Title: Packing properties of some classes of cubic squarefree monomial ideals

Let $I$ be an ideal in a Noetherian ring $R$. Its $n$-th symbolic power of I is defined as $$ I(n)= \bigcap_{ p \in Ass(R) } (I^n R_p \cap R). $$ Symbolic powers for several classes of ideals have been investigated for many years. The symbolic powers in general are not equal to the ordinary powers. Therefore, one interesting question here is for what classes of ideals ordinary and symbolic powers coincide? The answer for this question for squarefree monomial ideals may be packing property. In this talk we will survey briefly packing property for squarefree monomial ideals from combinatorial and algebraic aspects. Then we will focus on the cubic squarefree monomial ideals and we will see some new results.

Arindam Banerjee

Title: Regularity of powers of edge ideals: local to global

In this talk we shall discuss Castelnuovo-Mumford regularity of various powers of edge ideals of finite simple graphs. In particularly we shall discuss how those regularities behave when we have conditions on the induced subgraphs obtained from the underlying graphs after deleting neighborhoods of various vertices. Some recent new results will be mentioned and some open problems for future research will be posed.

Soumya Dipta Banerjee

Title: An \(S_n\)-module for \((-1)^{n-1} \nabla p_n\)

Let \(S_n\) denote the symmetric group of degree \(n\). We will present the construction of the \(S_n\)-module with Frobenius characteristic \((-1)^{n-1} \nabla p_n\). This is a joint work with Mahir Can and Adriano Garsia.

Rachelle Bouchat

Title: Minimal Free Resolutions of Domino Ideals

Domino ideals are a class of squarefree monomial ideals arising from domino tilings of a rectangular tableau using 2 x 1 tiles, that is disjoint arrangements of 2 x 1 tiles placed horizontally or vertically to completely cover the area of the rectangle. Domino tilings are a well-studied classical combinatorial object, and in this talk we will show that some of these nice combinatorial results are present in the study of the minimal free resolutions of the associated domino ideals. In particular, we will show that the domino tilings are independent of the characteristic of the underlying field, and we will demonstrate a natural splitting to these ideals that provides a recursive formula for the graded Betti numbers.

Elena Guardo

Title: On Waldschmidt constant and asymptotic resurgence for some ideals in $\mathbb P^n$ and in $\mathbb{P}^1 \times \mathbb{P}^1$

Abstract: We are interested in the ``ideal containment problem'': given a nontrivial homogeneous ideal $I$ of a polynomial ring $R = k[x_1,\ldots,x_n]$ over a field $k$, the problem is to determine all positive integer pairs $(m,r)$ such that $I^{(m)} \subseteq I^r$. Most of the work done up to now has been done for ideals defining 0-dimensional subschemes of projective space. Here, we focus on certain ideals defined by a union of lines in $\pr^3$ which can also be viewed as points in $\popo$. We also consider ideals of $s$ general lines in $\pr^n$. We give results in the case of squarefree monomial ideals. This talk is based on joint papers with B. Harbourne, A. Van Tuyl and MFO Group

Roy Joshua

Title: Equivariant derived categories for toroidal group imbeddings

Let X denote a projective variety over an algebraically closed field on which a linear algebraic group acts with finitely many orbits. Then, a conjecture of Soergel and Lunts in the setting of Koszul duality and Langlands' philosophy, postulates that the equivariant derived category of bounded complexes with constructible equivariant cohomology sheaves on X is equivalent to a full subcategory of the derived category of modules over a graded ring defined as a suitable graded Ext.

Only special cases of this conjecture have been proven so far.

The purpose of this talk is to outline a proof of this conjecture for toroidal imbeddings of complex reductive groups. Since every equivariant imbedding of such a group is dominated by a toroidal imbedding, the class of varieties for which our proof applies is quite large.

We also show that, in general, there exist a countable number of obstructions for this conjecture to be true and that half of these vanish when the odd dimensional equivariant intersection cohomology sheaves on the orbit closures vanish. This last vanishing condition had been proven to be true in many cases of spherical varieties by the speaker and Michel Brion in prior work. A significant part of the current work evolved as a joint project with Michel Brion.

Takayuki Hibi

Title : Extremal Betti numbers of edge ideals

Given integers $r$ and $b$ with $1 \leq b \leq r$, we exhibit a finite simple connected graph $G$ with ${\rm reg}(S/I(G)) = r$, where $S = K[x_1, \ldots, x_n]$ is the polynomial ring over a field $K$ whose variables are the vertices of $G$ and where $I(G)$ is the edge ideal of $G$, for which the number of extremal Betti numbers of $S/I(G)$ is equal to $b$. This is a joint work with Kyouko Kimura and Kazunori Matsuda.

Reuven Hodges

Title: A classification of spherical Schubert varieties in the Grassmannian

Over the last several years, in joint work with Venkatramani Lakshmibai and Mahir Bilen Can, I have been studying Levi subgroup actions on Schubert varieties. This project has two overarching goals. The first goal is the classification of those Schubert varieties that are spherical varieties for the action of a Levi subgroup. In the last decade spherical varieties have been fully classified in terms of combinatorial data referred to as their homogeneous spherical data. Hence the second goal of the project has been to determine the homogeneous spherical data of those Schubert varieties that are spherical varieties. In this talk, I will discuss our recent results that give a complete classification the Schubert varieties in the Grassmannian X_w and the Levi subgroups L for which X_w is a spherical L-variety. I will also discuss progress towards computing the homogeneous spherical data of spherical Schubert varieties in the Grassmannian.

Selvi Kara

Title: Asymptotic behavior of the regularity function of edge ideals

It is well-known that for a homogeneous ideal $I$ in a standard graded algebra over a field, the regularity function $\textrm{reg} I^s$ is asymptotically linear. In this talk, we will discuss the regularity function for edge ideals of graphs. We will explore several classes of graphs for which the regularity function can be explicitly described or bounded in terms of combinatorial data of the graph.

Kiumars Kaveh

Title: Piecewise linear functions and classification of toric families

We consider the notion of a valuation on an algebra/domain with values in an idempotent semialgebra. The case of interest in this talk is the semialgebra of piecewise linear functions in Q^n. We see that T-equivariant (flat) families over a toric variety base are classified by such valuations. In particular, we look at the case of equivariant vector bundles over a toric variety and Klyachko's classification of such bundles. Beside toric vector bundles, this construction includes many examples from combinatorial algebraic geometry. This is a work in progress with Chris Manon.

Kuei-Nuan Lin

Title: Multi-Rees algebras and toric dynamical systems

In this talk, I would explain what is a toric dynamical system associated to a chemical reaction network. In the study of chemical reaction networks, the toric methods introduced by Gatermann were formalized by Craciun, Dickenstein, Shiu and Sturmfels in the paper, Toric Dynamical Systems in 2009. We then recall the muti-Rees algebra associated to a direct sum of ideals. Finally, we explore the relation between multi-Rees algebras and ideals that arise in the study of toric dynamical systems from the theory of chemical reaction networks. This is joint work with David Cox and Gabriel Sosa.

András Lőrincz

Title: Iterated local cohomology groups and Lyubeznik numbers for determinantal rings

We present a recipe for determining iterated local cohomology groups with support in ideals of minors of a generic matrix in characteristic zero, expressing them as direct sums of indecomposable D-modules. For non-square matrices these indecomposables are simple, but this is no longer true for square matrices where the relevant indecomposables arise from the pole order filtration associated with the determinant hypersurface. Specializing our results to a single iteration, we determine the Lyubeznik numbers for all generic determinantal rings, thus answering a question of Hochster. This is joint work with Claudiu Raicu.

Uwe Nagel

Title: Sequences of symmetric ideals

Algebraic-statistical models often have a rich algebraic structure, reflected, for example, in a large symmetry group. It is natural to expect that the properties of the model remain somewhat invariant when one only changes its parameters. We discuss an algebraic framework in order to capture such phenomena. Related questions arise in representation theory and topology. A starting point are ascending chains of symmetric ideals in more and more variables. Finiteness results in a suitable category lead to a uniform description of properties of all but finitely many ideals in such a chain. This is based on joint work with Tim Römer.

Lex Renner

Title: Local invariants of group actions

Geometric invariant theory has many important applications. In particular, it is a fantastic tool for constructing moduli spaces. But, as a theory, it has some limitations.

a. the quotient morphism may have exceptional divisors

b. the quotient morphism may not be "visible"

c. the quotient morphism may not be locally "stable"

d. nonreductive groups are excluded

This presents us with a challenge. Indeed we want to see a theory that produces an appealing quotient morphism for any linear algebraic group action.

To accomplish this, we consider linear algebraic groups and algebraic varieties defined over the algebraically closed field k. Starting with an action G в X --> X, on the normal, quasi-affine variety X, we analyse the maximal G-finite subalgebra of k(X). We use our findings to assess the behaviour of the canonical map p:U --> U//G = Spec(O(U)^G) for a G-invariant, open subset U of X. It turns out that for any G-invariant divisor D of X, there is a G-invariant, open subset V of U such that V meets D, and the canonical morphism p:V --> V//G is "visible" with no exceptional divisors. By a theorem of Nagata and Zariski O(U)^G = O(W), the coordinate ring of some normal, quasi-affine variety W.

Conclusion:

Thanks in part to a theorem of Rosenlicht, every group action is almost locally visible. And, by the above discussion, there are no exceptional divisors locally. Using this, we expect to find a G-invariant, open subset U of X, such that X \ U has codimension two or more in X, and there is a visible quotient morphism p:U --> U/G with no exceptional divisors. This would involve solving some kind of separable gluing problem.

Tim Römer

Title: Asymptotic properties of sequences of symmetric ideals

Symmetric ideals in increasingly larger polynomial rings that form an ascending chain arise in various contexts like algebraic statistics, commutative algebra, and representation theory. In this talk we discuss some recent results and open questions on the asymptotic behavior of algebraic/homological invariants of ideals in such chains. This talk is based on joint work with Dinh Van Le, Uwe Nagel, and Hop D. Nguyen.

Rahul Singh

Title: The conormal variety of a Schubert variety.

Let N be the conormal variety of a Schubert variety X. In this talk, we discuss the geometry of N in two cases, when X is cominuscule, and when X is a divisor. In particular, we present a resolution of singularities and a system of defining equations for N, and also describe certain cases when N is normal, Cohen-Macaulay, and Frobenius split. We will also illustrate the close relationship between conormal varieties and orbital varieties, and discuss the consequences of the above constructions for orbital varieties.

Ozlem Ugurlu

Title: Counting Borel Orbits in Polarization of Type CI

Let $G$ be a complex reductive algebraic group and $B$ be a Borel subgroup of $G$. There are many situations where it is necessary to study the Borel orbits in $G/G^{\theta}$, where $\theta$ is an involutory automorphism. This is equivalent to analyze $K=G^{\theta}$ orbits in the flag variety $G/B$. In our previous work, we studied the combinatorics of the number of Borel orbits in symmetric spaces of classical types $SL(n,\mathds{C})/S(GL(p, \mathds{C})\times GL(q, \mathds{C}))$ (Type AIII), $Sp(2n,\mathds{C})/Sp(2p, \mathds{C})\times Sp(2q, \mathds{C})$ (Type CII) and $SO(2n+1, \mathds{C})/SO(2p, \mathds{C})\times GL(2q+1, \mathds{C})$ (Type BI). By finding their recurrence formulas, we determine their generating series. In particular, we find set of lattice paths weighted by appropriate statistics that are in bijection with the set of Borel orbits in these symmetric spaces. In this talk, we will focus on the polarizations $Sp(n,\mathds{C})/GL(n, \mathds{C})$ (Type CI).

Adam Van Tuyl

Title: The symbolic defect of an ideal

In this talk, I will review the definition of the m-th symbolic power of an ideal and give an overview of the recent problem of comparing the m-th symbolic power of an ideal to its regular power. I will introduce the symbolic defect of an ideal as a way to measure the difference between these two ideals. I'll explain this definition, and describe some of results that compute this invariant in the special case of star configurations. This talk is based upon a project with F. Galetto, A.V. Geramita, and Y.S. Shin.