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1:00-1:20 OPENING WORDS
Session chair: Kalina Mincheva
1:30-1:50 Samrith Ram
Affiliation: Indraprastha Institute of Information Technology, Delhi
Title: Subspace profiles of Linear Operators over Finite Fields.
Abstract: We introduce a family of univariate polynomials indexed by integer partitions. At prime powers, they count the number of subspaces in a finite vector space that transform under a regular diagonal matrix in a specified manner. At 1, they count set partitions with specified block sizes. At 0, they count standard tableaux of specified shape. At -1, they count standard shifted tableaux of a specified shape. These polynomials are generated by a new statistic on set partitions (called the interlacing number) as well as a polynomial statistic on standard tableaux. They allow us to express q-Stirling numbers of the second kind as sums over standard tableaux and as sums over set partitions. For partitions whose parts are at most two, these polynomials are the non-zero entries of the Catalan triangle associated to the q-Hermite orthogonal polynomial sequence. In particular, when all parts are equal to two, they coincide with the polynomials defined by Touchard that enumerate chord diagrams by number of crossings.
2:00-2:20 Nati Friedenberg
Affiliation: Tulane University
Title: Classifying valuated term orders
Abstract: To do valuated Gröbner theory, one needs a valuated term order. As an application of our recent work in semiring theory, we classify valuated term pre-orders. If time permits, we will explain how our classification can be used to identify which valuated term pre-orders are, in fact, orders. Joint work with Kalina Mincheva.
BREAK
Session chair: Gary Birkenmeier
3:00-3:20 Ralph P Tucci
Affiliation: Loyola University
Title: Rings in which every element is Idempotent or Nilpotent of Bounded Index.
Abstract: A BN ring (Boolan – nil ring) is a ring in which every element is idempotent or nilpotent. Such a ring is BN (k) if every nilpotent element of the ring has index of nilpotency k or less. In this talk we show that any BN (k) ring is BN (2). This is joint work with Dr. Mark Farag of Fairleigh Dickinson University.
3:30-3:50 Bach Nguyen
Affiliation: Xavier University, Louisiana
Title: Poisson Geometry and Azumaya loci of cluster Algebras.
Abstract: The two important objects in the theory of cluster algebra are the upper cluster algebras equipped with the Gekhtman-Shapiro-Vainshtein Poisson structure and their root of unity quantizations. In this talk, we will discuss the Poisson side of this picture, where we prove that the spectrum of every finitely generated upper cluster algebra always has a Zariski open orbit of symplectic leaves and give an explicit description of it. Time permitting, we will also tell the quantum side of the story in which the fully Azumaya loci of the root of unity upper quantum cluster algebras is described.
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4:00-4:20 Arik Wilbert
Affiliation: University of South Alabama
Title: Positivity in weighted Flag varieties.
Abstract: Let X be the flag variety of a complex connected reductive group. A classical result states that the classes of the Schubert varieties form a basis of the cohomology ring of X, and the structure constants in this basis are all nonnegative. In 2001, Graham showed how to generalize this positivity result to the torus equivariant cohomology of X. In this talk, we will discuss the case of weighted flag varieties. In general, weighted flag varieties are singular. But they are always projective, normal, and rationally smooth orbifolds. Furthermore, they carry an action of a torus with isolated fixed points and their equivariant cohomology rings admit a Schubert basis. In joint work with William Graham and Scott Larson, we obtain a positivity result in equivariant cohomology for all weighted flag varieties. This generalizes both Graham's result, as well as a positivity result by Abe-Matsumura for weighted Grassmannians from 2015.
4:30-4:50 Nestor Diaz Morera
Affiliation: Tulane University
Title: Spherical partition Schubert varities and Dyck paths
Abstract: We consider Ding's partition Schubert varieties whose members are very close to being a toric variety. More precisely, we characterize the spherical partition Schubert varieties in terms of Dyck paths. We introduce a notion of a nearly toric variety. We identify the nearly toric partition Schubert varieties as well as all singular nearly toric Schubert varieties. We find the cardinalities of the sets of these Schubert varieties.
5:00-7:30 Reception
Location: Mezzanie Area, Lavin-Bernick Center (LBC), Tulane University
Session chair: Jörg Feldvoss
9:30-9:50 Cornelius Pillen
Affiliation: University of South Alabama
` Title: New Generic Lower Bound for Donkin’s Tilting Module Conjecture.
Abstract: Let $G$ be a simple simply connected algebraic group over an algebraically closed field of positive characteristic $p$ and $G_1$ its first Frobenius kernel. We consider four questions of primary interest for the representation theory of $G$:
(i) Donkin’s Tilting Module Conjecture,
(ii) the Humphreys-Verma Question,
(iii) whether the Steinberg module censored with an irrreducible representation of p-restricted highest weight is a tilting module, and
(iv) whether the $G_1$-extensions between two simple modules with $p$-restricted highest weight are tilting modules for $G$.
We establish affirmative answers to each of these questions with a new uniform bound, namely $p \geq 2h -4$ where $h$ is the Coxeter number. Notably, this verifies these statements for infinitely many new cases.
10:00-10:20 Hongyu He
Affiliation: Louisiana State University
Title: Projection of adjoint orbits and Gan-Gross-Prasad Conjectures.
Abstract: In this talk, I will give an elementary introduction to projection of adjoint orbits of unitary groups preserving a Hermitian form. We will discuss linear spaces equipped with Hermitian forms and their associated linear transformations. Then we will see how a conjugacy class of a larger group is projected onto conjugacy classes of a smaller group. Finally, we will explain in the context of Kirillov-Kostant philosophy why the local Gan-Gross-Prasad conjecture must be true.
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Session chair: Mahir Bilen Can
11:00-11:20 Fernando L Pinero
Affiliation: University of Puerto Rico, Ponce
Title: Orbit Structure and decoder for Grassmann codes
Abstract: In this talk we present a decoder for Grassmann codes. Using the orbit structure of the Grassmannian under the natural action of multiplicative group of certain finite field extensions, we project the corresponding Grassmann code and obtain subcodes of certain Reed-Solomon codes. We recover the information bits from orbits containing an information set of the original Grassmann code. By improving Peterson's decoding algorithm for the projected subcodes, we prove that one can correct up to $\lfloor d-1/2\rfloor$ errors.
11:30-11:50 Sudhir Ghorpade
Affiliation: Indian Institute of Technology, Bombay
Title: Pure Resolutions, Betti numbers, and Linear Codes
Abstract: Since the seminal work of Hilbert on his syzygy theorem and related topics, it is known that the structure of a standard graded algebra over a field can be better understood from its graded minimal free resolution. These resolutions are particularly nice when they are pure. In 2013, Johnsen and Verdure used this paradigm to associate to a linear error-correcting code C a fine set of invariants called its Betti numbers, and showed that they determine the generalized Hamming weights of C,. Further research has shown that finding the Betti numbers of a linear code can be useful and interesting. But it is often a hard problem. It is, however, quite tractable if the corresponding graded minimal free resolution is pure. We shall briefly outline these developments, and then discuss the following recent results: (i) an intrinsic characterization of purity of graded minimal free resolutions of standard graded algebras associated to a linear code, and (ii) a complete characterization of (generalized) Reed-Muller as well as projective Reed-Muller codes of an arbitrary order for which the corresponding graded minimal free resolutions are pure. The contents of this talk are based on a joint work with Prasant Singh and also with Rati Ludhani.
LUNCH BREAK
Session chair: Bach Nguyen
1:30-1:50 Jörg Feldvoss
Affiliation: University of South Alabama
Title: Cohomology of solvable Leibniz algebras.
Abstract: Leibniz algebras were introduced by Bloh and Loday as non-commutative analogues of Lie algebras. Many results for Lie algebras have been proven to hold for Leibniz algebras, but there are also several results that are not true in this more general context. In a previous work we used an analog of the Hochschild-Serre spectral sequence to prove, among other things, the second Whitehead Lemma for Leibniz algebras. It turns out that the analog of this spectral sequence cannot be applied to many ideals different from the Leibniz kernel, and therefore it is not useful for computing the cohomology of non-semi-simple Leibniz algebras. In my talk I will explain how one can use similar tools as developed by Farnsteiner for Hochschild cohomology to work around this. In particular, this enables us to generalize the vanishing theorems of Dixmier and Barnes for nilpotent and (super)solvable Lie algebras to Leibniz algebras. This is joint work with Friedrich Wagemann.
2:00-2:20 Kalina Mincheva
Affiliation: Tulane University
Title: Tropical Adic Spaces.
Abstract: The process of tropicalization associates to an algebraic variety its combinatorial shadow - the tropical variety. Tropical varieties do not come naturally with extra structure such as scheme structure. Moreover, tropicalization as defined in the literature -- in any of its variants (algebraic or analytic) -- is not a morphism in any category. Working towards endowing tropical varieties with extra structure, we study the algebra of convergent tropical power series and the topological spaces (of prime congruences) it corresponds to. The construction so far allows us to see a tropicalization map as a natural transformation of functors taking values in the category of topological spaces.
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Session chair: Ralph Tucci
3:00-3:20 Gary F. Birkenmeier
Affiliation: University of Louisiana, Lafayette
Title: Extending properties of (bounded) linear operators on a (topological) vector space to (bounded) mappings on the (topological) vector space.
Abstract: In the this talk, we extend various properties of associative algebras of linear operators on a (topological) vector space T to right nearalgebras of (bounded) mappings on T. This is joint work with Nayil Kilic, Figen Takil Mutlu, Edanur Tastan, Adnan Tercan, and Ramazan Yasar.
3:30-3:50 Ayman Badawi
Affiliation: American University of Sharjah
Title: The n-pseudo valuation domain.
Abstract: Let R be an integral domain with quotient field K and n is a positive integer. A proper ideal I of R is an n-powerful ideal of R if whenever x^ny^n in I for x, y in K, then x^n in I or y^n in I. If every prime ideal of R is an n-powerful ideal, then R is called an n-pseudo valuation domain. In this talk, we study the above concept and relate it to several generalizations of pseudo-valuation domains.
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4:00-4:20 Simplice Tchamna-Kouna
Affiliation: Georgia College
Title: An introduction to Dirings
Abstract: We introduce the notion of diring as a generalization of rings and digroups. We treat basic properties of dirings and provide several examples. In addition, several notions of ring theory such as ideals and hmomorphisms are considered in the newly introduced framework. Finally, we construct Leibniz algebra and quasi-Jordan algebra structures on noncommutative algebras by using diring operations. This isa joint work with Guy Biyogmam and Jean Bernand Nganou.
6:00-8:30 Dinner
Location: Mezzanie Area, Lavin-Bernick Center (LBC), Tulane University
Session chair: Simplice Tchamna-Kouna
9:30-9:50 Jean B Nganou
Affiliation: University of Houston, Downtown
Title: Generalized semisimplicity of algebras of many value logics.
Abstract: We initiate a study of two general concepts of semisimplicity for MV-algebras by replacing the standard MV-algebra $[0,1]$ with an arbitrary MV-chain $\mathbf{C}$. These generalized notions are called $\mathbf{C}$-semisimple MV-algebras and $\mathfrak{m}$-semisimple MV-algebras. We obtain several of their characterizations and explore in more-depth the case of perfect MV-chains.
10:00-10:20 Dony Varghese
Affiliation: University of North Alabama
Title: Algebraic Characteristic Sets of Matroids.
Abstract: Matroids are combinatorial structures that generalize the concept of linear independence. The linear representations and characteristic sets of matroids are well known. But the algebraic representations and characteristic sets received much less attention, and the possible algebraic characteristic sets are still not completely known. We look at possible pairs of linear-algebraic characteristic sets of matroids and algebraic characteristic sets which are infinite but not co-finite. Based on joint works with Dustin Cartwright (arXiv:2212.00095).
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Session chair: Jean Nganou
11:00-11:20 Abu C. Thomas
Affiliation: Dickinson College
Title: Walschmidt constant of ideals of Points in Multiprojective space.
Abstract: Waldschmidt constant of a homogeneous ideal $I \subset k[x_0,\dots,x_n]$ is very useful in the study of ideal containment problems, that is knowing for what pairs $(m, r) \in \mathbb{N}^2$ does $I^{(m)} \subseteq I^r$, where $I^{(m)}$ is the $m$-th symbolic power of the ideal $I$. Given a finite set of points $Z \subset \mathbb{P}^n$, It also plays a very important role in determining the lower bounds on the least degree of a hypersurface containing $Z$ with a fixed multiplicity $m \in \mathbb{N}$. Motivated by this, we define and compute the Waldschmidt set of $0-$dimensional schemes in a multi-projective space.
11:30-11:50 Victor Bankston
Affiliation: Tulane University
Title: Nonlocal games on Pauli measurements.
Affiliation: Nonlocal games (or 2-prover, 1 round multiprover interactive protocols) are well-studied in communication complexity. The Pauli (Stabilizer) measurements are a natural class of measurements from quantum information. We will present a series of games based on these measurements and describe upper bounds on their best strategies via graph expansion. The games exhibit a specific task where quantum technologies can achieve a goal that cannot be replicated using only classical resources. Depending on the amount of time allotted, I will include some of the additional details about the use of the expander mixing lemma and association schemes.
LUNCH BREAK
Session chair: Cornelius Pillen
1:30-1:50 Jared Painter
Affiliation: University of North Alabama
Title: Always isomorphic to $\mathbb{Z}_n$ Direct Products and Summation Sets.
Abstract: We will discuss how studies of sets formed by consecutive sums of ordered elements, called summation sets, on partitions defined by cosets and subcosets of $\mathbb{Z}_n$ have led to a new binary operation imposed on direct products which will always be isomorphic to $\mathbb{Z}_n$. It is commonly known in modern algebra that if $n=n_1\cdots n_q$, then $\mathbb{Z}_n$ and $G=\mathbb{Z}_{n_1}\times \cdots \times \mathbb{Z}_{n_q}$ are isomorphic as groups, under componentwise addition in $G$, if and only if $n_i$ and $n_j$ are relatively prime for each $i\not=j$. We define a new additive binary operation, denoted $\boxplus$, on $G$ in which $\mathbb{Z}_n$ and $(G,\boxplus)$ are always isomorphic regardless of the factorization, $n_1\cdots n_q$, of $n$. Moreover, the isomorphism $v:\mathbb{Z}_n\rightarrow (G,\boxplus)$ is an order preserving isomorphism which allows us to classify our aforementioned summation sets.
2:00-2:20 Karl Heinrich Hofmann
Affiliation: Tulane University
Title: The Charm of Weakly Complete Vector Spaces
Abstract: A real topological vector space is {\it weakly complete} if it is isomorphic to $\R^J$ for a set $J$ with its product topology. The category of these topological vector spaces is utterly simple, yet achieves a surprising variety of tools of considerable sophistication in the representation and structure theory of compact groups (e.g. topological group rings up to Tannaka-Hochschild Duality) and of pro-Lie groups (including topological universal enveloping algebras of all Lie algebras of pro-Lie groups, including an effective Poincar\'e-Birkhoff-Witt formalism).
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Session chair: Kalina Mincheva
3:00-3:20 Layla Sorkatti
Affiliation: Southern Illinios University
Title: Minimal Nilpotent Sympletic Alternating Algebras.
Abstract: We first give some general overview of symplectic alternating algebras and then focus in particular on the structure of nilpotent symplectic alternating algebras with some recent results.
3:30-3:50 Samir Raouafi
Affiliation: Auburn University
Title: Pseudospectra of bounded operators and normality.
Abstract: Normal operators are well understood in operator theory because the spectral theorem holds for them. Several mathematicians have therefore paid attention to investigating some conditions under which certain operators or matrices are normal. It is well known that normal operators have minimal pseudospectra, and the converse is not valid in general. In this talk, we shall discuss some characterizations of the normality in terms of the pseudospectra. We will also examine the class of non-normal operators with minimal pseudospectra and derive some applications to the numerical range.
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4:00-4:20 John Carr
Affiliation: University of North Alabama
Title: Determining the Color-Trade-Spectrum of Graphs
Abstract: Two proper edge-colorings of a graph G are mate-colorings if and only if every vertex of G is incident to the same set of colors under each edge-coloring while each edge receives a different color under each edge-coloring. The color-trade-spectrum of a graph G is the set of all t for which there exist two mate-colorings of G using t colors. We present techniques for expanding the known color-trade-spectrum of a graph, along with a way to determine if two edge-colorings form a color trade. We also fully determine the color-trade-spectrum of several families of graphs, and introduce some preliminary findings on algebraically describing color trades of complete bipartite graphs.
6:00-8:30 Dinner
Location: Mezzanie Area, Lavin-Bernick Center (LBC), Tulane University
Session chair: Mahir Bilen Can
9:30-9:50 Lokendra Paudel
Affiliation: University of South Carolina, Salkehatchie
Title: Valuation Overrings of Polynomial Rings and Group Divisibility.
Abstract: Let k be a field and let x1, x2, ..., xn be indeterminates for k. In this presentation, we will discuss the types of valuation overrings of k[x1, x2, ..., xn] based on the rank and rational rank of value groups. Also, we describe the group of divisibility of a finite intersection of valuation overrings of k[x1, x2, ..., xn]. In particular, we focus on the case for n > 3.
10:00-10:20 Mark Greer
Affiliation: University of North Alabama
Title: Moufang Type loops and their non-commuting graphs.
Abstract: Geometric group theory has been well studied and there has been progress to expand these ideas into loop theory. We will discuss some results and difficulties. The main focus will be on the connection between a generalization of Moufang and Steiner loops and their non-commuting graph.
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Session chair: Nati Friedenberg
11:00-11:20 Charles Burnette
Affiliation: Xavier University , Louisiana
Title: Involution factorizations of Ewens random permutations.
Abstract: An involution is a bijection that is its own inverse. Given a permutation $\sigma$ in the symmetric group $S_n,$ let $\mathsf{invol}(\sigma)$ denote the number of ways to express $\sigma$ as composition of two involutions in $S_n.$ In this talk, I will explain why the statistic $\mathsf{invol}$ is asymptotically lognormal when each $S_n$ is equipped with Ewens Sampling Formula probability measures of some fixed parameter $\theta.$ I will also present some strengthenings and generalizations of previously determined results on the limiting distribution of $\log(\mathsf{invol})$ for uniform random permutations, i.e. the specific case of $\theta = 1.$ We will also investigate the first two moments of $\mathsf{invol}$ itself.
11:30-11:50 Guy Biyogmam
Affiliation: Georgia College
Title: Outer Derivations of Leibniz Algebras.
Abstract: In this talk, we will discuss the completeness of non-Lie Leibniz algebras by studying various conditions under which they admit outer derivations. Attention will we focused on the class of non-perfect Leibniz algebras whose center is not contained in the Leibniz kernel.
12:00 CLOSING WORDS (NEXT YEAR'S BUSINESS)
Session chair: Naufil Sakran
9:30-9:50 Thiago Holleben -- ZOOM
Affiliation: Dalhousie University, Canada
Title: Homological Invariants of Ternary Graphs
Abstract: In 2022, Jinha Kim proved a conjecture by Engstrom that states the independence complex of graphs with no induced cycle of length divisible by 3 is either contractible or homotopy equivalent to a sphere. These graphs are called ternary graphs. In algebraic terms, one consequence of the theorem is the minimal free resolution of the edge ideal of these graphs is characteristic-free. We apply this result to give a combinatorial description of projective dimension and depth of the edge ideal of these graphs. As a consequence, we give a complete description of the multigraded betti numbers of edge ideals of ternary graphs in terms of its combinatorial structure.
10:00-10:20 Sandra Ferreira -- ZOOM
Affiliation: University of Beira Interior, Portugal
Title: Different Types of Distributions in Additive Models.
Abstract: Motivated by classical cumulants and some properties, we explore models that are the sum of a fixed mean vector X \beta with w independent random terms X_i Z_i, i=1,…w. The random vectors Z_i, i=1,…,w will have c_1,…,c_w independent and identical distributed components, with variances \sigma^2_1, …, \sigma^2_w. It is often preferable to work with cumulants rather than moments, since the two are entirely equivalent and for independent random variables, a sum's cumulants are the cumulants’ sum. The types of the distributions of the component of vectors Z_i, i=1,…,w may be different, which makes the applications of these models not only centered on the normal type expanding its application.
BREAK
Session chair: Nestor Diaz Morera
11:00-11:20 Fouzia Shaheen -- ZOOM
Affiliation: United Arab Emirates University
Title: Invariant unitary group property of unital C*-algebras.
Abstract: The classification of operator algebras is an important topic in the field. The unitary groups play a significant role in the classifications of unital C*-algebras. A. Rawshdeh, Booth, and Giordano proved that for a large class of simple, amenable, unital, separable $C^*$-algebras, their unitary groups are complete invariant. In this paper, we introduce a new property called an invariant unitary group property(IUG-P). Indeed, we aim to discuss which property is invariant, if the unitary groups of C*-algebras are isomorphic. We prove that some properties are IUG-properties, topological IUG-properties, and also orthogonal IUG-properties for certain classes of $C^*$-algebras.
11:30-11:50 Madhushika Madduwe Hewalage -- ZOOM
Affiliation: Dalhousie University, Canada
Title: Homological Invariants of edge ideals of Cameron Walker Graphs.
Abstract: In “Homological Invariants of Cameron Walker Graphs,” Hibi and coauthors studied the lattice points in N^2 that appear as (depth(R/I (G)), dim(R/I (G))) when G is a graph on n vertices, and showed these points lie between two convex polytopes. When restricting to the class of Cameron Walker graphs, they showed that these pairs do not form a convex lattice polytope. In this talk we will show, for a fixed n, exactly how many pairs of N^2 appear as as (depth(R/I (G)), dim(R/I (G))) where G is a Cameron Walker graph on n vertices. We also determine the number of points in N^4 which are (depth(R/I(G)),reg(R/I(G)),dim(R/I(G)),deg h(R/I(G))) for a Cameron Walker G graph on n vertices. Moving from this points we try to determine for a given (depth(R/I(G), dim(R/I(G)) how many Cameron Walker graphs exists on a fixed number of vertices.
12:00 CLOSING WORDS (NEXT YEAR'S BUSINESS)
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