BCC Mathematics Department Research Seminar
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 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. Speakers rescheduled to the fall.|
Fall 2019 Research Seminar Schedule
Coordinator: Cormac O'Sullivan
|Sept 10||Bart Van Steirteghem, FAU Erlangen-Nürnberg and CUNY, Momentum polytopes of projective spherical varieties|
|Sept 17||Khalid Bou-Rabee, City College and the CUNY Graduate Center, Quantifying residual finiteness|
|Sept 24||No Seminar|
|Oct 1||Holiday – No Seminar|
|Oct 8||Holiday – No Seminar|
|Oct 15||Corina Calinescu, NY City Tech and the CUNY Graduate Center, On Applications of Vertex Algebras to Partitions of Numbers|
|Oct 22||Amita Malik, Rutgers, Zeros of derivatives of the completed Riemann zeta function|
|Oct 29||Bianca Santoro, City College and the CUNY Graduate Center, Bifurcation of periodic solutions to the singular Yamabe problem on spheres|
|Nov 5||Philipp Rothmaler, BCC and the CUNY Graduate Center, High and low formulas in modules|
|Nov 12||Uma Iyer, BCC and the CUNY Graduate Center, The algebra of Integro-Differential operators|
|Nov 19||Karen Taylor, BCC, Quadratic identities and Theta Series attached to Quadratic Fields|
|Nov 26||Philip Ording, Sarah Lawrence College, What we talk about when we talk about writing|
|Dec 3||Luis Fernandez, BCC and the CUNY Graduate Center, Why does the occidental musical scale have 12 notes?|
|Dec 10||Aleksandar Milivojevic, Stony Brook University, The rational topology of closed almost complex manifolds|
Spring 2019 Research Seminar Schedule
Coordinator: Cormac O'Sullivan
|Feb 12||Holiday – No Seminar|
|Feb 19||Roman Kossak, BCC and the CUNY Graduate Center, Kernels of directed graphs (and some logic)|
|Feb 26||Ronnie Nagloo, BCC, The Ax-Lindemann-Weierstrass theorem and the Fuchsian automorphic functions|
|Mar 5||Karen Taylor, BCC, Dirichlet Series and Generalized Hecke Groups|
|Mar 12||Corey Switzer, the CUNY Graduate Center, Independence in Arithmetic and Non-Standard Combinatorics|
|Mar 19||Scott Wilson, Queens College and the CUNY Graduate Center, Some problems and recent progress related to almost complex manifolds|
|Mar 26||Alex Martsinkovsky, Northeastern University, From Torsion to Cotorsion and Back|
|Apr 2||Gleb Pogudin, Courant NYU, Primitive Element Theorem for fields with commuting derivations and automorphisms|
|Apr 9||Ethan Cotterill, Federal Fluminense University, Brazil, Real inflection points on real (hyper)elliptic curves|
|Apr 16||Zhe Wang, BCC, An example of Lebesgue invariant Markov partitions and its Beurling-Ahlfors quasiconformal extension|
|Apr 23||Spring Break – No Seminar|
|Apr 30||Kevin O’Bryant, College of Staten Island and the CUNY Graduate Center, Rational Approximation: Numbers, Functions, and Numbers Again|
|May 7||Mingxian Zhong, Lehman College, Two recent results on coloring graphs with forbidden induced subgraphs|
|May 14||The Stanley Friedlander Lecture, Dennis Sullivan,
CUNY Graduate Center and Stony Brook University,
A notion of geometry that includes many more three dimensional manifolds
Previous seminars 1996 - 2018
Spring 2020 Abstracts
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
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.
Fall 2019 Abstracts
Momentum polytopes of projective spherical varieties
Speaker: Bart Van Steirteghem, FAU Erlangen-Nürnberg and CUNY
Tuesday, Sept 10
Abstract: Spherical varieties are a natural generalization of toric varieties, where the acting torus is replaced by a general connected reductive group. They also include flag varieties and symmetric varieties. Using examples, I will explain how their rich combinatorics allows us to characterize the momentum polytopes of projective spherical varieties. I will also discuss an application to the Kählerizability problem for compact multiplicity free Hamiltonian manifolds. This is joint work with S. Cupit-Foutou and G. Pezzini.
Title: Quantifying residual finiteness
Speaker: Khalid Bou-Rabee, City College CUNY
Tuesday, Sept 17
Abstract: The theory of quantifying residual finiteness assigns, to each finitely generated group, an invariant that indicates how well-approximated the group is by its finite quotients. We introduce this theory and survey the current state of the subject. There will be a strong emphasis on examples, open questions, and connections to other subjects.
On Applications of Vertex Algebras to Partitions of Numbers
Speaker: Corina Calinescu, NY City Tech and the CUNY Graduate Center
Tuesday, Oct 15
Abstract: A remarkable feature of the theory of vertex algebras has been its connection to Lie algebras, combinatorics and modular forms, which lead to fundamental results and conjectures in mathematics. Standard modules and certain subspaces, called principal subspaces, for affine Lie algebras have vertex operator constructions. These constructions have been studied in conjunction with combinatorial identities, such as the Rogers-Ramanujan partition identities. This talk is based on joint works with Lepowsky, Milas, Penn and Sadowski.
Title: Zeros of derivatives of the completed Riemann zeta function
Speaker: Amita Malik, Rutgers
Tuesday, Oct 22
Abstract: In his only paper from 1859 in Number Theory, Riemann studied the function that is now known as the completed Riemann zeta function. He stated that the zeros of this function are expected to lie on the critical line. Under this assumption, it can be shown that the zeros of its derivatives also lie on the critical line. In this talk, we discuss the vertical distribution of these zeros and prove discrepancy results analogous to those of the original function. If time permits, we show that in the limiting case, 100% of the zeros of certain combinations of these derivatives lie on the critical line in the spirit of Conrey's results on the zeros of these derivatives.
of periodic solutions to the singular Yamabe problem on spheres
Speaker: Bianca Santoro, City College and the Graduate Center
Tuesday, Oct 29
Abstract: In this talk, we describe how to obtain uncountably many periodic solutions to the singular Yamabe problem on a round sphere that blow up along a great circle. These are complete constant scalar curvature metrics on the complement of S^1 inside S^m, m ≥ 5, conformal to the round metric and periodic in the sense of being invariant under a discrete group of conformal transformations. These solutions come from bifurcating branches of constant scalar curvature metrics on compact quotients of S^m \ S^1. This is a joint work with R. Bettiol (Lehman College) and P. Piccione (USP-Brazil).
|Title: The algebra
of Integro-Differential operators
Speaker: Uma Iyer, BCC and the CUNY Graduate Center
Tuesday, Nov 12
Abstract: This algebra has been introduced and studied by V. Bavula. This algebra contains the Weyl algebra. I will talk about this algebra and a little bit about its representation theory (studied by Futorny, Bekket, Bavula). I will then introduce some questions for further study.
identities and Theta Series attached to Quadratic Fields
Speaker: Karen Taylor, BCC
Tuesday, Nov 19
Abstract: We will discuss identities between solutions to the generalized Pell's equation obtained from the coincidence of Theta series. Our emphasis will be on examples found in the literature.
|Title: What we talk
about when we talk about writing
Speaker: Philip Ording, Sarah Lawrence College
Tuesday, Nov 26
Abstract: What qualities does mathematical writing exhibit? Is it possible to describe the ways that mathematicians write independent of what they prove? And if so, how? Why do some writerly terms appear more relevant (e.g. style) than others (e.g. voice)? What might these loan words bring to a mathematical context from writing and criticism elsewhere? The goal of this talk is to motivate these questions through vivid examples, most of which I encountered while writing the book '99 Variations on a Proof' in the wilds of the math landscape, past and present.
|Title: Why does the
occidental musical scale have 12 notes?
Speaker: Luis Fernandez, BCC and the CUNY Graduate Center
Tuesday, Dec 3
Abstract: If we look at a piano keyboard we see that there is a pattern of 12 consecutive (white and black) notes. Why 12? What is the relation between consecutive notes? How was this scale constructed?
There is a purely mathematical explanation of this fact. I will explain why the 12-note chromatic scale, as well as the pentatonic scale, appear naturally. I will also show the construction of our current equal temperament chromatic scale.
Disclaimer: This will not be a research talk. It will be an expository talk and the only prerequisites are logarithms and a very basic knowledge of modular arithmetic, so it is appropriate for math majors. And I am pretty sure it will be fun.
|Title: The rational
topology of closed almost complex manifolds
Speaker: Aleksandar Milivojevic, Stony Brook University
Tuesday, Dec 10
Abstract: Given an even-dimensional oriented closed smooth manifold, there are topological obstructions to it admitting an almost complex structure (i.e. an endomorphism of its tangent bundle that squares to -1). The most prominent of these come from (a) the requirement that the total Pontryagin class must have a “square root” in the form of the Chern class, and (b) various congruences among the tentative Chern numbers coming from Hirzebruch-Riemann-Roch/Atiyah-Singer.
If we start with only a (simply connected) topological space with no additional structure, we can ask whether there exists a closed almost complex manifold rationally equivalent to the space (i.e. admitting a map to a “rationalized” version of the space which induces an isomorphism on rational homology). A first necessary condition is that the space satisfies Poincare duality on its rational cohomology, giving it a rational cohomological dimension which we will assume is at least six. If this topological space can support “Chern classes” in its rational cohomology whose associated Chern numbers satisfy the congruences from point (b) above, along with some further conditions on the middle degree nondegenerate pairing if the dimension is divisible by 4, then there is indeed such a closed almost complex manifold with these Chern classes. I will elaborate on this result and illustrate how number-theoretic problems quickly enter the discussion.
Spring 2019 Abstracts
|Title: Kernels of
directed graphs (and some logic)
Speaker: Roman Kossak
Tuesday, Feb 19
Abstract: I will outline the proof of the following recent theorem due to James Schmerl: Every resplendent directed acyclic graph having local finite height has a kernel.
If time permits, I will discuss an application of the theorem to a problem in model theory of arithmetic. For a definition and a short discussion of resplendent structures see https://www.ams.org/notices/201106/rtx110600812p.pdf
Ax-Lindemann-Weierstrass theorem and the Fuchsian automorphic functions
Speaker: Ronnie Nagloo
Tuesday, Feb 26
Abstract: In this talk, I will discuss recent work, joint with G. Casale and J. Freitag, around the problem of transcendence of the Fuchsian automorphic functions. These are the uniformizing functions for compact Riemann surfaces of genus zero that are obtained from quotients of the upper half plane by certain Fuchsian groups (a well-known example is the modular j-function).
I will place particular emphasis on the so called Ax-Lindemann-Weierstrass theorem and on a conjecture of Painlevé (1903) which this work resolves. If time permits, I will also explain how our work can be applied to answer a question in number theory (or more precisely to tackle an instance of the special points conjectures).
Series and Generalized Hecke Groups
Speaker: Karen Taylor
Tuesday, March 5
Abstract: Duke introduce the groups G(A) which naturally correspond to an ideal class, A, of a real quadratic field. These groups generalize the Hecke groups G(lambda), lambda>2. In this talk, I will describe two problems, which have been solved for the Hecke groups, and which I am currently working on generalizing. One problem, solved by Rosen, is the description of the elements of G(lambda) by continued fractions. The other is to prove the appropriate form of the Riemann-Bochner-Hecke converse theorem.
in Arithmetic and Non-Standard Combinatorics
Speaker: Corey Switzer, the CUNY Graduate Center
Tuesday, March 12
Abstract: Peano Arithmetic (PA) is the formalization of finite mathematics in first order logic. The standard model of PA is simply the natural numbers with their normal operations. However, in the setting of model theory, there are also non standard models of PA: discretely ordered semi-rings with infinite descending chains satisfying a version of induction. The existence of such models opens up the possibility of statements independent of arithmetic: true statements about the natural numbers not provable from standard finite methods. In this talk we will show how such statements are found and give some examples. Time permitting we'll discuss our recent work on (L, n)-models which codify such independence results.
problems and recent progress related to almost complex manifolds.
Speaker: Scott Wilson, Queens College and the CUNY Graduate Center
Tuesday, March 19
Abstract: An almost complex manifold is a smooth manifold whose tangent bundle has the structure of a complex vector bundle. Many interesting examples appear in complex geometry and symplectic topology. A fair amount is known concerning when a manifold has one of these structures, and yet there remain many important and illusive open questions. In this talk I'll survey some elementary results while exploring these questions, and report on some recent progress which develops new algebraic and analytic methods for studying these objects.
|Title: From Torsion
to Cotorsion and Back
Speaker: Alex Martsinkovsky, Northeastern University
Tuesday, March 26
Abstract: The notion of torsion is known to everybody who has ever seen an abelian group. The notion of cotorsion is not known to anybody; in fact, it doesn’t even exist. This claim does not contradict the fact that there are 597 entries in MathSciNet containing the word cotorsion. The goal of this talk is to explain this paradox and give the correct definition of cotorsion. To do this, we have to rethink the classical definition of torsion and generalize it to arbitrary rings and modules. The definition of cotorsion will also be done in complete generality — for arbitrary modules over arbitrary rings. The two concepts turn out to be related by a duality.
This is an expository talk and prerequisites are modest: basic properties of the tensor product and the Hom functors. I will freely use the notions of projective and injective modules and of derived functors, but these will be reviewed at the beginning of the lecture.
This is joint work with Jeremy Russell.
Element Theorem for fields with commuting derivations and automorphisms
Speaker: Gleb Pogudin, Courant NYU
Tuesday, April 2
Abstract: Primitive Element Theorem says that every finitely generated algebraic extension of fields of zero characteristic is generated by a single element. It is a fundamental tool in algebra and symbolic computation.
When we talk about fields generated by solutions of differential or difference equations, the field extensions obtained by adjoining the solutions of these equations are not longer algebraic. On the other hand, these field extensions have extra structure, namely differential or difference operators, and this structure might give us additional expressive power. Appropriate generalizations of the Primitive Element Theorem to field extensions equipped with commuting derivations or a difference operator were obtained by Kolchin (1942) and Cohn (1965), respectively.
In my talk, I will describe my recent result that generalizes theorems by Kolchin and Cohn in two directions simultaneously
1. Any number of derivations and difference operators is allowed given that they commute with each other;
2. All restrictions on the base field are removed. This makes the theorem applicable to many new important cases, for example, to extensions coming from autonomous equations.
inflection points on real (hyper)elliptic curves
Speaker: Ethan Cotterill, Federal Fluminense University, Brazil
Tuesday, April 9
Abstract: Understanding when a given abstract algebraic curve comes equipped with a degree-d embedding in r-dimensional projective space is a fundamental problem of algebraic geometry. A crucial subsidiary issue is how inflected the curve C is under a given projective embedding. When our curve is defined over the complex numbers, a celebrated classical result of Plucker establishes that the degree of total inflection is an explicit function of d, r, and the genus g of C. In this talk we will examine what Plucker-like analogues hold when we work over the real instead of complex numbers, and assuming that C is of hyperelliptic type. If time permits, we will also comment on generalizations to other base fields. (Joint with I. Biswas and C. Garay López)
|Title: An example
of Lebesgue invariant Markov partitions and its Beurling-Ahlfors
Speaker: Zhe Wang, BCC
Tuesday, April 16
Abstract: I will use an example to explain a research project with Yunchun Hu and Yunping Jiang (Queens College). We are working on symmetric conjugacy classes of uniformly quasisymmetric expanding covering maps of the unit circle. We want to show the uniqueness of the Lebesgue invariant map in each conjugacy class.
Approximation: Numbers, Functions, and Numbers Again
Speaker: Kevin O'Bryant, College of Staten Island and the CUNY Graduate Center
Tuesday, April 30
Abstract: Quantifying the density of rationals in the reals is a fundamental problem. Given a real number $x$, there are rationals close to it with large denominator, but what exactly is the relationship between ``close to $x$’’ and ``large denominator’’? Unsatisfying partial answers are given by measure theory, continued fractions, and often beautiful complex function arguments. In the last decades, the little-known theory of Pad\’e approximants (approximating an analytic function with a rational function, as opposed to with a mere polynomial) has provided concrete progress for a few $x$.
|Title: Two recent
results on coloring graphs with forbidden induced subgraphs
Speaker: Mingxian Zhong, Lehman College
Tuesday, May 7
Abstract: Graph coloring is a fundamental problem in graph theory and has a variety of applications. It is well known that deciding whether a graph can be colored within k colors is NP-complete if k is at least 3. Thus it is natural to ask whether the complexity of the problem changes if we know that the input graph does not contain a particular induced subgraph. In the first part of the talk, I will present a polynomial time algorithm for the 4-coloring problem and the 4-precoloring extension problem restricted to the class of graphs with no induced six-vertex path, thus proving a conjecture of Huang. Combined with previously known results this completes the classification of the complexity of the 4-coloring problem for graphs with a connected forbidden induced subgraph. This is joint work with Maria Chundnovsky and Sophie Spirkl. In the second part of the talk, I will discuss the the case when the forbidden graph is not connected. Specficically, I will present a polynomial tiame algorithm of list-three coloring on $P_6+rP_1$-free grpahs. This is joint work with Maria Chundnovsky, Shenwei Huang and Sophie Spirkl.
|Title: A notion of
geometry that includes many more three dimensional manifolds
Speaker: Dennis Sullivan, CUNY Graduate Center and Stony Brook University
Tuesday, May 14
Abstract: Beyond Riemann's notion of geometry, one has projective geometry on projective space which in turn can be generalized to coset spaces of any Lie group, Ehresmann 1950. The lecture will describe an extension of the verified Thurston-Poincaré conjecture that is possible because of further structure in that information (joint work with Alice Kwon, PhD CUNY 2019). The last main point of this work is to extend the scope of the Ehresmann conception of geometry to profit from the extra information mentioned above. Then all prime three manifolds will have geometry.