New Mexico State University

 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.

  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.

 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.

 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.

 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.

 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.  

 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.

A meta-claim in real algebraic geometry is that the description complexity controls geometric complexity. A concrete instance of this claim, named Kushnirenko's conjecture, asserts the number of real zeros of a sparse polynomial system with t terms in n variables is at most O(t^n). This claim remains open for all n (except n=1) for four decades. I'll explain the connections between degree notions in algebraic geometry and volume notion which allows developing a 'expected degree' framework in real algebraic geometry. Then I'll explain our results on Kushnirenko's conjecture. This is combinations of two joint works with Burgisser, Telek, and Tonelli Cueto.

 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.

 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.

 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.

 February 12

Speaker: Michael DiPasquale, New Mexico State University

Title:  An Introduction to Symbolic Powers of Ideals, Part II

 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.

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.

 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.)

 October 30

Speaker: Bruce Olberding, New Mexico State University

Title:  Connectedness in the Space of Valuation Rings of a Field, Part II

 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.

 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.

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 $a$.

Let $R=\mathbb K[x_1,\ldots,x_k]$ be the ring of polynomials with the standard grading given by the degree, and let $I(X)\subset R$ be the defining ideal of $X$. Since $\dim(C(X)_a)$ equals the Hilbert function of $R/I(X)$ evaluated at $a$, would it be possible to relate also $d(X)_a$ to some algebraic invariants of $R/I(X)$? Since 20 years ago mathematicians have explored this question and found interesting lower bounds for $d(X)_a$ that involve the $\alpha$-invariant, the minimum socle degree, or the Castelnuovo-Mumford regularity.

In this talk I will present joint work with John Pawlina where we find much better lower bounds for more general situations, by using also the three invariants mentioned above. We also have examples when these bounds are non-trivially attained.

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
general values of the parameters.

With Matthew Faust, we use ideas from combinatorial algebraic geometry to give an upper bound for the number of critical points at generic parameters, and also a criterion for when that bound is obtained. The dispersion relation has a natural compactification in a toric variety, and the criterion concerns the smoothness of the dispersion relation at toric infinity.

September 18

Speaker: Arvind Kumar, New Mexico State University

Title:  Koszul Algebras and Subadditivity of maximal shifts, Part II

 May 1

Speaker: Giulio Peruginelli, Università di Padova

Title:  Polynomial Dedekind Domains

Abstract: the link

 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

 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.

 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.

 March 27

Speaker: Laura Cossu, University of Graz, Austria

Title:  Factorization beyond Factorization Theory II

 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.

We combine the language of monoids and preorders to make first steps towards the construction of a "unified theory of factorization". In particular, we prove an abstract factorization theorem that recovers, among others, a classical theorem of Cohn on atomic factorizations in cancellative monoids and a classical result of Anderson and Valdes-Leon on "irreducible factorizations" in commutative monoids. Moreover, we generalize the notion itself of factorization and some of the most studied arithmetical invariants of the classical theory (including sets of lengths and elasticities). The talks are based on joint work with S. Tringali.

 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.

February 20

Speaker:  Bruce Olberding, New Mexico State University

Title:  Varieties as topological spaces, Part II

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. 

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.

November 14

Speaker: Louiza Fouli, New Mexico State University

Title: The Depth Function for Monomial Ideals, Part II

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. 

This is joint work with Tài Huy Hà and Susan Morey.

October 31

Speaker: Le Tran, New Mexico State University

Title: Initially regular sequences on cycles, Part II

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.

October 17

Speaker: Louiza Fouli, New Mexico State University

Title: On the core of ideals and modules, Part II

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.

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.

The content of this talk is part of joint work with Alexandra Seceleanu, sponsored by the 2021 AWM Mentoring Travel Grant.

September 12

Speaker:  Bruce Olberding, New Mexico State University

Title:  Local rings and connectedness 

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.

 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.

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.

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.  

 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. 

 November 8

Speaker: Sudipta Das, New Mexico State University

Title: A volume = multiplicity formula for Hilbert-Kunz multiplicity

 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.

This generalizes previous results on the structure of doubly idempotent semirings and, in the case when the distributive lattice is a Heyting algebra, it provides structural insight into unary-determined residuated lattices that are algebraic models of bunched implication logic. We also provide Kripke semantics for the algebras under consideration, which leads to more efficient algorithms for constructing finite models.

These results are joint work with Natanael Alpay and Melissa Sugimoto.

 October 4

Speaker: Jonathan Montano Martinez, New Mexico State University

Title: Blowup algebras of determinantal ideals in prime characteristic 

 September 27

Speaker: Bruce Olberding, New Mexico State University

Title: Quadratic Transforms, Part 2 

 September 20

Speaker: Bruce Olberding, New Mexico State University

Title: Quadratic Transforms 

 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.
Prominent examples are real closed fields (like the real numbers or the Puiseux series field over the reals), convex valuation rings of real closed fields (like Puiseux series with non-negative support) and rings of continuous functions.

Real closed rings occur in the topological study of semi-algebraic sets [i.e. sets described by polynomial inequalities like the closed unit disc, but not the graph of exponentiation], where they take on the role of coordinate rings, just like finitely generated algebras over a field K are coordinate rings of (classical) varieties defined over K. In this context they were introduced by Niels Schwartz in the early 1980s.
On a more algebraic side, real closed rings play a similar role in the class of partially ordered rings as real closed fields do in the class of ordered fields.

The talk will primarily be an introduction to the algebraic theory of real closed rings and their role in real algebraic geometry and the theory of rings of continuous functions. A particular emphasize will be on the prime spectrum of real closed rings in comparison to spectra of rings of continuous functions.

 August 30

Speaker: Vladislav Slyusarev, New Mexico State University

Title: Modal Logic of Cayley Graphs 

  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.

 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.

In a research project initiated at a BIRS workshop “Women in Commutative Algebra” in Fall 2019, the authors studied simplicial and cellular structures that produced resolutions of powers of monomial ideals. This talk will focus on powers of square-free monomial ideals of projective dimension one. Faridi and Hersey proved that a monomial ideal has projective dimension one if and only if there is an associated tree (one dimensional acyclic simplicial complex) that supports a free resolution of the ideal. The talk will show how, for each power r >1, to use the tree associated to a square-free monomial ideal I of projective dimension one to produce a cellular complex that supports a free resolution of I^r. Moreover, each of these resolutions will be minimal resolutions. These cellular resolutions can also be viewed as strands of the resolution of the Rees algebra of I.

 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.

 April 12

Speaker: James Lewis, New Mexico State University

Title: Binomial Edge Ideals of Graphs and Ideals of Konig-type 

 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.

 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.

In a recent dissertation A. Omairi generalized Olberding's theorem to the context of rings with zero-divisors and presented at FAU's Algebra Seminar. During the seminar the question of when a C(X) is an h-local ring arose. We shall discuss our findings. In a strong sense the class of spaces for which C(X) is h-local is connected to the class of almost P-spaces.

 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.

Macpherson and Steinhorn define in [3] the concept of one-dimensional asymptotic classes, which are classes of finite structures with strong conditions on the sizes of definable sets that imply nice model-theoretic behaviour of their ultraproducts. These classes include, among many others, the class of finite fields, the class of Paley graphs and the class of cyclic groups.

In this talk I will review the main concepts of pseudofinite structures, and present joint work with D. Macpherson and C. Steinhorn (cf. [1]) where we explored conditions on the (fine) pseudofinite dimension that guarantee good model-theoretic properties (simplicity or supersimplicity) of the underlying theory of an ultraproduct of finite structures, as well as a characterization of forking in terms of decrease of the pseudofinite dimension.

 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.

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.
I intend this to be a descriptive talk about work in progress. My main objective is to draw some diagrams illustrating the main relationships. If time permits, I will also discuss some of the algebraic ramifications to the f-rings of (localic) real-valued functions on these objects. This is joint work with Joanne Walters-Wayland.

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.