of the rigidity matrix, and the variety it defines consists of special embeddings of the framework with nongeneric behavior. We will discuss how the combinatorics of G can be used to understand the structure of PG.

Friday, January 8, 2016, 1:00 p.m. − 5:20 p.m. (Room 603, Convention Center)

AMS-AWM Special Session on Commutative Algebra and Its Interactions with Algebraic Geometry, I

1:00 p.m. F-thresholds of graded rings

Alessandro De Stefani*, University of Virginia; Luis Núñez-Betancourt, University of Virginia (1116-13-1096)

The F-pure threshold, the diagonal F-threshold, and the a-invariant are three important invariants for standard graded rings of positive characteristic. Hirose, Watanabe, and Yoshida conjectured some relations between these numbers for strongly F-regular rings. We prove their conjecture, only assuming that the ring is F-pure. Furthermore, we give an interpretation of the F -pure threshold of a standard graded Gorenstein algebra in terms of the maximal length of a regular sequence that preserves F-purity at each step.

1:30 p.m. Lower Semi-Continuity of the F-Signature

Thomas Polstra*, University of Missouri-Columbia (1116-13-538)

In this talk, we will discuss some strong Hilbert-Kunz length bounds found in all Noetherian characteristic p rings which are either F-finite or essentially of finite type over an excellent local ring. These numerical bounds will lead to a number of interesting applications. One such application will be an affirmative answer to if the F-signature function is lower semi-continuous on such rings, a problem that has been asked by several people.

2:00 p.m. Closure operations that induce big Cohen-Macaulay modules and algebras, module closures, and classification of singularities

Rebecca R.G.*, University of Michigan (1116-13-1838)

In this talk, we will discuss some strong Hilbert-Kunz length bounds found in all Noetherian characteristic p rings which are either F-finite or essentially of finite type over an excellent local ring. These numerical bounds will lead to a number of interesting applications. One such application will be an affirmative answer to if the F-signature function is lower semi-continuous on such rings, a problem that has been asked by several people.

2:30 p.m. Frobenius actions and a type of singularity in characteristic p > 0

Eric Canton*, University of Nebraska – Lincoln (1116-13-2005)

The singularities of rings with prime characteristic p > 0 are often studied using the p th-power (“Frobenius”) homomorphism. In this talk, we study certain mild singularities of hypersurfaces in characteristic p using a Frobenius homomorphism on a particular module associated to the hypersurface.

3:00 p.m. Lyubeznik numbers, connectivity, and cohomological dimension in mixed characteristic

Daniel J. Hernández, University of Michigan; Luis Núñez-Betancourt, University of Virginia; J. Felipe Pérez*, Georgia State University; Emily E. Witt, University of Kansas (1116-13-1450)

In this short talk we present an extension of the Second Vanishing Theorem for local cohomology modules over regular rings to the (unramified) mixed characteristic situation. We use this theorem, together with new results about Lyubeznik Numbers in mixed characteristic of cohomological complete intersection rings, to give results about the connectedness of the spectrum.

3:30 p.m. Rational singularities and Uniform Symbolic Topologies

Robert M. Walker*, University of Michigan, Ann Arbor (1116-13-1273)

A Noetherian ring R satisfies the uniform symbolic topology property (USTP) if there’s an integer D > 0 such that the symbolic power P(da) ⊆ Pa for all prime ideals P in R and all integers a > 0. In the previous decade, two classes of rings were shown to satisfy the USTP: regular rings of finite type over a field (the Ein-Lazarsfeld-Smith theorem); and reduced isolated singularities that either are F-finite and contain a field of positive characteristic, or are essentially of finite type over a field of characteristic zero (Huneke-Katz-Validashti). In contrast with the regular case, however, the proof in the isolated singularity case is nonconstructive, confirming that a D exists without giving an explicit, effective bound. In this talk, we explain how to find explicit multipliers D for a large class of algebro-geometric surface singularities R (e.g., toric, du Val (ADE)). By reinterpreting classical results of Lipman on rational singularities, we also deduce that all two-dimensional regular rings satisfy the USTP with D = 1, partially extending the Ein-Lazarsfeld-Smith theorem to mixed characteristic.

4:00 p.m. Derived S Systems

Alberto Chiecchio*, TASIS in Dorado; Lance E Miller, University of Arkansas (1116-14-2619)

In his work, A canonical linear system associated to adjoint divisors in characteristic p > 0, Schwede introduced the linear system S that in positive characteristic exhibits similar behaviors to the cohomology in characteristic 0. We discuss the possibility – and the issues – in deriving such systems.

4:30 p.m. Generic vanishing and classification of irregular surfaces in positive characteristics

Yuan Wang*, University of Utah (1116-14-1366)

Classification of surfaces is a very classical topic. A classification for surfaces with Kodaira dimension -1,0 and 1, known as the Enriques-Kodaira Classification, has been given in the last century. But after that the classification of surfaces of general type is much more difficult. There were results in characteristic 0 made between 1982 and 2003. My recent work provides a classification result for surfaces of general type with Euler characteristic 1 and Albanese dimension 4 in positive characteristics, and to the best of my knowledge this is the first explicit classification result for surfaces of general type in positive characteristics. The construction of this result is inspired by a paper of Hacon and Pardini but contains a lot of new ideas, including the construction of a generic vanishing theorem for surfaces that lift to W2(k), the second Witt vector space. In my talk I will present the generic vanishing theorem and explain how it helps in studying the structure of irregular surfaces.

5:00 p.m. Subadditivity of log-Kodaira dimension

Sándor J Kovács*, University of Washington; Zsolt Patakfalvi, Princeton University (1116-14-1507)

This is a report on joint work with Zsolt Patakfalvi. We prove several positivity results for fiber spaces whose general fiber is of log general type and use them to confirm the Iitaka-Viehweg conjecture on the subadditivity of log-Kodaira dimension.

Saturday, January 9, 2016, 8:00 a.m. – 10:55 a.m. (Room 603, Convention Center)

AMS-AWM Special Session on Commutative Algebra, II

8:00 a.m. Rees rings and singularities of curves

Claudia Polini*, University of Notre Dame (1116-13-834)

Let f1,…,fn be forms of the same degree in the polynomial ring R = k[x1,x2] that define a regular map Φ : ℙ1 → ℙn−1 . The bi-homogeneous coordinate ring of the graph of Φ as a subvariety of ℙ1 × ℙn−1 is the Rees algebra ℛ(I) = R[f1t, . . . , fnt] of the ideal I = (f1, . . . , fn) ⊂ R, whereas the homogeneous coordinate ring of the closed image of Φ, the curve X ⊂ P1 parametrized by f1, . . . , fn is the subalgebra k[f1t, . . . , fnt] ≅ ℛ(I) ⊗ k. It is a fundamental problem in elimination theory, commutative algebra, algebraic geometry, and applied mathematics to determine the defining ideals of these rings. Since this is a very ambitious goal, an important first step is to determine or at least bound the (bi)-degrees of the defining equations. In this talk I will survey several approaches to solve this problem. In addition, I will explain how features of the defining ideals correspond to the types and the constellation of the singularities of the curve X.

8:30 a.m. Stabilization of Boij-Söderberg decompositions of systems of ideals

Sarah Mayes-Tang*, Quest University Canada (1116-13-994)

While much work has been done to understand Boij-Söderberg decompositions, the meaning of the diagrams and coefficients that appear in them is not well understood in general. In this talk, we will discuss patterns in the decompositions of Betti tables of systems of related polynomial ideals. In particular, we will describe a stability that characterizes the asymptotic behaviour of these decompositions in certain cases.

9:00 a.m. Interior operations on the set of ideals of a ring

Janet Vassilev*, University of New Mexico (1116-13-1147)

In a recent paper, Epstein and Schwede introduced the tight interior operation which is defined as a dual operation to tight closure. We will discuss interior operations more generally on the set of ideals of a ring and properties that hold when an interior operation and a closure operation are dual to each other.

9:30 a.m. Nilpotent matrices having a given Jordan type as maximum commuting nilpotent orbit

Anthony Iarrobino, Northeastern;
Leila Khatami*, Union College ;
Bart Van Steirteghem, Medgar Evers College, City University of New York ;
Rui Zhao, University of Missouri, Columbia (1116-15-972)

The Jordan type of a nilpotent matrix is the partition giving the sizes of the Jordan blocks in the normal Jordan form of the matrix. In this talk we discuss pairs of partitions (P, Q), where Q = Q(P ) is the Jordan type of a generic element of the nilpotent commutator of the Jordan matrix of type P. In particular, we report on a joint work with A. Iarrobino, B. Van Steirteghem and R. Zha in which we prove a conjecture formulated by P. Oblak in 2012 concerning the cardinality of Q−1(Q) when Q has two parts. We also propose a generalization for an arbitrary partition Q.

10:00 a.m. Uniform Bounds of Artin-Rees type for free resolutions

Ian Aberbach, Department of Mathematics, University of Missouri, Columbia;
Aline Hosry, Department of Mathematics, Faculty of Sciences II, Lebanese University, Fanar, Lebanon;
Janet Striuli*, Department of Mathematics, Fairfield University, Fairfield, Connecticut (1116-13-1138)

Let (R, m) be a local noetherian ring of dimension d. Given a finitely generated R-module M we study a free resolution of M, which we denote by (F*M, ∂ M). We show that there exists a positive integer h such that InFiM ∩ Im( ∂ M ) ⊆
In – hIm(∂i+1M ) for all i ≥ 0, for all n > h, for all the ideals I ⊆ R and for all modules that are d-th syzygies. The proof of this statement involves the definition of Koszul annihilator sequence which we will introduce for the talk.

10:30 a.m. Matrix Schubert varieties and Gaussian conditional independence models

Alex Fink, Queen Mary University of London;
Jenna Rajchgot*, University of Michigan;
Seth Sullivant, North Carolina State University (1116-13-1853)

Matrix Schubert varieties are certain varieties in the affine space of square matrices determined by putting rank conditions on submatrices. I will discuss analogs of these varieties for the spaces of upper triangular and symmetric matrices and show that, as in the traditional matrix Schubert setting, defining ideals have nice Gröbner bases, and primary decomposition of sums of defining ideals can be computed combinatorially.
Our motivation for discussing these upper triangular and symmetric matrix Schubert varieties comes from algebraic statistics. I will explain how to use matrix Schubert varieties to solve two problems concerning Gaussian random variables.
This is joint work with Alex Fink and Seth Sullivant.

11:00 a.m. The parametric variation of A A-hypergeometric functions

Christine Berkesch Zamaere*, University of Minnesota;
Jens Forsgård, Texas A&M University;
Laura Felicia Matusevich, Texas A&M University (1116-13-1299)

A-hypergeometric systems are the D-module counterparts of toric ideals, and their behavior is linked closely to the combinatorics of toric varieties. I will discuss recent work that aims to explain the behavior of the solutions of these systems as their parameters vary. Our goal is to stratify the parameter space so that solutions are locally analytic within each (connected component of a) stratum, and this turns out to be closely related to certain local cohomology modules.

11:30 a.m. Algebraic Methods in Computer Vision

Rekha Thomas*, University of Washington (1116-14-540)

A foundational problem in computer vision is the reconstruction of 3-dimensional scenes from camera images of the scene. In the absence of noise, this reconstruction problem is often equivalent to the existence of a real solution to a system of polynomial equations. This allows one to study these problems using tools from algebraic geometry, commutative algebra, combinatorics and polynomial optimization. In this talk I will describe recent results that have been possible by approaching 3D reconstruction problems from such an algebraic point of view.

Saturday, January 9, 2016, 8:00 a.m. – 10:55 a.m. (Room 4C-3, Convention Center)

AWM Workshop: Special Session on Algebraic Combinatorics, I

8:00 a.m. On Demazure Crystals for the Quantum Affine Algebra Uq(sl^)n)

Margaret L. Rahmoeller*, Roanoke College (1116-08-234)

In 1968, Victor Kac and Robert Moody defined a class of infinite dimensional Lie algebras called affine Lie algebras.
An affine Lie algebra can be viewed as the universal central extension of the Lie algebra of polynomial maps from the unity circle to a finite dimensional simple Lie algebra. Kashiwara showed that irreducible modules for the q-deformed universal enveloping algebra of an affine Lie algebra admit crystal bases. Kang, Kashiwara, Misra, Miwa, Nakashima and Nakayashiki gave the path realizations of affine crystals as a semi-infinite tensor product of some finite crystals called perfect crystals in 1991. In this talk, we use this path realization and show that the union and intersection of certain Demazure crystals for the quantum affine algebra Uq(sl^)n) are finite tensors of the corresponding perfect crystals.

8:30 a.m. Demazure Flags, Chebyshev Polynomials, Mock and Partial Theta Functions

Rekha Biswal, The Institute of Mathematical Sciences, Chennai, India; Vyjayanthi Chari, University of California, Riverside;
Lisa Schneider*, Susquehanna University; Sankaran Viswanath, The Institute of Mathematical Sciences, Chennai, India (1116-05-221)

In this talk, I will present recent joint work with Rekha Biswal, Vyjayanthi Chari, and Sankaran Viswanath concerning the multiplicities associated to Demazure flags of Demazure modules for the current algebra sl2[t]. I will first introduce the notion of a Demazure flag and the associated q-multiplicities. Then I will define generating series which encode these q-multiplicities. Using previous results in representation theory, I will present recursive formulae for these series. Then I will discuss the interesting combinatorics that arise from special cases and the specialization to q = 1. In particular, I will relate these series to Chebyshev polynomials, partial theta functions, and fifth order mock theta functions of Ramanujan.

9:00 a.m. From the weak Bruhat order to crystal graphs as posets

Patricia Hersh*, North Carolina State University, Cristian Lenart, SUNY Albany (1116-05-1386)

Crystal graphs give a combinatorial approach to studying the representation theory of Kac-Moody algebras, and often can be regarded as partially ordered sets. We prove that fundamental properties of the weak Bruhat order transfer to lower intervals in these crystal posets, but that even in type A these properties do not always hold for arbitrary intervals. In particular, for lower intervals we give a crystal theoretic analogue for the statement that any two reduced expressions for the same Coxeter group element are connected by a series of (long and short) braid moves, and we prove that the Moebius function only takes the values 0,1,-1. This Moebius function determination is a consequence of a stronger homotopy theoretic statement. We will also discuss the role of the key of a crystal in this story.

9:30 a.m. Parking Functions, Sandpiles, and Gessel’s Fundamental Basis

Angela S Hicks*, Stanford University (1116-05-900)

The more than decade old shuffle conjecture ties the bi-graded Frobenius characteristic of the diagonal harmonics to two classical statistics (area and dinv) on parking functions, each with an associated quasisymmetric function. It has been previously shown that when we look at only single grading (i.e. only considering the simpler of the two statistics, area) the conjecture is true and the action on parking functions in this case has been given explicitly. A separate bijection (the “phi map”) gives that these two statistics are equidistributed on the parking functions with pmaj and area, but the associated quasisymmetric function is not calculated in the same way as in the first sum. We explain how to calculate it and think about the associated action (in the singly graded case) in the context of a natural statistic on sandpile models.

10:00 a.m. Walking on Representation Graphs and Generalized Hyperbolic Functions

Georgia Benkart*, University of Wisconsin-Madison (1116-05-1120)

A finite group G and a representation of G on a finite-dimensional vector space V determine a certain graph (the so-called representation graph). For example, when G is a product of n copies of the integers modulo 2, V could be taken to be the n-cube. This talk will focus on counting the number of walks from one node to another on such graphs. For any G and V, we give an expression for the number of walks in terms of group characters and show for any abelian group that the exponential generating function for the number of walks can be expressed using generalized hyperbolic functions. The number of walks determines the dimension of the irreducible modules for the centralizer algebra of the action of G on tensor powers of V, so the expressions give those dimensions as well. This is joint work with Dongho Moon.

10:30 a.m. A Random Walk on Sn generated by Random Involutions

Megan M Bernstein*, Georgia Institute of Technology (1116-05-730)

The involution walk is a random walk on the symmetric group generated by random involutions with 2-cycles distributed binomially with parameter p. Using spectral analysis, the involution walk is shown in this paper to mix for p ≥ 1 fixed,
n sufficiently large in between log1/p(n) steps and log2/(1+p)(n) steps. The paper introduces a new technique for finding
eigenvalues of random walks generated by many conjugacy classes using the character polynomial for the characters of the representations of the symmetric group. This is especially efficient at calculating the large eigenvalues of walks with generators with pattered cycle decompositions. The smaller eigenvalues are handled by developing monotonicity relations. These relations also give after sufficient time the likelihood order, the order from most likely to least likely state in the walk. The walk was introduced to study a conjecture about a random walk on the unitary group from the information theory of back holes.

Saturday January 9, 2016, 1:00 p.m.-5:20 p.m. Room 603, Washington State Convention Center

AMS-AWM Special Session on Commutative Algebra and Its Interactions with Algebraic Geometry, II

1:00 p.m. Morita Equivalence Revisited

Paul Frank Baum*, Penn State (1116-16-674)

Let X be a complex affine variety and k its coordinate algebra. Equivalently, k is a unital algebra over the complex numbers which is commutative, finitely generated, and nilpotent-free. A k-algebra is an algebra A over the complex numbers ℂ which is a k-module (with an evident compatibility between the algebra structure of A and the k-module structure of A). A is not required to have a unit. A k-algebra A is of finite type if as a k-module A is finitely generated. This talk will introduce — for finite type k-algebras —a weakening of Morita equivalence called geometric equivalence. The new equivalence relation preserves the primitive ideal space (i.e. the set of equivalence classes of irreducible A- modules) and the periodic cyclic homology of A. However, the new equivalence relation permits a tearing apart of strata in the primitive ideal space which is not allowed by Morita equivalence. The ABPS (Aubert-Baum-Plymen-Solleveld) conjecture asserts that if G is a connected split reductive p-adic group, then the finite type algebra which Bernstein assigns to any given Bernstein component is geometrically equivalent to the coordinate algebra of the associated extended quotient.

1:30 p.m. Centers of endomorphism rings of modules

Haydee M Lindo*, The University of Utah and Williams college (1116-13-2407)

In my talk, I will present some new results concerning the center of the endomorphism ring of a finitely generated module over a commutative noetherian ring. The gist of the results is that the properties of the center are closely related to those of the module itself, especially in the context of one-dimensional Gorenstein rings. Trace ideals of modules are a key ingredient in the proofs.

2:00 p.m. Generalized Associated Primes and Depth in the Perfect Closure

George E Whelan*, George Mason University (1116-13-2486)

Let (R,m) be a local ring and M a module over R. If R is not Noetherian, the concept of an associated prime of M generalizes to definitions of weakly associated primes and strong Krull primes. Similarly the notion of depth requires finer definitions, which were first investigated by Barger and Hochster in the 1970s. In this talk we will establish the relationship between these prime ideals and their corresponding types of depth.
Here we will let (R, m) be a Noetherian ring of characteristic p > 0, and we investigate the perfect closure R∞. The extension R → R∞ has two relevant features: 1) the map Spec(R∞) → Spec(R) is an order isomorphism, and 2) R∞ is almost always Non-Noetherian. We will discuss an arbitrary ideal I ⊂ R and find a direct correspondence between the strong Krull primes of the cyclic module R∞/IR∞and the associated primes of R/(I[pe])F for e ∈ N. We will consider corresponding notions of depth, and generalize the results to (R∞ ⊗ M) for a finitely generated R-module M.

2:30 p.m. Finite F-type and F-abundant Modules

Hailong Dao, University of Kansas;
Tony Se*, University of Kansas (1116-13-897)

We introduce and study basic properties of two types of modules over a commutative Noetherian ring R of positive prime characteristic. The first is the category of modules of finite F-type. They include reflexive ideals representing torsion elements in the divisor class group. The second class is what we call F-abundant modules. These include, for example, the ring R itself and the canonical module when R has positive splitting dimension. We are able to prove many facts about these two categories and how they are related, for example that HomR(M,N) is maximal Cohen-Macaulay when M is of finite F-type and N is F-abundant, plus some extra conditions. Our methods allow us to extend previous results by Patakfalvi-Schwede, Yao and Watanabe. They also afford a deeper understanding of these objects, including complete classifications in many cases of interest.

3:00 p.m. An optimal upper bound on the regularity of powers of edge ideals

Ali Alilooee*, Western Illinois University;
Arindam Banerjee, Purdue University;
Selvi Beyarslan, Tulane University;
Huy Tai Ha, Tulane University (1116-13-1079)

Let R = K[x1,…,xn] be a polynomial ring over a field K and I ⊂ R an ideal. It is well known that if I is a homogeneous ideal whose generates all have the same degree, reg(Is) is asymptotically linear for s ≫ 0. One question that arises here is to find the exact form of this linear function.
Beyarslan, Há and Trung identified this linear function for the edge ideals of trees and cycles. They finished their paper with the following question.
Question 1 Let G be a graph with edge ideal I(G). Let ν(G) denote the induced matching number of G. For which graphs G are the following true?

reg(I(G)s)=2s+ν(G)−1 for s ≫ 0.

Here in this talk we first give an upper bound for the regularity of powers of edge ideals and then we partially answer this question.

3:30 p.m. A McKay correspondence for reflection groups

Ragnar-Olaf Buchweitz, University of Toronto Scarborough;
Eleonore M Faber*, University of Michigan;
Colin Ingalls, University of New Brunswick (1116-14-1953)

The classical McKay correspondence relates the geometry of so-called Kleinian surface singularities with the representation theory of finite subgroups of SL(2, C). There is also an algebraic version of the correspondence, initiated by M. Auslander: let G be a finite subgroup of SL(2,K) for a field K whose characteristic does not divide the order of G. The group acts linearly on the polynomial ring S = K[x, y] and then the so-called skew group algebra A = G ∗ S can be seen as an incarnation of the correspondence.
We want to establish an analogous result when G in GL(n,K) is a finite group generated by reflections, assuming that the characteristic of K does not divide the order of the group. Therefore we consider again the skew group algebra A = G ∗ S, where S is the polynomial ring in n variables, and its quotient A/AeA, where e is the idempotent in A corresponding to the trivial representation. With D the coordinate ring of the discriminant of the group action on S, we show that the ring A/AeA is the endomorphism ring of the direct image of the coordinate ring of the associated hyperplane arrangement.
In this way one obtains a noncommutative resolution of singularities of that discriminant, a hypersurface that is singular in codimension one.

4:00 p.m. Resolution of singularities: new invariants in positive characteristic.

Angelica Benito*, Instituto de Ciencias Matemáticas (ICMAT)
Orlando E. Villamayor U., Universidad Autónoma de Madrid (1116-14-903)

In this talk we present some new invariants of the singularities defined in the case of positive characteristic using com- mutative algebra tools. These invariants are an extension of some others defined in previous jobs (used, in particular, to prove resolution of singularities of 2-dimensional schemes). The improvement of these new ones is that they are now upper semicontinuous. This is a joint work with Orlando E. Villamayor U.

4:30 p.m. Lower bound cluster algebras: presentation and properties

Greg Muller*, University of Michigan;
Jenna Rajchgot, University of Michigan;
Bradley Zykoski, University of Virginia (1116-13-2489)

Cluster algebras are generated by a set of cluster variables which are produced by a recursive process called mutation. Unfortunately, these generating sets are often infinite, even when the algebra can be finitely generated. One workaround is to truncate the recursive process after a finite number of steps; the resulting algebra is called a lower bound cluster algebra.
This talk will review recent work which produced a uniform presentation of every lower bound cluster algebra. We consider a degeneration of the ideal of relations, which allows us to use techniques from combinatorics to prove that lower bound cluster algebras are always normal and Cohen-Macaulay.

5:00 p.m. Notched Arcs of Cluster Algebras from Punctured Surfaces

Emily Gunawan*, University of Minnesota (1116-08-2411)

Cluster algebras, introduced by Sergey Fomin and Andrei Zelevinsky in 2000, are commutative algebras which are defined combinatorially by an iterated process. The notion of cluster algebra links together diverse fields of study, e.g. discrete dynamical systems, Riemann surfaces and Teichmu ̈ller theory, algebraic geometry, and representation theory of quivers. An important class of cluster algebras arise from triangulations of surfaces with marked points. We generalize Ralf Schiffler and Hugh Thomas’ combinatorial T-path formula for arcs of unpunctured surfaces to tagged arcs (possibly with decorations called notchings at their endpoints) of punctured surfaces, and use this to investigate the existence of atomic bases for cluster algebras arising from punctured surfaces.

Saturday January 9, 2016, 2:00 p.m.-4:55 p.m. Room 4C-3, Washington State Convention Center

AWM Workshop: Special Session on Algebraic Combinatorics, II

2:00 p.m. Littlewood-Richardson rules for symmetric skew quasisymmetric Schur functions

Stephanie van Willigenburg*, University of British Columbia (1116-05-827)

Symmetric skew quasisymmetric Schur functions are a generalization of skew Schur functions and contain skew Schur functions as a special case. One way of expanding skew Schur functions in terms of Schur functions is to use the famed version of the classical Littlewood-Richardson rule involving Yamanouchi words. This given, a natural question to consider is whether there exists an analogous rule for symmetric skew quasisymmetric Schur functions.
In this talk we will give two Littlewood-Richardson rules for symmetric skew quasisymmetric Schur functions that are analogous to the aforementioned version of the classical Littlewood-Richardson rule. Furthermore, both our rules have the property that they contain the classical version as a special case. We will then apply our rules to classify symmetric skew quasisymmetric Schur functions diagrammatically. This talk is based on joint work with Christine Bessenrodt and Vasu Tewari.

2:30 p.m. The combinatorics of quasisymmetric (k,l)-hook Schur functions

Elizabeth Niese*, Marshall University;
Sarah K Mason, Wake Forest University (1116-05-150)

In this talk we introduce a refinement of Berele and Regev’s (k, l)-hook Schur functions using quasisymmetric Schur func- tions and row-strict quasisymmetric Schur functions. The quasisymmetric (k, l)-hook Schur functions can be defined as the generating function for a certain set of composition tableaux on two alphabets. We will present combinatorial proper- ties of the quasisymmetric (k, l)-hook Schur functions, including a decomposition into super-fundamental quasisymmetric functions, an analogue of the RSK algorithm, and a generalized Cauchy identity.

3:00 p.m. Discrete Homotopy and Homology Groups

Hélène Barcelo*, Mathematical Sciences Research Institute (1116-05-2166)

Discrete homotopy theory is a (refined) discrete analogue of homotopy theory, associating a (bigraded) sequence of groups to a simplicial complex, capturing its combinatorial structure, rather than its topological structure. It can be defined for graphs, resulting in algebraic invariants that differ substantially from the classical homotopy groups. One can also define discrete homology groups in analogy to the continuous case. We will review these notions and discuss a surprising application.

3:30 p.m. Peak Sets of Classical Coxeter Groups

Alexander Diaz-Lopez, University of Notre Dame;
Pamela Estephania Harris*, United States Military Academy;
Erik Insko, Florida Gulf Coast University;
Darleen Perez-Lavin, Florida Gulf Coast University (1116-05-111)

We say a permutation π = π1π2 ···πn in the symmetric group Sn has a peak at index i if πi−1 < πi > πi+1 and we let P(π) = {i ∈ {1,2,…,n}|i is a peak of π}. Given a set S of positive integers, we let P(S;n) denote the subset of Sn consisting of all permutations π, where P(π) = S. Billey, Burdzy, and Sagan proved |P(S;n)| = p(n)2n−|S|−1, where p(n) is a polynomial of degree max(S) − 1 and Castro-Velez et al. considered the Coxeter group of type Bn as the group of signed permutations on n letters and showed that |PB(S;n)| = p(n)22n−|S|−1 where p(n) is the same polynomial of degree max(S) − 1. In this talk, we embed the Coxeter groups of Lie type Cn and Dn into S2n and partition these groups into bundles of permutations π1π2 · · · πn|πn+1 · · · π2n such that π1π2 · · · πn has the same relative order as some permutation σ1σ2 · · · σn ∈ Sn. This allows us to count the number of permutations in types Cn and Dn with peak set S by reducing the enumeration to calculations in the symmetric group and sums across rows of Pascal’s triangle.

4:00 p.m. The geometry behind permutations and their subwords

Julianna Tymoczko*, Smith College (1116-05-2775)

The geometry of the flag variety and its subvarieties is intimately connected to the combinatorics of the permutation group. On a basic level, the permutations can be viewed as fixed points of the flag variety under a very natural group action. This extends to deeper structural connections involving subwords of permutations and subsets of roots negated by permutations. For instance Billey’s formula defines polynomials that depend on pairs of permutations; these polynomials determine local features of the tangent space of Schubert varieties, as well as their cohomology classes. Other results use subwords of permutations to characterize the Betti numbers of Springer varieties. We will discuss these and other related results.

4:30 p.m. Toric matrix Schubert varieties

Laura Escobar*, U Illinois Urbana-Champaign;
Karola Meszaros, Cornell University (1116-05-755)

Given a matrix Schubert variety Xπ, it can be written as Xπ = Yπ × ℂq (where q is maximal possible). We characterize when Yπ is toric (with respect to a (ℂ∗)2n−1-action) and study the associated polytope of its projectivization. We construct regular triangulations of this polytope which we show are geometric realizations of a family of subword complexes. Subword complexes were introduced by Knutson and Miller, who also showed that they are homeomorphic to balls or spheres and raised the question of their polytopal realizations. Based on joint work with Karola Meszaros.