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Spencer Cattalani, Stony Brook University

Ahlfors currents and symplectic non-hyperbolicity

Rational curves are one of the main tools in symplectic geometry and provide a bridge to algebraic geometry. Complex lines are a more general class of curve that has the potential to connect symplectic and complex analytic geometry. These curves are non-compact, which presents a serious difficulty in understanding their symplectic aspects. In this talk, I will explain how Ahlfors currents can be used to resolve this difficulty and produce a theory parallel to that of rational curves. In particular, Ahlfors currents can be constructed via a continuity method, they control bubbling of holomorphic curves, and they form a convex set.

MC 5417

Jarry Gu, University of Waterloo

The Boyd-Lawton Formula

While Mahler measure gives us a quantification of geometric means of polynomials, the Boyd-Lawton formula provides a link between singlevariate and multivariate Mahler measures. In this talk, we will focus on how Lawton proved this formula, and discuss how we can approximate continuous functions on the unit torus with trigonometric polynomials.

MC 5479

Muhammad Afifurrahman, University of New South Wales

Number Theory Seminar:Counting multiplicatively dependent integer vectors on a hyperplane

We give several asymptotic formulae and bounds for the number of multiplicatively dependent integer vectors of bounded height that lies on a hyperplane, extending the work of Pappalardi, Sha, Shparlinski and Stewart. Joint work with Valentio Iverson and Gian Cordana Sanjaya (University of Waterloo). 

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Julian Cheng, University of Waterloo

An Introduction to Random Binary Sequences

In this week, our goal is to cover two definitions of what it means for an infinite binary sequence to be random: one definition will come from the perspective of computability and the other from measure theory. We will show that they are equivalent. As an example, we will show that Chaitin's constant (also known as the halting probability) is random. This will cover parts of section 6.1 and 6.2 from Downey and Hirschfeldt.

MC 5403

Chris Schulz, Postdoc at the University of Waterloo

A Cobham theorem for scalar multiplication

The famous Cobham-Semenov theorem states that for k, l > 1 multiplicatively independent, if a subset of N^d is definable both using the base-k representations of its elements and using the base-l representations, then it is definable from only the addition function on N. We present an analogous result in the real case: for alpha, beta quadratic irrationals from distinct field extensions of Q, if a subset of R^d is definable by expanding the real ordered group with Z both using the multiplication by alpha and using the multiplication by beta, then it is definable from only the real ordered group with Z itself. This talk is based on joint work with Philipp Hieronymi and Sven Manthe.

MC 5403 

Chris Karpinski, McGill University

Relativizing computable categoricity

A metric space is hyperbolic if geodesic triangles in the metric space are uniformly slim. To any hyperbolic metric space, one can associate a boundary at infinity, a topological space called the Gromov boundary. A group acting on a hyperbolic metric space by isometries induces an action on the associated Gromov boundary by homeomorphisms. Given a hyperbolic space equipped with an action of a group, one can then study the orbit equivalence relation of the boundary action. We show that a class of groups of interest in geometric group theory, defined using graphical small cancellation theory, induce hyperfinite orbit equivalence relations on the boundaries of their natural hyperbolic Cayley graphs, meaning roughly that the orbits look like lines. This is joint work with Damian Osajda and Koichi Oyakawa.

MC 5403

Nathaniel Sagman, University of Toronto

Complex harmonic maps and the Atiyah-Bott-Goldman symplectic form

The field of higher Teichmuller theory has developed alongside the theories of Higgs bundles and harmonic maps to symmetric spaces of non-compact type. In this talk, I aim to give an introduction to harmonic maps and Hitchin components for SL(n,R) (examples of so-called higher Teichmuller spaces), and to present a new development: complex harmonic maps to holomorphic Riemannian symmetric spaces. Along with surveying the basic theory, old and new, I will explain how we used complex harmonic maps to prove that the Atiyah-Bott-Goldman symplectic form determines a pseudo-Kahler structure on the Hitchin component for SL(3,R). This is joint work with Christian El Emam.

MC 5417

Alexander Teeter, University of Waterloo

Slice Knots and Knot Concordance

In this talk, we explore an overview of the interplay between Knot Theory and Four Dimensional Topology. Specifically, we look at both Topologically and Smoothly Slice Knots, which are Knots in S^3 that bound (smoothly) embedded disks in B^4. We explore some of the techniques in the proof that the conway knot is not smoothly slice, and look at some of the ideas involved the construction of exotic R^4s using such knots.

MC 5403

Junichiro Matsuda, Department of Pure Mathematics, University of Waterloo

Quantum graphs violate the classical characterization of the existence of d-regular graphs.

Quantum graphs are a non-commutative analogue of classical graphs that replace the function algebra on vertices with C*-algebras. It is known that classical simple d-regular graphs on n points exist if and only if dn is even. This is false for quantum graphs in both directions. We provide a necessary condition on the number of quantum edges between quantum vertices (matrix summands) to make it a d-regular quantum graph. Using this technique, we also describe 1-regular or 2-regular quantum graphs on general tracial quantum sets. 1-regular quantum graphs have quantum edges only between summands of the same size. Centrally connected $2$-regular quantum graphs are classified into 8 families by their central skeleton. This is a joint work with Matthew Kennedy and Larissa Kroell.

Hybrid - QNC 1507, https://uwaterloo.zoom.us/j/94186354814?pwd=NGpLM3B4eWNZckd1aTROcmRreW96QT09

Ali Alsetri, University of Kentucky

Burgess-type character sum estimates over generalized arithmetic progressions of rank 2

We extend the classical Burgess estimates to character sums over proper generalized arithmetic progressions (GAPs) of rank 2 in prime fields. The core of our proof is a sharp upper bound for the multiplicative energy of these sets, established by adapting an argument of Konyagin and leveraging tools from the geometry of numbers. This is joint work with Xuancheng Shao.

Online - https://uwaterloo.zoom.us/j/98942212227?pwd=huSbGSNTP1ODaePFVsXb4FJy6Deite.1