Events

Julius Frizzell, University of Waterloo

The Unfair 0-1 Polynomial Conjecture

The unfair 0-1 polynomial conjecture states that if you have two monic polynomials, with non-negative real coefficients, and their product has only zeros and ones as coefficients, then the original two polynomials also have only zeros and ones as coefficients. In this talk, I will introduce the problem and some of the basic considerations about it. Then I will discuss the current techniques being used to make progress on this conjecture, as well as possible future approaches. Along the way, we will see how facts about zeros of polynomials, resultants, Taylor series, and binary sequences are related. We will also discuss the computational steps required in the current work on the problem.

MC 5479

Joey Lakerdas-Gayle, University of Waterloo

Weakly uniform computable categoricity

A computable structure A is Y-computably categorical if for every computable copy B of A, some isomorphism between A and B is computable in the oracle Y. Motivated by results about isomorphism spectra of computable structures, we introduce a relativized notion of uniform computable categoricity: For sets X and Yof natural numbers, we say that a computable structure A is weakly X-uniformly Y-computably categorical if A is Y-computably categorical and we can find Y-computable isomorphisms uniformly in X. We will investigate some natural questions about weakly uniform computable categoricity and compute some particular examples.

MC 5403

Casey Blacker, Augusta University

Geometric and algebraic reduction of multisymplectic manifolds

A symplectic Hamiltonian manifold consists of a Lie group action on a symplectic manifold together with an associated moment map. In special cases, the moment map distinguishes a smooth submanifold to which the Lie group action restricts, and the quotient inherits the structure of a symplectic manifold. In every case, it is possible to construct a reduced Poisson algebra that plays the role of the space of smooth functions on the reduced space.

In this talk, we will discuss an adaptation of these ideas to the multisymplectic setting. Specifically, we will exhibit a geometric reduction scheme for multisymplectic manifolds in the presence of a Hamiltonian action, an algebraic reduction procedure for the associated L-infinity algebras of classical observables, and a comparison of these two constructions. This is joint work with Antonio Miti and Leonid Ryvkin.

MC 5417

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

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

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

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 

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

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