VIRTUAL ONLY: USTARS

Institute for Computational and Experimental Research in Mathematics (ICERM)

April 29, 2021 - April 30, 2021
Thursday, April 29, 2021
  • 11:40 - 11:45 am EDT
    ICERM Welcome
    Welcome - Virtual
    • Brendan Hassett, ICERM/Brown University
  • 11:45 am - 12:00 pm EDT
    Welcome
    Virtual
    • Ryan Moruzzi Jr, California State University, East Bay
  • 12:00 - 12:30 pm EDT
    Small finite rings satisfying some ring properties
    Virtual
    • Speaker
    • Henry Chimal-Dzul, Ohio University
    • Moderator
    • Ryan Moruzzi Jr, California State University, East Bay
    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 non-semicommutative ring, whose cardinality has been shown to be 256.
  • 12:00 - 12:30 pm EDT
    Annular Khovanov homology and meridional disks
    Virtual
    • Speaker
    • Gage Martin, Boston College
    • Moderator
    • Javier Ronquillo Rivera, Bridge to Enter Advanced Mathematics
    Abstract
    The relationship between quantum link invariants and topological invariants of a link has been a motivating question in low-dimensional topology. For annular Khovanov homology, the wrapping conjecture of Hoste-Przytycki suggests that the maximal non-zero 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 non-zero annular Khovanov grading and the maximum non-zero annular Floer-theoretic 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 EDT
    Let's Talk About... Lattices
    Virtual
    • Speaker
    • Paige Helms, University of Washington
    • Moderator
    • Javier Ronquillo Rivera, Bridge to Enter Advanced Mathematics
    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.
  • 12:30 - 1:00 pm EDT
    Decompositions of the $h^*$-polynomial for rational polytopes
    Virtual
    • Speaker
    • Andres Vindas Melendez, University of Kentucky
    • Moderator
    • Ryan Moruzzi Jr, California State University, East Bay
    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).
  • 1:00 - 2:00 pm EDT
    Lunch/Free Time
    Virtual
  • 2:00 - 3:00 pm EDT
    Mentoring & Career Panel
    Panel Discussion - Virtual
    • Moderator
    • Anisah Nu’Man, Spelman College
    • Panelists
    • Ranthony Edmonds, The Ohio State University
    • Garett Jones, Marshfield Clinic Health System
    • Ashlee Kalauli, University of California, Santa Barbara
    • Roberto Soto, California State University Fullerton
  • 3:00 - 3:30 pm EDT
    On discrete gradient vector fields and Laplacians of simplicial complexes
    Virtual
    • Speaker
    • Andrew Tawfeek, University of Washington
    • Moderator
    • Christopher O'Neill, San Diego State University
    Abstract
    Discrete Morse theory, a cell complex-analog 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.
  • 3:00 - 3:30 pm EDT
    Entangled Polynomials
    Virtual
    • Speaker
    • Ashley Pallone, Ohio University
    • Moderator
    • Paige Helms, University of Washington
    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 $K$, non-trivial 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 so-called $m$-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.
  • 3:30 - 4:00 pm EDT
    Graphs Associated with the Factorization Sets of Numerical Semigroups
    Virtual
    • Speaker
    • Mariah Moschetti, San Diego State University
    • Moderator
    • Anisah Nu’Man, Spelman College
    Abstract
    A numerical semigroup is a submonoid of the non-negative 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.
  • 3:30 - 4:00 pm EDT
    Writing Nilpotent Matrices as Commutators
    Virtual
    • Speaker
    • Emily Hoopes-Boyd, Kent State University
    • Moderator
    • Paige Helms, University of Washington
    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.
  • 4:05 - 5:05 pm EDT
    Combinatorial Techniques for the Study of Toric Rings
    Virtual
    • Speaker
    • Gabriel Sosa Castillo, Colgate University
    • Moderator
    • Shannon Talbott, Moravian College
    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 multi-Rees 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 multi-Rees, 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 multi-Ress, algebra possess nice combinatorial properties.
  • 5:05 - 5:45 pm EDT
    Closing Remarks
    Virtual
    • Christopher O'Neill, San Diego State University
Friday, April 30, 2021
  • 11:45 am - 12:00 pm EDT
    Networking/ Coffee Break
    Coffee Break - Virtual
  • 12:00 - 12:30 pm EDT
    On Computing Modular forms over Imaginary Quadratic Fields
    Virtual
    • Speaker
    • Kalani Thalagoda, University Of North Carolina at Greensboro
    • Moderator
    • Shannon Talbott, Moravian College
    Abstract
    Classical Modular Forms are holomorphic functions on the complex upper half-plane 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.
  • 12:00 - 12:30 pm EDT
    Optimizing Polytopal Norms with Respect to Numerical Semigroups
    Virtual
    • Speaker
    • Brittney Marsters, San Diego State University
    • Moderator
    • Christopher O'Neill, San Diego State University
    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 non-negative 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 non-negative 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.
  • 12:30 - 1:00 pm EDT
    Fiber Bundles and Group Extensions
    Virtual
    • Speaker
    • Jonathan Alcaraz, University of California, Riverside
    • Moderator
    • Shannon Talbott, Moravian College
    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.
  • 12:30 - 1:00 pm EDT
    An Introduction to Parking Functions
    Virtual
    • Speaker
    • Kimberly Hadaway, Williams College
    • Moderator
    • Christopher O'Neill, San Diego State University
    Abstract
    In 1966, Alan G. Konheim and Benjamin Weiss defined ``parking functions'' as follows: We have a one-way, one-lane 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 $(n+1)^{n-1}$. In this presentation, we describe a recursive formula, expound Pollak's succinct six-sentence proof of an explicit formula, and conclude with a discussion of other parking function generalizations.
  • 1:00 - 2:00 pm EDT
    Lunch/Free Time
    Virtual
  • 2:00 - 2:30 pm EDT
    Break/ Informal Networking
    Coffee Break - Virtual
  • 2:30 - 3:00 pm EDT
    A Monoid Structure on the Set of all Binary Operations over a Fixed Set and some of its Properties
    Virtual
    • Speaker
    • Asiyeh Rafieipour, Ohio University
    • Moderator
    • Anisah Nu’Man, Spelman College
    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 user-friendly representation of the elements of $\mathcal{M}(S)$. \\ This talk reports some results from a paper with S.R. L\'opez-Permouth and Isaac Owusu Mensah that is currently under consideration for publication.
  • 2:30 - 3:00 pm EDT
    Some Aspects of Generalized Covering Space Theory
    Virtual
    • Speaker
    • Jacob Garcia, University of California, Riverside
    • Moderator
    • Paige Helms, University of Washington
    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 EDT
    Algebras such that The Basic Modules are Isomorphic Then the Bases Are Mutually Congenial
    Virtual
    • Speaker
    • Majed Zailaee, Ohio University
    • Moderator
    • Anisah Nu’Man, Spelman College
    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 EDT
    An Exploration of the Curve Complex
    Virtual
    • Speaker
    • Brandis Whitfield, Temple University
    • Moderator
    • Paige Helms, University of Washington
    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 EDT
    DGS- Trisections of 4-manifolds
    Virtual
    • Speaker
    • Jose Aranda Cuevas, University of Iowa
    • Moderator
    • Ty Frazier, University of MInnesota
    Abstract
    In 2016, D. Gay and R. Kirby proved that $M$ can be decomposed as the union of three 4-dimensional 1-handlebodies with pairwise intersection 3-dimensional 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 4-manifold 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 4-manifold $M$ with fundamental group $G$ with trisection genus achieving Chu-Tillmann's lower bound.
  • 4:45 - 5:00 pm EDT
    Closing Remarks
    Virtual
    • Anisah Nu’Man, Spelman College
  • 5:00 - 5:45 pm EDT
    Networking Event
    Networking Event - Virtual

All event times are listed in ICERM local time in Providence, RI (Eastern Daylight Time / UTC-4).

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