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BCC Mathematics Department Research Seminar
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Talks at the Research Seminar are given by BCC department faculty members and guest speakers from other institutions.
The seminar meets on Tuesdays from 12:05-1:00 pm in CP 305. Everyone is welcome to attend.
BCC Math Dept. research interests

Previous seminars 1996 - 2018, 2019



 Spring 2023 Research Seminar Schedule
Coordinator: Cormac O'Sullivan

Feb 14 Nikos Apostolakis, BCC, Pegging cacti and the dominance Dyck lattice
   
Feb 21 Holiday – No Seminar
   
Feb 28 No Seminar
   
Mar 7 Luis Fernandez, BCC and the CUNY Graduate Center, Kahler identities for almost complex manifolds
   
Mar 14 Cormac O'Sullivan, BCC and the CUNY Graduate Center, Powers of polynomials
   
Mar 21 Owen Sweeney, the CUNY Graduate Center, Fermat's Last Theorem for cyclotomic fields
   
Mar 28 Tony Weaver, BCC, Phylogenetic Trees I
   
Apr 4 Tony Weaver, BCC, Phylogenetic Trees II
   
Apr 11 Spring Break – No Seminar
   
Apr 18 Miodrag Iovanov, University of Iowa, Incidence algebras: an interplay of combinatorics, algebra and representation theory
   
Apr 25 Karen Taylor, BCC, Explicit lift of a hyperbolic Poincare series to a locally harmonic Maass function
   
May 2 Nikos Apostolakis, BCC, Pegging bicycles on tori and unicycles on annuli
   
May 9 Kealey Dias, BCC, Characterization of the separatrix graph of a rational vector field on the Riemann sphere
   

 



 Spring 2020 Research Seminar Schedule
Coordinator: Cormac O'Sullivan

Feb 11 Abhijit Champanerkar, College of Staten Island and the CUNY Graduate Center, Graphs, growth and geometry
   
Feb 18 Karen Taylor, BCC, Towards a generalization of the Shintani Lift
   
Feb 25 Robert Thompson, Hunter College and the CUNY Graduate Center, Completion in unstable homotopy theory
   
Mar 3 Jorge Pineiro, BCC, Hyperbolic polarizations, pairs of inverse maps and the Dirichlet property of compactified divisors​
   
Mar 10 Hans-Joachim Hein, Fordham University, From harmonic functions to hyper-Kähler geometry
   
  No further seminars this semester due to the coronavirus.
   





 Spring 2023 Abstracts

Title: Pegging cacti and the dominance Dyck lattice  
Speaker: Nikos Apostolakis
Tuesday, Feb 14

Abstract: We give a combinatorial proof of the formula that gives the number of minimal transitive factorizations of the identity permutation as a product of transposition in the symmetric group S_n, i.e. the Hurwitz number h_{0,n}. We prove a formula for the number of conjugacy classes of such factorizations that involves the Catalan numbers.
The set of such conjugacy classes is in bijection with the set of certain edge labeled graphs that we call cacti. A cactus with n vertices can be properly embedded (pegged) in a sphere with n holes and can therefore be interpreted as a map in the sphere. We study cacti via their minimal spanning trees and this leads to a certain lattice structure on the set of Dyck words of length n-1, thus explaining the appearance of the Catalan numbers.
This is joint work with Cormac O’Sullivan and still in progress.
 
Title: Kahler identities for almost complex manifolds
Speaker: Luis Fernandez
Tuesday, Mar 7

Abstract: In a Kahler manifold there are certain commutator relations between some operators, called the “Kahler identities”. They are the key to prove some important topological properties of Kahler manifolds. There are generalizations of these identities to complex manifolds. We show that these generalizations also work for almost complex manifolds.
Although the previous paragraph is very technical, I will do my best to explain and define everything from basic ideas, giving the fundamental concepts without getting into boring details.
  
Title: Powers of polynomials
Speaker: Cormac O'Sullivan
Tuesday, Mar 14

Abstract: It is a nice exercise to find a good way to take powers of polynomials. De Moivre did this in 1697 and his basic result has been rediscovered regularly - though sometimes with poor notation or obscuring normalizations. In this talk I set everyone straight and also show some of the many applications to power series and number theory. Highlights include: the "book" proof of Faa di Bruno's formula, historical corrections - including the real discoverer of that formula, new discoveries about De Moivre's old system, and a surprising expansion of Ramanujan.
 
Title: Fermat's Last Theorem for cyclotomic fields
Speaker: Owen Sweeney
Tuesday, Mar 21

Abstract: In the nineteenth century, Kummer proved Fermat's last theorem for "regular" prime exponents, not just over the rational numbers but over certain cyclotomic fields. Whether this is true for all primes is still an open problem. We discuss the development of Kummer's cycylotomic approach and more recent criteria applicable in the more general setting when the prime exponent does not satisfy the regularity hypothesis. Time permitting, we say a few words about the application of the uniform abc conjecture to Fermat's last theorem over more general number fields, and in particular the cyclotomic fields.
   
Title: Phylogenetic Trees I, II
Speaker: Tony Weaver
Tuesday, Mar 28, Apr 4

Abstract: Evolutionary biologists try to infer or reconstruct the evolutionary "tree of life" from incomplete information available in the present. In two expository talks, I will discuss some of the combinatorial, probabilistic, and algebraic techniques that have been brought to bear on this task. The presentation is based on a beautiful survey paper by Mike Steel in the American Mathematical Monthly (Vol. 121, No. 2, November, 2014).
   
Title: Incidence algebras: an interplay of combinatorics, algebra and representation theory
Speaker: Miodrag Iovanov
Tuesday, Apr 18

Abstract: Incidence algebras were introduced by Rota more than 50 years ago and have found many applications in various fields of mathematics. We show how they appear naturally in number theory and linear algebra, and how, due to the combinatorial nature of their ideals, they show up in algebra and representation theory, and have even found applications in topological data analysis. One of the main results I will present gives characterizations of finite dimensional incidence algebras in terms of combinatorial properties of their lattice of ideals (finiteness/distributivity), and from this perspective, they turn out to be the finite dimensional counterpart of Prufer rings from commutative algebra. The main tool is the introduction of a deformation theory which brings in homological and topological methods. As applications, we show how incidence algebras control a part of representation theory called "thin" - that is, those modules where the multiplicity (in the composition series) of each simple is at most 1. We give other applications of the main method, to some outstanding open questions and to linear algebra problems, such as a canonical form conjugation by diagonal matrices, which appears in some instances (such as a problem from cryptography). This leads to yet another interplay between graphs, incidence matrices and basic cohomology tools.
   
Title: Explicit lift of a hyperbolic Poincare series to a locally harmonic Maass function
Speaker: Karen Taylor
Tuesday, Apr 25

Abstract: In 2014, Bringmann, Kane, and Kohnen (BKK) introduced locally harmonic Maass functions. They gave an explicit lift of a certain hyperbolic Eisenstein series to a locally harmonic Maass function. In this talk, we discuss BKK's construction and the question of generalizing it to obtain the locally harmonic lift of hyperbolic Poincare series. The definitions will be given in the talk. This is work in progress with Larry Rolen and Andreas Mono.
    
Title: Pegging bicycles on tori and unicycles on annuli
Speaker: Nikos Apostolakis
Tuesday, May 2

Abstract
   
Title: Characterization of the separatrix graph of a rational vector field on the Riemann sphere
Speaker: Kealey Dias
Tuesday, May 9

Abstract: We consider the flow of a rational vector field \xi_{R(z)} = R(z)(d/dz) on the Riemann sphere, where R is given by the quotient of two polynomials without common factors. We characterize the properties of a planar directed graph to be the separatrix graph of such a rational flow. This talk will largely focus on interesting examples and the development of the ideas. (Joint work [2020] with Antonio Garijo.)
 


 
 
 


Spring 2020 Abstracts

Title: Graphs, growth and geometry
Speaker: Abhijit Champanerkar, College of Staten Island and the CUNY Graduate Center
Tuesday, Feb 11

Abstract: We study the growth rate of the number of spanning trees of a sequence of planar graphs which diagrammatically converge to a biperiodic planar graph. We relate this growth rate to the Mahler measure of 2-variable polynomials and hyperbolic volume of link complements. We use this circle of ideas to study an interesting conjecture in knot theory.
 
Title: Towards a generalization of the Shintani lift
Speaker: Karen Taylor, BCC
Tuesday, Feb 18

Abstract: In this talk we will describe the Shintani lift and discuss, work in progress with Larry Rolen, a possible generalization using hyperbolic Poincare series.
 
Title: Completion in unstable homotopy theory
Speaker: Robert Thompson, Hunter College and the CUNY Graduate Center
Tuesday, Feb 25

Abstract: Localization and completion have played a central role in homotopy theory for many decades. I will begin this talk by discussing a classical construction due to Bousfield and Kan of the completion of a topological space with respect to a ring. Then I will discuss completion with respect to a generalized homology theory and summarize the "chromatic approach" to homotopy theory, in which the homotopy category of spaces is fractured into layers based on a sequence of higher K-theories. Then I will focus on an example - the K-theory completion of a sphere, where K-theory is complex topological K-theory.
 
Title: Hyperbolic polarizations, pairs of inverse maps and the Dirichlet property of compactified divisors​
Speaker: Jorge Pineiro, BCC
Tuesday, Mar 3

Abstract: We explore some intersection properties of divisors associated to polarized dynamical systems on algebraic surfaces. As a consequence, we obtain necessary geometric conditions for the existence of polarizations of hyperbolic type and exhibit compactified divisors associated to automorphisms on K3 surfaces that do not have the Dirichlet property as defined by Moriwaki.​
 
Title: From harmonic functions to hyper-Kähler geometry
Speaker: Hans-Joachim Hein, Fordham University
Tuesday, Mar 10

Abstract: Given a positive harmonic function on an open subset of R^3, or of any flat 3-manifold, the Gibbons-Hawking ansatz produces a Ricci-flat Riemannian metric on the total space of a circle bundle over the harmonic function's domain. This Riemannian metric is actually hyper-Kähler, i.e., Kähler with respect to a triple of complex structures satisfying the quaternion relations. I will go over some classical and new applications of this ansatz, and will show how simple examples of harmonic functions give rise to Riemannian 4-manifolds of considerable complexity. I will also explain the role of these examples in some recent work on degenerations of K3 surfaces.