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Sunil Naik, Queen's University

On a question of Christensen, Gipson and Kulosman

The study of irreducible polynomials in various polynomial rings is an important topic in mathematics. In this context, polynomials with restricted exponents have become the focus of considerable attention in recent years. Motivated by these considerations, Matsuda introduced the ring $F[X;M]$ of polynomials with coefficients in a field $F$ and exponents in a commutative, torsion-free, cancellative (additive) monoid $M$ and began an inquiry into the irreducibility of various polynomials in these rings. For any prime $\ell$, we say that $M$ is a Matsuda monoid of type $\ell$ if for each indivisible $\alpha$ in $M$, the polynomial $X^{\alpha}-1$ is irreducible in $F[X;M]$ for any field $F$ of characteristic $\ell$.

Let $M$ be the additive submonoid of non-negative integers generated by 2 and 3. In a recent work, Christensen, Gipson, and Kulosman proved that $M$ is not a Matsuda monoid of type 2 and type 3 and they have raised the question of whether $M$ is a Matsuda monoid of type $\ell$ for any prime $\ell$. Assuming the Generalized Riemann Hypothesis (GRH), Daileda showed that $M$ is not a Matsuda monoid of any positive type. In this talk, we will discuss an unconditional proof of the above result using its connection with Artin’s primitive root conjecture.

Tobias Shin, University of Chicago

Almost complex manifolds are (almost) complex

What is the difference topologically between an almost complex manifold and a complex manifold? Are there examples of almost complex manifolds in higher dimensions (complex dimension 3 and greater) which admit no integrable complex structure? We will discuss these two questions with the aid of a deep theorem of Demailly and Gaussier, where they construct a universal space that induces almost complex structures for a given dimension. A careful analysis of this space shows the question of integrability of complex structures can be phrased in the framework of Gromov's h-principle. If time permits, we will conclude with some examples of almost complex manifolds that admit a family of Nijenhuis tensors whose sup norms tend to 0, despite having no integrable complex structure (joint with L. Fernandez and S. Wilson).

MC 5417

Valeriya Kovaleva, University of Montreal

Correlations of the Riemann Zeta on the critical line

In this talk we will discuss the behaviour of the Riemann zeta on the critical line, and in particular, its correlations in various ranges. We will prove a new result for correlations of squares, where shifts may be up to size T^{3/2-\epsilon}. We will also explain how this result relates to Motohashi’s formula for the fourth moment, as well as the moments of moments of the Riemann Zeta and its maximum in short intervals.

MC 5479

Diophantine Results for Shimura Varieties

Welcoming remarks: Dean Mark Giesbrecht (University of Waterloo)

Laudation: Professor Peter Sarnak (Institute for Advanced Study and Princeton University)

Ostrowski Lecture: Professor Jacob Tsimerman (University of Toronto)

Shimura Varieties are higher dimensional analogues of modular curves, and they play a foundational role in modern number theory. The most familiar Shimura varieties are the moduli spaces of Abelian varieties, and in this context we have a wealth of diophantine results, both in the number field and function field setting: Finiteness of S-rational points, the Tate conjecture, the Shafarevich conjecture, semisimplicity of Galois representations, and others. These results constitute a blueprint for what we expect to be true in other settings but is largely out of reach.

DC1302

Tristan Collins, University of Toronto

A free boundary Monge-Ampere equation with applications to Calabi-Yau metrics.

I will discuss a free boundary Monge-Ampere equation that arises from an attempt to construct complete Calabi-Yau metrics. I will explain how this equation can be solved and its connections with optimal transport. This is joint work with F. Tong and S.-T. Yau.

MC 5501

Camila Sehnem, University of Waterloo

A characterization of primality for reduced crossed products

In this talk I will discuss ideal structure of reduced crossed products by actions of discrete groups on noncommutative C*-algebras. I will report on joint work with M. Kennedy and L. Kroell, in which we give a characterization of primality for reduced crossed products by arbitrary actions. For a class of groups containing finitely generated groups of polynomial growth, we show that the ideal intersection property together with primality of the action is equivalent to primality of the crossed product. This extends previous results of Geffen and Ursu and of Echterhoff in the setting of minimal actions.

MC 5417 

Zev Friedman, University of Waterloo

N-cohomologies on non-integrable almost complex manifolds

I will define an N-cohomology and compute some interesting examples, showing the different isomorphism classes on certain almost complex manifolds.

MC 5479

Kyle Pereira, University of Waterloo

Fundamentals of Computability Theory 5

We will look at Post's Theorem and related hierarchies, following Robert Soare's textbook.

MC 5403

Sumun Iyer, Carnegie Mellon University

Knaster continuum homeomorphism group

Knaster continua are a class of compact, connected, metrizable spaces. Each Knaster continuum is indecomposable-- it cannot be written as the union of two proper nontrivial sub continua. We consider the group Homeo(K) of all homeomorphisms of the universal Knaster continuum; this is a non-locally compact Polish group. We will describe some "large" topological group phenomena that occur in this group, in relation to the group's universal minimal flow and its generic elements.

MC 5479

Mark Hamilton, Mount Allison University

Toric degenerations and independence of polarization

In the theory of geometric quantization, one essential ingredient is the choice of a "polarization"; a natural question is then whether the resulting quantization depends on this choice.  One recent approach to the question of "independence of polarization" is using a deformation of complex structure to "deform" one polarization into another.  Originally applied to smooth toric varieties, this has also been applied to a broader class of examples, such as flag varieties, by using a toric degeneration. 

In this talk I will present an overview of this program (including a short introduction to the key ideas of geometric quantization), and mention several examples of its application, including flag manifolds, more general varieties, and moduli spaces of flat connections (work in progress).

MC 5403