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 

Title: The
AxLindemannWeierstrass 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 wellknown example is the modular
jfunction). I will place particular emphasis on the so called
AxLindemannWeierstrass 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). 

Title: Dirichlet
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
RiemannBochnerHecke converse theorem. 

Title: Independence
in Arithmetic and NonStandard 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
semirings 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. 

Title: Some
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. 

Title: Primitive
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. 

Title: Real
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 degreed embedding in rdimensional 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 Pluckerlike 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 BeurlingAhlfors
quasiconformal extension 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. 

Title: Rational
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 littleknown 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 NPcomplete 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 4coloring problem and the 4precoloring
extension problem restricted to the class of graphs with no induced
sixvertex path, thus proving a conjecture of Huang. Combined with
previously known results this completes the classification of the
complexity of the 4coloring 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 listthree 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
ThurstonPoincaré 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. 