VIRTUAL ONLY: USTARS
Institute for Computational and Experimental Research in Mathematics (ICERM)
April 29, 2021  April 30, 2021
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Thursday, April 29, 2021

11:40  11:45 am EDTICERM WelcomeWelcome  Virtual
 Brendan Hassett, ICERM/Brown University

11:45 am  12:00 pm EDTWelcomeVirtual
 Ryan Moruzzi Jr, California State University, East Bay

12:00  12:30 pm EDTSmall finite rings satisfying some ring propertiesVirtual
 Speaker
 Henry ChimalDzul, Ohio University
 Moderator
 Ryan Moruzzi Jr, California State University, East Bay (Virtual)
Abstract
Various ring properties trivially satisfied by commutative rings have given place to various classes of rings. These include reflexive, semicommutative, abelian, and reversible rings. In this talk we present various examples of finite rings with the smallest possible cardinality satisfying those ring properties. In particular, special attention is devoted to a reflexive abelian nonsemicommutative ring, whose cardinality has been shown to be 256.

12:00  12:30 pm EDTAnnular Khovanov homology and meridional disksVirtual
 Speaker
 Gage Martin, Boston College
 Moderator
 Javier Ronquillo Rivera, Bridge to Enter Advanced Mathematics (Virtual)
Abstract
The relationship between quantum link invariants and topological invariants of a link has been a motivating question in lowdimensional topology. For annular Khovanov homology, the wrapping conjecture of HostePrzytycki suggests that the maximal nonzero annular grading is related to embeddings of meridional disks. We provide further evidence for this conjecture by exhibiting infinite families of annular links for which the gap between the maximum nonzero annular Khovanov grading and the maximum nonzero annular Floertheoretic gradings is arbitrarily large. We also show this gap exists at the decategorified level for some of the infinite families. Additionally, we show that certain satellite operations can not be used to construct counterexamples to the categorified wrapping conjecture.

12:30  1:00 pm EDTDecompositions of the $h^*$polynomial for rational polytopesVirtual
 Speaker
 Andres Vindas Melendez, University of Kentucky
 Moderator
 Ryan Moruzzi Jr, California State University, East Bay (Virtual)
Abstract
A rational convex polytope, i.e., the convex hull of finitely many rational points in \mathbb{Q}^d, its lattice point count, and its associated combinatorial invariants provide information on quantities of geometric and algebraic interest. In algebraic geometry, a polytope $P$ corresponds to a projective toric variety $X_P$ and an ample line bundle $L$, whose Hilbert polynomial enumerates the lattice points in $P$. In commutative algebra, lattice point enumeration appears in the guise of Hilbert series of graded rings. In the representation theory of semisimple Lie algebras, the Kostant partition function enumerates the lattice points in flow polytopes. The lattice point enumerator results to be a quasipolynomial, known as the Ehrhart quasipolynomial of a rational polytope P, and encodes the number of integer lattice points in dilates of $P$. The $h^*$ polynomial of $P$ is the numerator of the accompanying generating function. We provide two decomposition formulas for the $h^*$polynomial of a rational polytope. The first decomposition generalizes a theorem of Betke and McMullen for lattice polytopes. We use our formula to provide a novel proof of Stanley's Monotonicity Theorem for the $h^*$polynomial of a rational polytope. The second decomposition generalizes a result of Stapledon, which we use to provide rational extensions of the Stanley and Hibi inequalities satisfied by the coefficients of the $h^*$polynomial for lattice polytopes. Lastly, we apply our results to rational polytopes containing the origin whose duals are lattice polytopes. This is joint work with Matthias Beck (SFSU & FU Berlin) and Ben Braun (Univ. of Kentucky).

12:30  1:00 pm EDTLet's Talk About... LatticesVirtual
 Speaker
 Paige Helms, University of Washington
 Moderator
 Javier Ronquillo Rivera, Bridge to Enter Advanced Mathematics (Virtual)
Abstract
Notable examples of periodic sphere packings include the hexagonal lattice in $\mathbb{R}^2$ which Gauss proved is the densest periodic planar circle packing, and the Leech lattice and $E_8$ in $\mathbb{R}^3$. In this talk, we will explore what a lattice is, and introduce a result by M. Faulhuber and S. Steinerberger which proves another extremal property of the hexagonal lattice, using the fact that it is a critical point in the space of lattices.

1:00  2:00 pm EDTLunch/Free TimeVirtual

2:00  3:00 pm EDTMentoring & Career PanelPanel Discussion  Virtual
 Moderator
 Anisah Nu’Man, Spelman College (Virtual)
 Panelists
 Ranthony Edmonds, The Ohio State University (Virtual)
 Garett Jones, Marshfield Clinic Health System (Virtual)
 Ashlee Kalauli, University of California, Santa Barbara (Virtual)
 Roberto Soto, California State University Fullerton (Virtual)

3:00  3:30 pm EDTEntangled PolynomialsVirtual
 Speaker
 Ashley Pallone, Ohio University
 Moderator
 Paige Helms, University of Washington (Virtual)
Abstract
Using a standard embedding of $K[x]$, the algebra of polynomials with coefficients in a field $K$, into the algebra of row and column finite matrices over $K$, nontrivial factorizations of some irreducible polynomials in $K[x]$ are possible in the larger context. The row and column finite matrices involved in those factorizations resemble polynomials in several ways. We call these matrices \emph{entangled polynomials} as each one of them is induced by a finite number of polynomials. For every fixed $m \in \Z_+$, entangled polynomials induced by $m$ polynomials (the socalled $m$nomials) form a ring, $K^{m}[x]$. The ring of $1$nomials is precisely $K[x]$. When $n$ divides $m$, $K^{n}[x]$ is a subring of $K^{m}[x]$. In particular, all rings of $m$nomials include the polynomials.
Given an algebra $A$ over a field $K$, a basis $\cB$ for $A$ is said to be amenable if one can naturally extend the $A$module structure on the $K$vector space $\bigoplus_{b \in \cB} Kb= K^{(B)}$ to the $K$vector space $\prod_{b \in \cB} Kb = K^{\cB}$. A basis $\cB$ is congenial to another one $\cC$ if infinite linear combinations of elements of $\cB$ translate in a natural way to infinite linear combinations of elements of $\cC$. An amenable basis $\cB$ is simple if it is not properly congenial to any other amenable basis. We will present amenable bases in the ring of all $m$nomials as well as some results about congeniality in the ring of all $2$nomials. This will culminate to presenting a simple basis in the ring of all $2$nomials. 
3:00  3:30 pm EDTOn discrete gradient vector fields and Laplacians of simplicial complexesVirtual
 Speaker
 Andrew Tawfeek, University of Washington
 Moderator
 Christopher O'Neill, San Diego State University (Virtual)
Abstract
Discrete Morse theory, a cell complexanalog to smooth Morse theory, has been developed over the past few decades since its original formulation by Forman in 1998. One of the main objects of concern in discrete Morse theory are discrete gradient vector fields on simplicial complexes. We prove that the characteristic polynomial of the Laplacian of a simplicial complex is a special generating function for gradients when the complex is either any graph or a triangulation of an orientable manifold of any dimension.

3:30  4:00 pm EDTWriting Nilpotent Matrices as CommutatorsVirtual
 Speaker
 Emily HoopesBoyd, Kent State University
 Moderator
 Paige Helms, University of Washington (Virtual)
Abstract
The relationship between nilpotent elements and commutators over rings has been studied in detail over the years. We will show that every nilpotent N in M_n(D), the ring of square matrices over a division ring, can be presented as a single commutator, that is, N = AB − BA for some matrices A, B in M_n(D). We will also construct an example illustrating that there exists a prime ring with unity over which some nilpotent matrices cannot be presented as commutators.

3:30  4:00 pm EDTGraphs Associated with the Factorization Sets of Numerical SemigroupsVirtual
 Speaker
 Mariah Moschetti, San Diego State University
 Moderator
 Anisah Nu’Man, Spelman College (Virtual)
Abstract
A numerical semigroup is a submonoid of the nonnegative integers under addition. An important property of numerical semigroups is that their elements may have multiple factorizations. Given an element $n$ in a numerical semigroup $S$, graphs can be constructed using the multiple factorizations of $n$ as vertices. In this talk, we explore the minimal trade graphs of $n$ and the factorization support graph of $n$. The rank of the fundamental group of these graphs can be found by counting how many edges are present in each.

4:05  5:05 pm EDTCombinatorial Techniques for the Study of Toric RingsVirtual
 Speaker
 Gabriel Sosa Castillo, Colgate University
 Moderator
 Shannon Talbott, Moravian College (Virtual)
Abstract
Given a field $K$, a polynomial ring $R=K[X_1, \dots, X_n]$, and a set of monomials $m_i=X_1^{a_{i,1}}\dots X_n^{a_{i,n}}$ of $R$, the subring of $R$ generated by $\{m_i, \dots, m_s\}$, i.e. $S=K[m_1,\dots, m_s]$, is called a toric ring.
The existence of the epimorphism \[ \begin{array}{rccc} \varphi: & K[T_1, \dots, T_s] & \rightarrow & K[m_1,\dots, m_s] \\ & T_i & \rightarrow & m_i \end{array} \] allows for the study of the toric ring $S$ by instead focusing on the ideal $I=\ker \varphi$. This ideal, known as the toric ideal, happens to be prime and is generated by binomials.
Rees, and multiRees algebras, of monomial ideals are toric rings whose study is of special interest because of its connection to Algebraic Geometry. A fundamental (open) question concerns describing explicitly a set of binomial generators (i.e. the defining equations) for the toric ideal associated to a Rees, or multiRees, algebra.
In this talk, we will discuss techniques that have allowed for a complete description of the defining equations when the monomial ideals associated to the Rees, or multiRess, algebra possess nice combinatorial properties. 
5:05  5:45 pm EDTClosing RemarksVirtual
 Christopher O'Neill, San Diego State University
Friday, April 30, 2021

11:45 am  12:00 pm EDTNetworking/ Coffee BreakCoffee Break  Virtual

12:00  12:30 pm EDTOptimizing Polytopal Norms with Respect to Numerical SemigroupsVirtual
 Speaker
 Brittney Marsters, San Diego State University
 Moderator
 Christopher O'Neill, San Diego State University (Virtual)
Abstract
Fix a polytope P. The polytopal norm of a point with respect to P is the smallest dilation factor t such that tP contains this point. A numerical semigroup S is a subset of the nonnegative integers that contains zero and is closed under addition. Elements of S can be expressed as linear combinations of the generators of S where coefficients are taken to be nonnegative integers. To each of these expressions, we associate a point that we call a factorization of this element in S. During this talk, we will discuss optimizing polytopal norms defined on sets of factorizations of elements of numerical semigroups. We will present results classifying the eventually quasilinear relationship for max and min polytopal norms for rational polytopes of dimension k.

12:00  12:30 pm EDTOn Computing Modular forms over Imaginary Quadratic FieldsVirtual
 Speaker
 Kalani Thalagoda, University Of North Carolina at Greensboro
 Moderator
 Shannon Talbott, Moravian College (Virtual)
Abstract
Classical Modular Forms are holomorphic functions on the complex upper halfplane satisfying functional equations with respect to congruence subgroups of SL(2, Z). Bianchi Modular forms are a generalization of this to imaginary quadratic fields. Similar to the classical case, there is a Hecke module isomorphism between this space and certain classes in the cohomology of the corresponding congruence subgroup. This gives us a technique to compute modular forms as Hecke eigensystems.
In this talk, I will go over the techniques used to compute classical modular forms and how some of those can also be modified to work for the Bianchi case. With explicit examples, I will demonstrate some similarities and differences between the Bianchi case to the classical case. 
12:30  1:00 pm EDTAn Introduction to Parking FunctionsVirtual
 Speaker
 Kimberly Hadaway, Williams College
 Moderator
 Christopher O'Neill, San Diego State University (Virtual)
Abstract
In 1966, Alan G. Konheim and Benjamin Weiss defined ``parking functions'' as follows: We have a oneway, onelane street with a dead end and $n$ parking spaces, numbered in consecutive order from 1 to $n$, and we have $n$ cars in line waiting to park. Each driver has a favorite (not necessarily distinct) parking spot, which we call its \emph{preference}. We order these preferences in a \emph{preference vector}. As each car parks, it drives to its preferred spot. If that spot is open, the car parks there; if not, it parks in the next available spot. If a preference vector allows all cars to park, we call it a \emph{parking function}. In 1974, Henry O. Pollak proved the total number of parking functions of length $n$, meaning there are $n$ parking spots and n cars, to be $(n+1)^{n1}$. In this presentation, we describe a recursive formula, expound Pollak's succinct sixsentence proof of an explicit formula, and conclude with a discussion of other parking function generalizations.

12:30  1:00 pm EDTFiber Bundles and Group ExtensionsVirtual
 Speaker
 Jonathan Alcaraz, University of California, Riverside
 Moderator
 Shannon Talbott, Moravian College (Virtual)
Abstract
In the world of topology, we like to study objects called fiber bundles, which act as a twisted version of a product. Through the fundamental group, these objects usually induce group extensions, which are a twisted version of the direct product of groups. We will use this connection to explore examples of spaces which admit multiple distinct fiberings.

1:00  2:00 pm EDTLunch/Free TimeVirtual

2:00  2:30 pm EDTBreak/ Informal NetworkingCoffee Break  Virtual

2:30  3:00 pm EDTA Monoid Structure on the Set of all Binary Operations over a Fixed Set and some of its PropertiesVirtual
 Speaker
 Asiyeh Rafieipour, Ohio University
 Moderator
 Anisah Nu’Man, Spelman College (Virtual)
Abstract
Given a set $S$, we consider an operation $\triangleleft$ on the set $\mathcal{M}(S)$ where $\mathcal{M}(S)=\{ \ast  \ast \text{ is a binary operation on } S\}$ such that $(\mathcal{M}(S), \triangleleft)$ is a monoid. We consider several properties of this monoid including the fact that it has all subsets of the form $out(\ast)=\{ \circ \in \mathcal{M}(S) \ast \text { distributes over } \circ \}$ as submonoids, a complete characterization of its group of units and of a subgroup of its group of automorphisms, induced by permutations. In addition, we introduce for the case when $S< \infty$, a userfriendly representation of the elements of $\mathcal{M}(S)$. \\ This talk reports some results from a paper with S.R. L\'opezPermouth and Isaac Owusu Mensah that is currently under consideration for publication.

2:30  3:00 pm EDTSome Aspects of Generalized Covering Space TheoryVirtual
 Speaker
 Jacob Garcia, University of California, Riverside
 Moderator
 Paige Helms, University of Washington (Virtual)
Abstract
Covering space theory is a classical tool used to characterize the geometry and topology of spaces. It seeks to separate the main geometric features from certain algebraic properties. For each conjugacy class of a subgroup of the fundamental group, it supplies a corresponding covering of the underlying space and encodes the interplay between algebra and geometry via group actions. The full applicability of this theory is limited to spaces that are, in some sense, locally simple. However, many modern areas of mathematics, such as fractal geometry, deal with spaces of high local complexity. This has stimulated much recent research into generalizing covering space theory by weakening the covering requirement while maintaining most of the classical utility. This talk will focus on the relationships between generalized covering projections, fibrations with unique path lifting, separation properties of the fibers, and continuity of the monodromy.

3:00  3:30 pm EDTAlgebras such that The Basic Modules are Isomorphic Then the Bases Are Mutually CongenialVirtual
 Speaker
 Majed Zailaee, Ohio University
 Moderator
 Anisah Nu’Man, Spelman College (Virtual)
Abstract
The study of recently introduced notion of amenability and congeniality of infinite dimensional algebras is furthered. A basis $\mathcal{B}$ of infinite dimensional $F$ algebra is said to be amenable if $F^{\mathcal{B}}$ can be made into an $A$module in a natural way. Mutual congeniality is a relation that serves to identify cases when different bases yields isomorphic $A$modules. If a basis $\mathcal{B}$ is congenial to the basis $\mathcal{C}$ but $\mathcal{C}$ is not congenial to the basis $\mathcal{B}$, then we say $\mathcal{B}$ is properly congenial to $\mathcal{C}$. An amenable basis $\mathcal{B}$ is said to be simple if is not properly congenial to any other amenable basis. Here we study the relations between the isomorphic basic modules and the simple basis.

3:00  3:30 pm EDTAn Exploration of the Curve ComplexVirtual
 Speaker
 Brandis Whitfield, Temple University
 Moderator
 Paige Helms, University of Washington (Virtual)
Abstract
Given a surface $S$, its associated curve complex, $\mathcal{C}(S)$, is a simplicial complex which encodes topological information on the set of homotopy classes of loops on S. In this introductory talk, we'll explore combinatorial properties of its underlying graph and its value to the topology, geometry and mapping class group of a hyperbolic surface.

3:45  4:45 pm EDTDGS Trisections of 4manifoldsVirtual
 Speaker
 Jose Aranda Cuevas, University of Iowa
 Moderator
 Ty Frazier, University of MInnesota (Virtual)
Abstract
In 2016, D. Gay and R. Kirby proved that $M$ can be decomposed as the union of three 4dimensional 1handlebodies with pairwise intersection 3dimensional handlebodies and triple intersection a closed orientable surface of genus $g$. Such decomposition is called a trisection of genus $g$ of $M$. In 2018, M. Chu and S. Tillmann used this to give a lower bound for the trisection genus of a closed 4manifold in terms of the Euler characteristic of $M$ and the rank of its fundamental group. In this talk, we show that given a group $G$, there exist a 4manifold $M$ with fundamental group $G$ with trisection genus achieving ChuTillmann's lower bound.

4:45  5:00 pm EDTClosing RemarksVirtual
 Anisah Nu’Man, Spelman College

5:00  5:45 pm EDTNetworking EventNetworking Event  Virtual
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