## Algebra Seminar

Mondays, 12:30 – 1:20 PM

### Spring 2023

 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.

### Fall 2022

 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 UniversityTitle: 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

### Spring 2022

 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.

### Fall 2021

 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

### Spring 2021

 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.