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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

 

Fall 2024 Research Seminar Schedule
Coordinator: Cormac O'Sullivan

Oct 1   Karen Taylor, BCC, Modular Forms on SO(4,3)
   
Oct 8  
   
Oct 15   (Monday schedule)
   
Oct 22   Philipp Rothmaler, BCC and the CUNY Graduate Center, Levels of prevalence of free structures
   
Oct 29   Zhe Wang, BCC, Symbolic space and Fibonacci type sequences
   
Nov 5   Cormac O'Sullivan, BCC and the CUNY Graduate Center, Topographs and some infinite series
   
Nov 12  
   
Nov 19   Jorge Pineiro, BCC, Spaces with simple Picard group
   
Nov 26  
Dec 3  
   
Dec 10  
   

 

 

Spring 2024 Research Seminar Schedule
Coordinator: Cormac O'Sullivan

Feb 20   Karen Taylor, BCC, In search of modular integrals for a new class of period functions
   
Feb 27   Roman Kossak, BCC and the CUNY Graduate Center, Undefinability and Absolute Undefinability
   
   
Apr 2   Cormac O'Sullivan, BCC and the CUNY Graduate Center, Conway magic: topographs for quadratic forms
   
Apr 9   Tony Weaver, BCC, Zero-sum multisets mod p
   

 


 Fall 2023 Research Seminar Schedule
Coordinator: Cormac O'Sullivan

Oct 3   Cormac O'Sullivan, BCC and the CUNY Graduate Center, A hidden link between two of Ramanujan's approximations
   
Oct 10   No seminar - Monday schedule
   
Oct 17   Teaching Tea - Karen Taylor, What are you doing in MTH 28.5?
  Speakers: Rony Gouraige, Winslow Jones, Yanil De La Rosa-Walcott
   
Oct 24   Manuel Cortés Izurdiaga, Universidad de Malaga, Spain, Homotopy categories of N-complexes of modules
   
Oct 31   Michael Pandazis, the CUNY Graduate Center, Ergodicity of the geodesic flow on symmetric surfaces
   
Nov 7   Mehdi Lejmi, BCC and the CUNY Graduate Center, The existence of canonical metrics in almost-Kahler geometry
   
Nov 14   No seminar this week
   
Nov 21   Philipp Rothmaler, BCC and the CUNY Graduate Center, What trinomials can be totally factored?
   
Nov 28   Karen Taylor, BCC, An example of a new class of period functions
   
Dec 5   Kealey Dias, BCC, Greenhouse gas monitoring program at BCC: a descriptive time-series analysis

 



 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 this week
   
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.
   


Fall 2024 Abstracts

Title: Modular Forms on SO(4,3)
Speaker: Karen Taylor
Tuesday, Oct 1

Abstract: The goal of this talk is to describe a fourier expansion of modular forms on SO(4,3). We first review the group theoretic description of classical modular forms.
Students who are in (or have completed) linear algebra are welcome to attend.
  
Title: Levels of prevalence of free structures
Speaker: Philipp Rothmaler
Tuesday, Oct 22

Abstract: Consider a variety V in the sense of universal algebra, that is, an equationally defined class of structures of a fixed similarity type, which may, for simplicity, be assumed (finite or) countable. It is a fact of nature that V contains free structures of any given finite or infinite rank. If this rank is uncountable, the free structure is unique up to isomorphism, and we say the subclass of all free structures is categorical in every uncountable cardinal. Examples are free groups, abelian groups, vector spaces, modules. Note that by Steinitz’ Theorem, the class of algebraically closed fields of any fixed characteristic is also uncountably categorical, however it does not form a variety.

Let F be the class of all free structures in V and let N be an intermediate subclass. To say that N is categorical in every uncountable cardinal is the same as saying that every uncountable structure in N is free. The latter is a common phenomenon in algebra, e.g., it is a famous question as to when all projective modules are free. So such questions can be seen as categoricity questions.

I will explore categoricity and weakenings thereof in general varieties, and give concrete answers for the variety of all modules over arbitrary associative rings.
  
Title: Symbolic space and Fibonacci type sequences
Speaker: Zhe Wang
Tuesday, Oct 29

Abstract: Given a uniformly quasisymmetric circle endomorphism preserving the Lebesgue measure, Yunchun Hu, Yunping Jiang and I constructed an infinite martingale sequence which has an L1 limit on the dual symbolic space. In this talk, I will discuss the relation between this dual symbolic space and Fibonacci type sequences. Using these Fibonacci type sequences, for any fixed $w_n$, we can show that the total measure of disjoint neighborhoods $[w_n w_m]$ for all $w_m$ in dual symbolic space is 1. This gives a proof of Furstenberg's 2-3 conjecture for the case of finite martingales.
 
Title: Topographs and some infinite series
Speaker: Cormac O'Sullivan
Tuesday, Nov 5

Abstract: The Fibonacci numbers are a familiar recursive sequence. Topographs are a kind of two dimensional version conjured up by J.H. Conway in his study of integral binary quadratic forms. These forms are ax^2 + bxy + cy^2 with integer coefficients. We'll review Conway's classification of topographs into 4 types and look at some new discoveries. Applications are to new class number formulas and a simplification of a proof of Gauss related to sums of three squares. We'll also see how certain infinite series over all the numbers in a topograph may be evaluated explicitly. This generalizes and extends results of Hurwitz and more recent authors and requires modular forms.
   
Title: Spaces with simple Picard group
Speaker: Jorge Pineiro
Tuesday, Nov 19

Abstract: The Picard group is an abelian group associated to either a ring A or an algebraic variety X. The Picard group measures, in both cases, to what extent locally free objects on A or X are in fact free. We study situations where the Picard group is particularly simple: trivial or finite, as well as some applications.
    
    

 


Spring 2024 Abstracts

Title: In search of modular integrals for a new class of period function
Speaker: Karen Taylor
Tuesday, Feb 20

Abstract: Polynomial and rational period functions have been well studied. Using an observation of Knopp, we can construct period functions in a larger class of functions. In this talk, I will review key results on polynomial and rational period functions and discuss attempts to find modular integrals associated with the new more general period functions.
  
Title: Undefinability and Absolute Undefinability
Speaker: Roman Kossak
Tuesday, Feb 27

Abstract: I call a relation on a countable structure absolutely undefinable if it has uncountably many automorphic images. By a theorem of Kueker and Reyes, every relation that is not absolutely undefinable is definable in a modest infinitary extension of first-order logic. I will explain all the concepts mentioned above and I will illustrate them using simple examples.
  
Title: Conway magic: topographs for quadratic forms
Speaker: Cormac O'Sullivan
Tuesday, Apr 2

Abstract: Quadratic forms are just the familiar q(x,y) = ax^2+bxy+cy^2 with integers a, b, c. Two forms are equivalent if they match under a linear change of variables, and class numbers count the inequivalent forms of each discriminant. Though deceptively simple looking, quadratic forms intrigued the greats such as Lagrange, Gauss and Dirichlet, with problems about class numbers still open today. John H. Conway gave a simple and beautiful way to picture these classes of forms that sidesteps many technicalities. He called these pictures topographs as they give a bird's eye view, containing rivers, lakes and wells. We explore this terrain and make some new discoveries.
 
Title: Zero-sum multisets mod p
Speaker: Tony Weaver
Tuesday, Apr 9

Abstract: I’ll show how to solve a problem in enumerative combinatorics which is equivalent to counting topological types of certain group actions on compact Riemann surfaces. However, I won’t talk much about surfaces. I’ll concentrate on the combinatorial problem and the elementary number theory behind it. The objects of interest are certain R-multisets of columns from the 2-dimensional vector space over the field with p elements, p an odd prime. The multisets must span the space, and the columns must sum to zero mod p. The general linear group acts on these multisets, preserving the rank and zero-sum conditions. How many orbits are there? The basic tool is Burnside’s Lemma. The problem can be carved into manageable pieces by identifying a partition of R that goes with each multiset, and restricting the action to just those multisets of a given partition type. The Ferrers diagram of the partition provides a concise way to organize (and visualize!) the computations.
   




 Fall 2023 Abstracts

Title: A hidden link between two of Ramanujan's approximations 
Speaker: Cormac O'Sullivan
Tuesday, Oct 3

Abstract: In consecutive notebook entries, Ramanujan gave asymptotic approximations to the exponential function and the exponential integral. The first is reasonable and understandable. The second is not. I discovered a hidden link between these approximations and we will look at the interesting ingredients the proof of this link needed. These include Stirling numbers, second-order Eulerian numbers and modifications of both of these. An anonymous referee provided a second proof (of similar length) using completely different methods.
  
Title: Homotopy categories of N-complexes of modules
Speaker: Manuel Cortés Izurdiaga
Tuesday, Oct 24

Abstract: Given a natural number N greater than or equal to 2, a not-necessarily commutative ring with unit and an additive subcategory of the category of right modules, one can consider N-complexes of modules in the subcategory and the corresponding N-homotopy category. This is an extension of the classical homotopy category constructed from 2-complexes. In the talk we will see how some results of classical homotopy categories of 2-complexes extend to homotopy categories of N-complexes. The methods for extending these results are some techniques from module theory, such us the deconstruction of a class of modules.
  
Title: Ergodicity of the geodesic flow on symmetric surfaces
Speaker: Michael Pandazis
Tuesday, Oct 31

Abstract: We consider conditions on the Fenchel-Nielsen parameters of a Riemann surface X that guarantee the surface X is of parabolic type. An interesting class of Riemann surfaces for this problem is the one with finitely many topological ends. In this case, the length part of the Fenchel-Nielsen coordinates can go to infinity for parabolic X. When the surface X is end symmetric, we prove that X being parabolic is equivalent to the covering group being of the first kind. Then we give necessary and sufficient conditions on the Fenchel-Nielsen coordinates of a half-twist symmetric surface X such that X is parabolic. As an application, we solve an open question from the prior work of Basmajian, Hakobyan, and Saric. This is joint work with Dragomir Saric.
  
Title: The existence of canonical metrics in almost-Kahler geometry
Speaker: Mehdi Lejmi
Tuesday, Nov 7

Abstract: In the Kahler geometry, some obstructions to the existence of Kahler-Einstein are known. Among these obstructions, we have the structure of the automorphism group of the complex manifold and the existence of Kahler-Ricci solitons. In this talk, we extend some of these to the almost-Kahler geometry and we obtain some obstructions to the existence of Chern-Einstein almost-Kahler metrics. This is a joint work with M. Albanese and G. Barbaro.
   
Title: What trinomials can be totally factored?
Speaker: Philipp Rothmaler
Tuesday, Nov 21

Abstract: I will report on some high school student’s work on monic quadratic trinomials that factor no matter what sign combination of the linear and constant terms, like the well-known x^2+5x+6. It is easy to find infinitely many, but it takes some work to see what are the “primitive” ones and that there are infinitely many of those. Come and see how.
   
Title: An example of a new class of period functions
Speaker: Karen Taylor
Tuesday, Nov 28

Abstract: In this talk we use Knopp's construction of a rational period function to give an example of a more general period function.
   
Title: Greenhouse gas monitoring program at BCC: a descriptive time-series analysis
Speaker: Kealey Dias
Tuesday, Dec 5

Abstract: It is well-known that human activities have dramatically increased the concentration of greenhouse gasses (GHGs) in the atmosphere, contributing significantly to climate change. Three key GHGs are methane (CH4), carbon dioxide (CO2), and water vapor (H2O). A Picarro Cavity Ring-Down Spectroscopy analyzer was used to measure the concentrations of these GHGs from the top of Meister Hall on the BCC campus for 2016 and 2017. I propose two descriptive time-series models to establish a baseline for a monitoring program: 1. A naïve decomposition model plus ARIMA errors and 2. A dynamic harmonic regression model plus ARIMA errors. This project is done in collaboration with chemistry faculty at BCC and LaGCC.
    

 




 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.