Mondays, 12:30PM – 1:30 PM
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Spring 2025
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April 28 |
Speaker: Vinh Pham, Tulane University Title: Multiplicity = Volume Formula and Newton Non-degenerate Ideals in Regular Local Domains Abstract: The concept of Newton non-degenerate (NND) ideals in rings of holomorphic germs was introduced by M. J. Saia in 1996 to understand geometric invariants of complex-valued functions with an isolated singularity. We extend this notion to regular local domains and investigate algebraic invariants and properties of graded families of NND ideals in terms of associated convex bodies. We use the limiting body $\mathcal{C}(\mathcal{I})$ associated to a graded family of ideals $\mathcal{I}$ to provide a new understanding of the "Multiplicity = Volume" formula. Particularly, we prove that for a Noetherian graded family $\mathcal{I}$ of $m$-primary ideals in a regular local ring $(R,m)$ of dimension $d$, the equality $e(\mathcal{I})=n!\mathrm{co-vol}_d(\mathcal{C}(\mathcal{I})$ holds if and only if $\mathcal{I}$ contains a certain subfamily of NND ideals. This is joint work with Tai Huy Ha and Thai Thanh Nguyen. |
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April 7 |
Speaker: Matthew Weaver, Arizona State University Title: Implicitization Of Tensor Product Surfaces Via Syzygies Abstract: It is a classical problem in algebraic geometry to find the implicit equations of the image or graph of a rational map between projective spaces. This so-called implicitization problem has been studied to great length by geometers, algebraists and, more recently, the geometric modeling community, for its applications to computer-aided geometric design. As one such application, we consider tensor product surfaces and the implicitization of these varieties, using the syzygies among their parametric equations. We discuss previous and ongoing work, borrowing tools from the study of Rees algebras and using syzygy methods, when these syzygies have low degree. |
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March 31 |
Speaker: Richard Sayanagi, NMSU Title: Non-Noetherian Cohen-Macaulayness in Directed Unions of Power Series Rings, Part 2 Abstract: Power series rings over non-Noetherian rings can exhibit unexpected behavior, like possessing an infinite chain of prime ideals even when the base ring is local and 0-dimensional. We discuss a construction of directed unions of power series rings over non-Noetherian 0-dimensional rings that have properties seen in Noetherian Cohen-Macaulay rings, such as Krull's height theorem for finitely generated ideals and the unmixedness theorem. We make connections to several different notions of non-Noetherian Cohen-Macaulayness in the literature. |
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March 24 |
Speaker: Richard Sayanagi, NMSU Title: Non-Noetherian Cohen-Macaulayness in Directed Unions of Power Series Rings Abstract: Power series rings over non-Noetherian rings can exhibit unexpected behavior, like possessing an infinite chain of prime ideals even when the base ring is local and 0-dimensional. We discuss a construction of directed unions of power series rings over non-Noetherian 0-dimensional rings that have properties seen in Noetherian Cohen-Macaulay rings, such as Krull's height theorem for finitely generated ideals and the unmixedness theorem. We make connections to several different notions of non-Noetherian Cohen-Macaulayness in the literature. |
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March 17 |
Speaker: Vladislav Sliusarev, NMSU Title: Criteria of Local Finiteness for Products of Modal Logics Abstract: A variety of algebras is locally finite, if any finitely generated algebra of this variety is finite. A modal algebra is a Boolean algebra endowed with a family A of unary operators (modalities) satisfying ⋄0 = 0 and ⋄(x∨y) = ⋄x ∨ ⋄y for all ⋄∈A. Segerberg-Maksimova criterion gives an equational condition that is equivalent to local finiteness for the varieties of modal algebras with one modality satisfying ⋄⋄p ≤ ⋄p. No such criterion is known for the general case. In this talk we discuss the varieties defined by modal products of relational structures. We give new criteria of local finiteness of these varieties and describe new previously unknown locally finite varieties. The talk is based on recent joint work with Ilya Shapirovsky. |
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March 3 |
Speaker: Reid Buchanan, Oklahoma State University Title: A Duality between Hilbert Functions of Lex Ideals and Quotients Abstract: We study the Macaulay coefficients induced by the ideal and quotient segments of a degree-δ monomial in n variables. We give explicit formulas for these coefficients and establish a duality between the two theories. Our main result is that the ideal and quotient coefficients form a set partition of {0,1,…,n+δ−2}. |
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February 24 |
Speaker: Michael DiPasquale, NMSU Title: Hyperplane Arrangements, Part 2: Module of Derivations and Jacobian Ideal Abstract: In the first part we discussed combinatorial invariants of arrangements of hyperplanes - in particular the intersection lattice of a hyperplane arrangement. In this second talk we will introduce the module of polynomial vector fields tangent to an arrangement - called the module of logarithmic derivations of the arrangement. We will discuss how this module is related to the Jacobian ideal of the hyperplane arrangement and then discuss some joint work with Jessica Sidman (Amherst College) and Will Traves (Naval Academy) that identifies the top graded component of the saturation of the Jacobian ideal of a hyperplane arrangement with the vector space of 'parallel drawings' of the arrangement. We will focus in particular on how the space of parallel drawings is or is not determined by the intersection lattice. |
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February 17 |
Speaker: Michael DiPasquale, NMSU Title: Hyperplane Arrangements, Part 1: Combinatorics Abstract: Arrangements of hyperplanes are unions of codimension one linear spaces in some vector space over a field. In this talk, I'll give a quick overview of some of the combinatorial aspects of hyperplane arrangements (including things like intersection lattices, characteristic/poincare polynomials, and chamber counting), and may also start to introduce some of the algebraic objects that I am interested in. |
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February 10 |
Speaker: Arvind Kumar, NMSU Title: Detailed Insights into Symbolic Powers of Stanley-Reisner Ideals of Matroids Abstract: In the previous seminar, we unveiled a striking connection between generalized Hamming weights in coding theory and the symbolic powers of Stanley-Reisner ideals of matroids. In this talk, I delve further into the algebraic foundations of this interplay. I will present detailed proof and concrete examples demonstrating how squarefree monomials capture the entire structure of these symbolic powers. |
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February 3 |
Speaker: Arvind Kumar, NMSU Title: Exploring Matroids: From Hamming Weights to Symbolic Powers Abstract: Matroids connect combinatorics, algebra, and geometry in fascinating ways! In this talk, we uncover a surprising link between generalized Hamming weights in coding theory and symbolic powers of Stanley-Reisner ideals of matroids. |
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January 24 |
Speaker: Dustin Ross, San Francisco State University Title: Ehrhart Fans Abstract: Motivated both by lattice point counting in polytopes and by Euler characteristics of sheaves on algebraic varieties, we introduce a new class of polyhedral fans that we call Ehrhart fans. In this talk, we will familiarize ourselves with Ehrhart fans and discuss how they provide a setting in which Euler characteristics of matroids, which were recently introduced by Larson, Li, Payne, and Proudfoot, can be generalized and studied alongside Euler characteristics of sheaves on smooth complete toric varieties. Using a classical lattice point counting interpretation of the latter, we will discuss how ideas from Ehrhart theory can be used to study positivity of Euler characteristics of matroids. This is joint and ongoing work with Melody Chan, Emily Clader, and Carly Klivans. |
Fall 2024
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December 5 |
Speaker: Gabriel Sosa Castillo, Colgate University Title: Algebraic Invariants of Rees and Multi-rees Algebras of Principal (Pure) Lexsegment Ideals Abstract: Using tools from computational and combinatorial commutative algebra such as Gröbner basis, initial ideals, the Stanley Reisner ideal and the Alexander dual, we determine the exact values of algebraic invariants such as the regularity, the reduction number and the analytic spread of the special fiber, and associated ideal/module, of a Rees (or multi-Rees) algebra of a direct sum of copies of a fixed principal (pure) Lexsegment ideal. Our results expand on bounds that Jonathan Montaño had determined for the reduction number of a principal Lexsegment ideal in his 2015 thesis and have implications regarding the core of monomial ideals. Our methods also allow us to characterize when these Rees algebras are Gorenstein. This is joint work with Alessandra Costantini and Kuei-Nuan Lin. |
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November 21 |
Speaker: Arvind Kumar, New Mexico State University Title: Rees Algebra of Almost Complete Intersection Binomial Edge Ideals Abstract: his talk will focus on almost complete intersection binomial edge ideals. We will explore their structure and then discuss the defining equations of the presentation ideal for their Rees algebras, emphasizing key relations and algebraic properties. |
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November 14 |
Speaker: Janet Vassilev, University of New Mexico Title: The Differential Powers Of The Homogeneous Maximal Ideal In Affine Semigroup Rings Abstract: We will discuss the Saito-Traves description of the ring of differential operators of an affine semigroup ring with complex coefficients and present how we can use this description of the ring of differential operators to determine the differential powers of the homogeneous maximal ideal. |
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November 8 |
Speaker: Tan Nhat Tran, Binghampton University Title: Worpitzky-compatible Sets And The Freeness Of Arrangements Between Shi And Catalan Abstract: Given an irreducible root system, the Worpitzky-compatible subsets are defined by a geometric property of the alcoves inside the fundamental parallelepiped of the root system. This concept is motivated and mainly understood through a lattice point counting formula concerning the characteristic and Ehrhart quasi-polynomials. In this talk, we show that the Worpitzky-compatibility has a simple combinatorial characterization in terms of roots. As a byproduct, we obtain a complete characterization by means of Worpitzky-compatibility for the freeness of the arrangements interpolating between the extended Shi and Catalan arrangements. This completes the earlier result by Yoshinaga in 2010 which was done for simply-laced root systems. This is joint work (arXiv:2403.17274) with Takuro Abe (Tokyo). |
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October 31 |
Speaker: Giovanni Secreti, New Mexico State University |
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October 24 |
Speaker: Lance E. Miller, University of Arkansas Title: Frobenius And Tight Buchsbaum Conditions Abstract: Buchsbaum rings are direct generalizations of Cohen-Macaulay rings, specifically a ring is Buchsbaum if and only if the difference between a parameter ideal's Hilbert-Samuel multiplicity and its colength is a constant depending only on the ring. Using definitions from tight closure theory in positive characteristic, Ma and Quy studied a similar condition called tight Buchsbaum, which requires that the Hilbert-Samuel multiplicity of a parameter ideal minus the colength of the tight closure of the parameter ideal be constant depending only on the ring, and is thus related to F-rational singularities. We continue the story by studying a similar condition which utilizes Frobenius closure in place of tight closure, and is thus related to F-injective singularities. This is work in progress with Kriti Goel, Kyle Maddox, and Austyn Simpson. |
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October 17 |
Speaker: Richard Sayanagi, New Mexico State University Title: Power Series Rings Over Zero-dimensional Rings Abstract: Properties of power series rings over Noetherian rings, such as its Krull dimension and flatness, are well-understood, but when the base ring is non-Noetherian, the power series ring can fail to be flat over the base ring, and its dimension can be infinite, even when the dimension of the base ring is finite. We study the case when the base ring has Krull dimension 0, and consider a version of the power series ring which preserves flatness and whose dimension remains finite. |
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October 10 |
Speaker: Aycin Iplikci Arodirik, Ohio State University Title: A Way to View Quasi-Coherent Sheaves of Ideals as Ideals of a Ring Abstract: A blow-up $X$ of an integral domain along an ideal is a projective scheme. In a naive sense, we want to consider this projective scheme as an affine scheme. We construct an associated ring $X^*$ and present a way to view the quasi-coherent sheaves of ideals belonging to the scheme $X$ as ideals of the ring $X^*$. The construction is based on the Nagata extension, which is a particular localization of polynomial rings of a given ring. First, we extend the notion of homogeneous ideals from polynomial rings to Nagata extension rings and then to $X^*$. In this extended sense, homogeneous ideals will be relevant to our purpose and we can establish a one-to-one correspondence between the set of relevant ideals of $X^*$ and the set of quasi-coherent sheaves of ideals on $X$. No knowledge of algebraic geometry is required to follow this talk, as our goal is obtaining insight into the projective schemes that correspond to blow-ups from a ring theory perspective. |
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October 03 |
Speaker: Michael DiPasquale, New Mexico State University Title: A Duality For Sequences Arising From Apolarity Abstract: In this talk we discuss a duality for sequences of natural numbers which is a discrete analogue of the notion of a pseudo inverse function. We discuss manifestations of this duality in the context of commutative algebra and algebraic geometry. The first is a duality between regularity of ideals generated by powers of linear forms and the initial degree of symbolic powers (these are linked via apolarity). The second (discussed if time permits) is a duality between the jet separation sequence of a set of points and the regularity of symbolic powers of the ideal of the same set of points. This is joint work with Thai Nguyen and Alexandra Seceleanu. |
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September 24 |
Speaker: Michael DiPasquale, New Mexico State University Title: Inverse Systems And Symbolic Powers Abstract: Suppose S and R are both polynomial rings over a field in the same number of variables. The apolarity action of R on S is defined by letting polynomials in R act as partial differential operators on S. Under this action, the inverse system of an ideal I in R (first introduced by Macaulay) is the collection of all elements of S annihilated by all polynomials in the ideal I. In this talk we introduce the apolarity action and a beautiful result of Emsalem and Iarrobino, which characterizes the inverse system of a symbolic power of an ideal (symbolic powers will be defined!). This is useful background for a subsequent talk on work with Seceleanu and Nguyen. |
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September 12 |
Speaker: Louiza Fouli, New Mexico State University Title: Blowup Algebras Of Ladder Determinantal Modules, Part II Abstract: In this second part we will examine blowup algebras of modules that are direct sums of ladder determinantal ideals. For these algebras, we compute several key invariants, including regularity, a-invariant, and reduction number. Additionally, we establish bounds on their multiplicity and provide both necessary and sufficient conditions for the algebras to be Gorenstein. This is based on joint work with A. Costantini, K. Goel, K.-N. Lin, H. Lindo, W. Liske, and M. Mostafazadehfard. |
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September 5 |
Speaker: Louiza Fouli, New Mexico State University Title: Blowup Algebras Of Ladder Determinantal Modules, Part I Abstract: In the first part of the talk we will discuss the general theory of blowup algebras. This part is meant to be introductory. In the second part we will examine blowup algebras of modules that are direct sums of ladder determinantal ideals. For these algebras, we compute several key invariants, including regularity, $a$-invariant, and reduction number. Additionally, we establish bounds on their multiplicity and provide both necessary and sufficient conditions for the algebras to be Gorenstein. This is based on joint work with A. Costantini, K. Goel, K.-N. Lin, H. Lindo, W. Liske, and M. Mostafazadehfard. |
Spring 2024
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April 15 |
Speaker: Jake Kettinger, Colorado State University Title: New Perspectives on Geproci Sets Abstract: The geproci property is a recent development in the world of geometry. We call a set of points Z ⊆ P3 k an (a, b)-geproci set (for GEneral PROjection is a Complete Intersection) if its projection from a general point P to a plane is a complete intersection of curves of degrees a and b. Examples known as grids have been known since 2011 and nondegenerate non-grids have been known only since 2018. Previously, the study of the geproci property has taken place within the characteristic 0 setting; prior to the work in my thesis, a procedure was known for creating specific nondegenerate non-grid (a, b)-geproci sets for 4 ≤ a ≤ b, but it was not known what other examples there can be. Furthermore, before the work in my thesis, almost all examples of geproci sets that were known were contained in unions of b disjoint lines (known as half grids) and there was no known way to generate new examples of non-half grids. Here, I will discuss how to use geometry in positive characteristics to find new methods of producing geproci half grids and non-half grids. |
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April 4 |
Speaker: Sankhaneel Bisui, Arizona State University Title: Rational Powers and Summation Formula Abstract: In commutative algebra, the summation formula is very well studied. Mustaţă proved the summation formula for multiplier ideals. In recent work, Hà, Jayanthan, Kumar, and Nguyen proved the summation formula for symbolic powers. Banerjee and Hà proved that the summation formula holds for rational powers of monomial ideals. In a recent project, S. Das, T.H.Hà, J. Montaño, and I also investigated the summation formula for rational powers. In that project, we introduced the concept of the Rees package, which ensures the summation formula for rational powers. In this talk, I will describe the necessary terms with examples. I will also describe the classes of ideals that satisfy the summation formula for rational powers. I will discuss the results from the joint work with S. Das, T.H.Hà, and J. Montaño. |
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March 25 |
Speaker: Alperen Ergur, University of Texas at San Antonio Title: Counting Zeros of Sparse Real Polynomial Systems Abstract: Descartes' rule of signs gives an upper bound for the number of real zeros using only number of terms, and is independent of the degree. |
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March 18 |
Speaker: Hoai Dao, Oklahoma State University Title: A new symmetric resolution for the (pinched) nth Veronese Abstract: Let S = k[x_1,..., x_n] be a polynomial ring over an arbitrary field k. We construct a new symmetric polytopal minimal resolution of (x_1,..., x_n)^n and a symmetric polytopal minimal resolution of an equigenerated monomial ideal obtained by removing x_1x_2...x_n from the generators of (x_1,..., x_n)^n. |
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February 27 |
Speaker: Michele Torielli, Northern Arizona University Title: Lefschetz Properties and Arrangements Abstract: In this talk we will discuss the notion of Weak and Strong Lefschetz properties for graded ring. We will then connect these properties to the study of generic initial ideals. Finally, we will discuss almost complete intersection and hyperplane arrangements from the point of view of Lefschetz properties. This is partially based on a collaboration with S. Marchesi and E. Palezzato. |
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February 19 |
Speaker: Le Tran, New Mexico State University Title: Associated Primes of Monomial Ideals and Depth of Edge Ideals of Unicyclic Graphs Abstract: In this talk, we will investigate the associated primes of monomial ideals. In particular, for an arbitrary monomial ideal I, we describe the minimal associated primes of ideals of the form in_>(I,f), where f is a linear form, in terms of the minimal associated primes of I. Moreover, we will describe the embedded associated primes of arbitrary monomial ideals using the notion of star neighbor sets. The analysis of associated primes of monomial ideals will be used to establish initially regular sequences (introduced by Fouli, Hà, and Morey) on cycles C_{3n+2} for n \ge 1. In the last part of the talk, we will discuss the depth of edge ideals of certain unicyclic graphs. |
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February 12 |
Speaker: Michael DiPasquale, New Mexico State University Title: An Introduction to Symbolic Powers of Ideals, Part II |
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February 5 |
Speaker: Michael DiPasquale, New Mexico State University Title: An Introduction to Symbolic Powers of Ideals Abstract: The Waring rank of a homogeneous form is the number of terms needed to write the form as a sum of powers of linear forms. I will give an introduction and overview of some of the questions around Waring rank. First, motivated by conjectures in complexity theory, it is of interest to determine and compare the Waring ranks of the determinant and permanent. At this time very little is known; in fact, there are very few forms whose Waring rank is known. I will discuss some of the known lower bounds for Waring rank and approaches to determining Waring ranks. The maximum value of Waring rank is unknown in all but a handful of cases; it is even unknown in most cases whether forms with higher than generic rank exist. I will describe ongoing work on apolarity of general hyperplane arrangements. The symmetric analogue of Strassen's conjecture asserts that Waring rank is additive for forms in independent variables. This remains open, despite Shitov's recent counterexample for Strassen's conjecture in the tensor (non-symmetric) case. Time permitting, I will discuss some sufficient conditions for forms to satisfy the assertion of the conjecture. |
Fall 2023
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November 13 |
Speaker: Zach Teitler, Boise State University Title: Invitation to Waring rank Abstract: The Waring rank of a homogeneous form is the number of terms needed to write the form as a sum of powers of linear forms. I will give an introduction and overview of some of the questions around Waring rank. First, motivated by conjectures in complexity theory, it is of interest to determine and compare the Waring ranks of the determinant and permanent. At this time very little is known; in fact, there are very few forms whose Waring rank is known. I will discuss some of the known lower bounds for Waring rank and approaches to determining Waring ranks. The maximum value of Waring rank is unknown in all but a handful of cases; it is even unknown in most cases whether forms with higher than generic rank exist. I will describe ongoing work on apolarity of general hyperplane arrangements. The symmetric analogue of Strassen's conjecture asserts that Waring rank is additive for forms in independent variables. This remains open, despite Shitov's recent counterexample for Strassen's conjecture in the tensor (non-symmetric) case. Time permitting, I will discuss some sufficient conditions for forms to satisfy the assertion of the conjecture. |
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November 6 |
Speaker: Matt Mastroeni, Iowa State University Title: Koszul Graded Möbius Algebras and Strongly Chordal Graphs Abstract: The graded Möbius algebra of a matroid is a graded commutative algebra that encodes the combinatorics of the lattice of flats of the matroid. As a special subalgebra of the augmented Chow ring of the matroid, these rings played an important role in the recent proof of the Dowling-Wilson Top Heavy Conjecture, which interpolates between the good algebraic properties of the augmented Chow ring and the combinatorics of the graded Möbius algebra. Recently, Mastroeni and McCullough proved that the Chow ring and augmented Chow ring of a matroid are also Koszul. We will discuss joint work with Jason McCullough and Irena Peeva studying when graded Möbius algebras are Koszul, drawing parallels with the much better studied case of Orlik-Solomon algebras throughout. In the case of graphic matroids, our results suggest a new characterization of strongly chordal graphs via edge orderings. (No prior experience with matroids will be necessary for this talk.) |
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October 30 |
Speaker: Bruce Olberding, New Mexico State University Title: Connectedness in the Space of Valuation Rings of a Field, Part II |
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October 23 |
Speaker: Bruce Olberding, New Mexico State University Title: Connectedness in the Space of Valuation Rings of a Field, Part I Abstract: Zariski’s Connectedness Theorem asserts connectedness for fibers of closed points of certain projective morphisms. This talk focuses on a lifting of this theorem to the space of valuation rings, with the goal of giving a topological criterion for when an intersection of valuation rings is a local ring. We also discuss the case of valuation overrings of a two-dimensional regular local ring, where the topological criterion can be strengthened to a kind of Nullstellensatz for the space of valuation rings. This is joint work with W. Heinzer, A. Loper and M. Toeniskoetter. |
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October 16 |
Speaker: Ilya Shapirovskiy, New Mexico State University Title: Syntactic and semantic conditions for local finiteness of modal algebras Abstract: A modal algebra is a Boolean algebra endowed with a unary operation that distributes with respect to finite joints. Important examples are powerset algebras of directed graphs, where the modal operation is induced by the edges. We will be interested in conditions for local finiteness of such algebras. Some known results and open problems will be discussed. |
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October 2 |
Speaker: Stefan Tohaneanu, University of Idaho Title: The geometry of the minimum distance Abstract: Let $\mathbb K$ be any field and let $X\subset\mathbb P^{k-1}$ be a set of $n$ distinct $\mathbb K$-rational points, not all contained in a hyperplane. Let $a\geq 1$ be an integer and let $C(X)_a$ be the evaluation code of order $a$ on $X$. This linear code has minimum distance, $d(X)_a$, equal to $n$ minus the maximum cardinality of a proper subset of $X$ that is contained on a hypersurface of degree $a$. |
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September 22 |
Speaker: Frank Sottiler, Texas A&M Title: Critical Points Of Discrete Periodic Operators Abstract: It is believed that the dispersion relation of a Schrodinger operator with a periodic potential has non-degenerate critical points, for general values of the potential and interaction strengths. In work with Kuchment and Do, we considered this for discrete operators on a periodic graph G, for then the dispersion relation is an algebraic hypersurface. We showed how, for a given periodic graph G, this may be established from a single numerical verification, if we knew the number of critical points for |
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September 18 |
Speaker: Arvind Kumar, New Mexico State University Title: Koszul Algebras and Subadditivity of maximal shifts, Part II |
Spring 2023
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May 1 |
Speaker: Giulio Peruginelli, Università di Padova Title: Polynomial Dedekind Domains Abstract: the link |
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April 24 |
Speaker: Pat Morandi, New Mexico State University Title: A unified approach to Gelfand and de Vries dualities: Comparing various dualities with the category of compact Hausdorff spaces, Part II |
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April 17 |
Speaker: Pat Morandi, New Mexico State University Title: A unified approach to Gelfand and de Vries dualities: Comparing various dualities with the category of compact Hausdorff spaces Abstract: There are several categories of algebras that are dually equivalent to the category of compact Hausdorff spaces and continuous maps and are therefore equivalent to each other. The proof of each duality requires the axiom of choice. In these talks we will discuss the various categories of algebras and how each type of algebra arises from a topological space. We will then discuss the tools we use to give choice-free arguments to show that the various categories are equivalent. This is joint work with Guram Bezhanishvili, Luca Carai, and Bruce Olberding. |
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April 10 |
Speaker: Kuei-Nuan Lin, Penn State University, Greater Allegheny Title: Rees algebras of determinantal ideals Abstract: We will give an introduction to SAGBI basis theory and its application to the Rees algebra. We then present a couple of recent results using SAGBI basis theory to find the presentation equations of Rees algebras of determinantal ideals and their properties. |
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March 27 |
Speaker: Laura Cossu, University of Graz, Austria Title: Factorization beyond Factorization Theory II |
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March 20 |
Speaker: Laura Cossu, University of Graz, Austria Title: Factorization beyond Factorization Theory I Abstract: Many problems in algebra involve the decomposition of certain elements of a ring (or more generally of a monoid) into a product of certain other elements (hereinafter generically referred to as "building blocks") that are in some sense minimal. The classical theory of factorization investigates factorizations in which the building blocks are atoms, i.e., non-unit elements of a monoid that are not products of two non-units. For example, it is well known that every non-zero non-unit of a Dedekind domain (more generally, of a Noetherian domain) can be written as a finite product of atoms and that in general such decompositions are not unique. On the other hand, examples of factorizations that lie beyond the scope of the classical theory include additive decompositions into multiplicative units in rings; cyclic decompositions of permutations in the symmetric group of degree n; idempotent factorizations of the "singular elements'' of a monoid; and so on. |
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February 27 |
Speaker: Janet Vassilev, University of New Mexico Title: Constructing new closures and interiors Abstract: In extending a closure operation defined on the ideals of a ring R to a closure on the submodules R-modules, typically 3 different methods have been used. We will discuss these methods along with some standard examples of these extensions. Then we will show that these constructions work to produce new operations from old for a broader class of operations which include interior operations. This talk is based on joint work with Neil Epstein and Rebecca R.G. |
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February 20 |
Speaker: Bruce Olberding, New Mexico State University Title: Varieties as topological spaces, Part II |
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February 13 |
Speaker: Bruce Olberding, New Mexico State University Title: Varieties as topological spaces, Part I Abstract: We discuss recent work of Kollar, Lieblich and Olsson on the underlying topology of varieties, as well as previous work of R. Wiegand on this topic. |
Fall 2022
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November 28 |
Speaker: Bruce Olberding, New Mexico State University Title: A very non-Noetherian ring in a Noetherian setting Abstract: There exist a great variety of non-Noetherian rings (and Noetherian rings, for that matter) but some of the textbook non-Noetherian examples occur in settings that are less familiar. In this talk we do a side-by-side comparison of a certain Noetherian ring and a certain non-Noetherian ring that occur together in a very natural setting. One is familiar (a polynomial ring), and the other not so much (a holomorphy ring). We use these examples to mention some different approaches to commutative algebra that share familiar origins but different goals. |
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November 14 |
Speaker: Louiza Fouli, New Mexico State University |
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November 7 |
Speaker: Louiza Fouli, New Mexico State University Title: The Depth Function for Monomial Ideals, Part I Abstract: Let $R$ be a polynomial ring and $I$ a monomial ideal in $R$. One important invariant associated to the ideal $I$ is the depth of $R/I$ or more generally, the depth function $f(t)=\depth R/I^t$ for $t\in \mathbb{N}$. We will discuss known and new results regarding this function. |
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October 31 |
Speaker: Le Tran, New Mexico State University Title: Initially regular sequences on cycles, Part II |
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October 24 |
Speaker: Le Tran, New Mexico State University Title: Initially regular sequences on cycles, Part I Abstract: Let $I$ be a homogeneous ideal in a polynomial ring $R$. Initially regular sequences on $R/I$ are a type of sequence that behaves like regular sequences and whose length provides a lower bound for the depth of $R/I$. We discuss the notion of initially regular sequences on $R/I$ and give an explicit description of an initially regular sequence of length equal to the depth of $R/I$, where $I$ is the edge ideal of any cycle $C_{n}$, for $n\ge 3$. We examine the associated primes of the initial ideal of the form $\text{in}_>(I,f)$, where $f$ is a trinomial and show that they arise from the associated primes of the ideal $I$. I will also discuss some results for the depth of edge ideals of graphs that are extensions of cycles. |
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October 17 |
Speaker: Louiza Fouli, New Mexico State University Title: On the core of ideals and modules, Part II |
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October 10 |
Speaker: Louiza Fouli, New Mexico State University Title: On the core of ideals and modules, Part I Abstract: In this talk we will discuss the notions of reductions of an ideal and core of an ideal in a Noetherian local ring. We will review basic properties and known results on this topic. As time permits, we will discuss the generalization to minimal reductions of modules and cores of modules. |
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September 19 |
Speaker: Alessandra Costantini, Oklahoma State University Title: Ordinary and symbolic powers of symmetric strongly shifted ideals Abstract: Symmetric strongly shifted ideals are a special class of monomial ideals, which is invariant under the action of the symmetric group by permutation of the variables. In this talk, I will describe how their combinatorial structure dictates the algebraic properties of their ordinary and symbolic powers. |
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September 12 |
Speaker: Bruce Olberding, New Mexico State University Title: Local rings and connectedness |
Spring 2022
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April 4 |
Speaker: Thai T Nguyen, Tulane University Title: Chudnovsky's Conjecture for General Points Abstract: Chudnovsky’s Conjecture suggests lower bounds for the degrees of elements in the symbolic powers of the defining ideal of a set of finitely many points in projective space. In this talk, we will discuss some recent developments on this conjecture with focus on results for general points. I will also present a result for sufficiently large number of general points from our joint work with Sankhaneel Bisui, Eloísa Grifo and Tài Huy Hà; and a more recent result for a smaller number of general points from our joint work with Sankhaneel Bisui. |
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March 28 |
Speaker: Janet Vassilev, University of New Mexico Title: Dualizing operations defined via colons Abstract: Building on the duality for submodule selectors developed by Epstein and R.G., we develop a duality between pairs of modules, which we use to produce a dual interior operation for basically full closure on a pair of Artinian modules called basically empty interior which also has a nice formula in terms of colons. Through our duality, we are also able to develop criteria for when a submodule of the injective hull of the residue field is integrally open and formulas to compute the integral-hull of some submodules of the injective hull of the residue field. This is talk is based on joint work with Epstein and R.G. |
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March 14 |
Speaker: Marco Abbadini, University Of Salerno, Italy Title: Free extension of universal algebras Abstract: Given an equational class of algebras (such as groups, Boolean algebras, etc.), and a fixed sublanguage of this class (such as monoid operations, lattice operations, etc.), we can show the equivalence of two properties. The first, which is called free extension property, is more semantic: it concerns extensions of certain partial functions to homomorphisms. Whereas the second, called expressibility of equations, is concerned with terms and identities, thus being more syntactic. |
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February 28 |
Speaker: Art Duval, UT El Paso Title: Enumerating simplicial spanning trees of shifted and color-shifted complexes, using simplicial effective resistance Abstract: Simplicial electrical networks generalize electrical networks from graphs to higher dimensional simplicial complexes, where resistances, currents, and voltages on the facets of the complex satisfy a generalized Ohm’s law. Simplicial effective resistance, developed by Kook and Lee, generalizes to this setting the notion of effective resistance, which is the resistance of a new facet required to replace a network of resistors. We use simplicial effective resistance to enumerate the simplicial spanning trees of shifted complexes, reproving a known result, and of color-shifted complexes, proving a previously conjectured result. |
Fall 2021
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November 15 |
Speaker: Igor Arrieta, University of Coimbra Title: A new diagonal separation and its relations with the Hausdorff property Abstract: Let P be a property of subobjects relevant in a category C. An object X in C is P-separated if the diagonal in X × X has P; thus e.g. closedness in the category of topological spaces (resp. locales) induces the Hausdorff (resp. strong Hausdorff) axiom. Moreover, a topological space is T_1 if and only if the diagonal is an intersection of open subspaces. In this talk we consider locales whose diagonal is fitted (i.e., an intersection of open sublocales - we speak about F-separated locales). Among others, we will show that F-separatedness is a property strictly weaker than fitness. Moreover, we will explore a pleasant parallel with the strong Hausdorff axiom, including a Dowker-Strauss type theorem and a characterization in terms of certain relaxed morphisms (preframe homomorphisms preserving covers in a suitable sense). We will also compare F-separatedness with other point-free separation axioms and finish with some open questions. This is a joint work with Jorge Picado and Aleš Pultr. |
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November 8 |
Speaker: Sudipta Das, New Mexico State University Title: A volume = multiplicity formula for Hilbert-Kunz multiplicity |
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October 18 |
Speaker: Peter Jipsen, Chapman University Title: Unary-determined distributive l-magmas Abstract: A distributive lattice-ordered magma (dl-magma) (A,^,v,⬝) is a distributive lattice with a binary operation⬝ that preserves joins in both arguments, and when ⬝ is associative then (A,v,⬝) is an idempotent semiring. A dl-magma with a top T is unary-determined if xy = (xT ^ y) v (x ^ Ty). These algebras are term-equivalent to a subvariety of distributive lattices with T and two join-preserving unary operations p,q. We obtain simple conditions on p,q such that xy = (p(x) ^ y) v (x ^ q(y)) is associative, commutative, idempotent and/or has an identity element. Furthermore, we show that a dl-magma with a Boolean lattice reduct and idempotent ⬝ (xx=x) is always unary-determined. |
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October 4 |
Speaker: Jonathan Montano Martinez, New Mexico State University Title: Blowup algebras of determinantal ideals in prime characteristic |
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September 27 |
Speaker: Bruce Olberding, New Mexico State University Title: Quadratic Transforms, Part 2 |
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September 20 |
Speaker: Bruce Olberding, New Mexico State University Title: Quadratic Transforms |
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September 13 |
Speaker: Marcus Tressl, University of Manchester Title: An Introduction to Real Closed Rings Abstract: A real closed ring is -- roughly speaking -- an abstract version of a ring of continuous real valued functions on a topological space. |
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August 30 |
Speaker: Vladislav Slyusarev, New Mexico State University Title: Modal Logic of Cayley Graphs |
Spring 2021
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May 3 |
Speaker: Nick Galatos, University of Denver Title: Almost minimal varieties of commutative integral residuated lattices (joint work with P. Agliano and M. Marcos) Abstract: Residuated structures include lattice-ordered groups, relation algebras, and ideal lattices of rings. They also form algebraic semantics of various non-classical logics, such as linear, relevance, many-valued and intuitionistic; therefore Boolean and Heyting algebras also form examples. Maximal consistent logics correspond to minimal non-trivial varieties of residuated lattices. We focus on the case of integral and commutative residuated lattices and we study their minimal and almost minimal subvarieties, showing that even at these very low levels there is a lot of complexity. We will start with a quick review of the necessary tools from universal algebra. |
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April 26 |
Speaker: Susan Morey, Texas State University Title: Resolutions of Powers of Monomial Ideals Abstract: Using combinatorial structures to obtain resolutions of monomial ideals is an idea that traces back to Diana Taylor’s thesis, where a simplex associated to the generators of a monomial ideal was used to construct a free resolution of the ideal. This concept has been expanded over the years, with various authors determining conditions under which simplicial or cellular complexes can be associated to monomial ideals in ways that produce a free resolution. |
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April 19 |
Speaker: Jim Madden, Louisiana State University Title: Conjunctive Join Semilattices Abstract: A join-semilattice L is said to be conjunctive if it has a top element 1 and it satisfies the following first-order condition: for any two distinct a, b in L, there is c in L such that EXACTLY one of the two suprema a∨c and b∨c is equal to 1. Equivalently a join-semilattice is conjunctive if every principal ideal is an intersection of maximal ideals. This talk discusses the history of the concept and numerous applications in topology, frame theory, and lattice theory. |
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April 12 |
Speaker: James Lewis, New Mexico State University Title: Binomial Edge Ideals of Graphs and Ideals of Konig-type |
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March 29 |
Speaker: Robert Walker, University of Wisconsin Title: Uniform Asymptotic Growth of Symbolic Powers of Ideals Abstract: Algebraic geometry (AG) is a major generalization of linear algebra which is fairly influential in mathematics. Since the 1980's with the development of computer algebra systems like Mathematica, AG has been leveraged in areas of STEM as diverse as statistics, robotic kinematics, computer science/geometric modeling, and mirror symmetry. Part one of my talk will be a brief introduction to AG, to two notions of taking powers of ideals (regular vs symbolic) in Noetherian commutative rings, and to the ideal containment problem that I study in my thesis. Part two of my talk will focus on stating the main results of my thesis in a user-ready form, giving a "comical" example or two of how to use them. At the risk of sounding like Paul Rudd in Ant-Man, I hope this talk will be awesome. |
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March 15 |
Speaker: Warren McGovern, Florida Atlantic University Title: When is C(X) an h-local Ring Abstract: The h-local domains were originally studied by Matlis in an attempt to generalize results on Dedekind domains and local domains. The domain D is said to be h-local if every non-zero prime ideal is contained in a unique maximal ideal and every non-zero element has finite character. Olberding (2007) has a list of 20 equivalent conditions characterizing h-local domains. |
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March 1 |
Speaker: Darío García, Universidad de los Andes Title: Pseudofinite Dimensions and Cardinalities of Definable Sets in Finite Structures Abstract: The fundamental theorem of ultraproducts ( Los' Theorem) provides a transference principle between the finite structures and their limits. It states that a formula is true in the ultraproduct M of an infinite class of structures if and only if it is true for ”almost every” structure in the class, which presents an interesting duality between finite structures and their infinite ultraproducts. This kind of finite/infinite connection can sometimes be used to prove qualitative properties of large finite structures using the powerful known methods and results coming from infinite model theory, and in the other direction, quantitative properties in the finite structures often induce desirable model-theoretic properties in their ultraproducts. These ideas were used by Hrushovski (cf. [2]) to apply ideas from geometric model theory to additive combinatorics, locally compact groups and linear approximate subgroups. |
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February 22 |
Speaker: Keller Vandebogert, University of South Carolina Title: Linear Strands Supported on Cell Complexes Abstract: In this talk, we will consider linear strands of ideals that can be supported on a cell complex. It turns out that certain classes of "rainbow" monomial ideals will always have this property; as a consequence, any such ideal with linear resolution must have cellular resolution. We will then look at some conditions ensuring linearity (that can be turned into an equivalence with an extra assumption), and apply these results to say some words about polarizations of Artinian monomial ideals. Some of this work is joint with Ayah Almousa. |
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February 8 |
Speaker: Rick Ball, University of Denver Title: The Anatomy of A Completely Regular Frame Abstract: The apparatus of point free topology can be fruitfully brought to bear on the most classical of topological subjects: namely completely regular frames. Such a frame is a (canonical) union of an ascending cardinally indexed sequence of kappa-frames, starting with the cozero part. In addition, every such frame has a (canonical) descending cardinally indexed sequence of minimal dense kappa Lindelof sublocales, ending with its booleanization. Taken together, these features form a kind of scaffolding which is descriptive of many of the salient attributes of the frame. |
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February 1 |
Speaker: Morgan Sinclaire, New Mexico State University Title: Formally Verifying Peano Arithmetic Abstract: Gentzen's consistency proof is a central result in proof theory that demonstrates the consistency of Peano arithmetic (PA) using a technique known as cut-elimination. The proof can be carried out in the weaker finitist system of primitive recursive arithmetic (PRA), if one extends that system with the principle of transfinite induction over the ordinal ε_0. In our work, we have partially implemented a version of this proof as a computer program in the Coq theorem prover. Consequently, the key steps in the proof have been computer verified, and many of its interesting structures-such as Cantor normal form ordinals and infinitary proof trees-have been built as constructive, finitistic objects. |
