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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
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