Catherine St-Pierre, University of Waterloo
Organizational meeting
We will organize the seminars for the summer.
MC 5403
Jennifer Zhu, University of Waterloo
Morphisms of Quantum Confusability Graphs
It would be unrealistic to have an information channel — quantum or classical — that always sends information with absolute accuracy; that is, we must expect a channel to have noise. In 1956, Shannon introduced the notion of zero-error capacity of a noisy (classical) channel using the confusability graph of this channel. In 2010, Duan, Severini, and Winter developed the analogous notion (quantum confusability graphs) for quantum channels and show that one can recover various types of zero-error capacities of quantum channels. In the first half of this talk, we will see how these quantum confusability graphs are derived and how they subsume Shannon's notion of classical confusability graphs.
QNC 1201
Facundo Camano, University of Waterloo
Boundary Conditions for Non-Euclidean Monopoles
In this talk, I will discuss the heuristic behind defining asymptotics for monopoles. Specifically, the asymptoticsshould be abelian solutions embedded into the gauge group. I will first go over this heuristic for Euclideanmonopoles and then move on to non-Euclidean situations such as hyperbolic and singly periodic.
MC 5403
Paul Cusson, University of Waterloo
Monopoles with rotational symmetry
We will first look at SU(2)-monopoles invariant under the action of a circle subgroup of SO(3) about the z-axis.The polynomials cutting out their spectral curves in TP^1 will be derived, and these will be used to describe thespectral curves of S^1-invariant SU(N)-monopoles for arbitrary N.
MC 5403
Elan Roth & William Dan, University of Waterloo
Randomness in the Arithmetic Hierarchy
We will introduce a problem posed by Bienvenu, Csima, and Harrison-Trainor about transforming indices of random left c.e. reals to optimal machines with specific halting probabilities. Then, we will prove two results that have been useful in our attempts to resolve the open problem.
MC 5403
Alex Pawelko, University of Waterloo
Morse Theory via Harmonic Oscillators
We will discuss the approach to Morse Theory originally due to Witten, where one constructs deformed Laplaceoperators whose low-energy eigenvectors concentrate near the critical points of one's Morse function, and thenuses Hodge theory to relate this to de Rham cohomology.
MC 5403
Ila Varma, University of Toronto
Counting Number Fields by P^1 height
When do two irreducible polynomials with integer coefficients define the same number field? One can define an action of GL_2 × GL_1 on the space of polynomials of degree n so that for any two polynomials f and g in the same orbit, the roots of f may be expressed as rational linear transformations of the roots of g; thus, they generate the same field. In this article, we show that almost all polynomials of degree n with size at most X can only define the same number field as another polynomial of degree n with size at most X if they lie in the same orbit for this group action. (Here we measure the size of polynomials by the greatest absolute value of their coefficients.)
This improves on work of Bhargava, Shankar, and Wang, who proved a similar statement for a positive proportion of polynomials. Using this result, we prove that the number of degree n fields such that the smallest polynomial defining the field has size at most X is asymptotic to a constant times X^{n+1} as long as n \ge 3. For n = 2, we obtain a precise asymptotic of the form 27/(pi^2) * X^2
This is joint work with Arango-Pineros, Gundlach, Lemke Oliver, McGown, Sawin, Serrano Lopez, and Shankar.
MC 5479
Henrique Sà Earp, Unicamp
Updates on flows of geometric structures
Aiming at a public with interests among Riemannian and complex geometry, Lie groups and parabolic PDEs, Iwill recall some results towards a general analytical theory for flows of H-structures, obtained with Fadel,Loubeau and Moreno. Then, as a concrete example, I will present some recent progress on SU(2)-flows on 4-manifolds, initiating a classification programme of ‘parabolic’ flows, based on a representation-theoretic methodoriginally due to Bryant in the context of G2 geometry, obtained with Fowdar.
MC 5501
Fateme Peimany, University of Waterloo
Definable Groups in Differentially Closed CCM-structures (DCCM)
How do definable groups behave when a compact complex manifold is equipped with a generic derivation? Inthis talk, I will discuss an ongoing project to understand definable groups within the theory of DifferentiallyClosed CCM-structures (DCCM). The presentation centers on a foundational model-theoretic analogy: just asadding a derivation to algebraic geometry bridges ACF and DCF, a parallel expansion maps CCM to DCCM inthe analytic setting. By reviewing established results for definable groups in ACF, DCF, and CCM, we willexplore how these classifications translate into the differential-analytic framework of DCCM.
MC 5417
Jules Ribolzi, University of Waterloo
A Chevalley structure theorem for meromorphic groups
We continue to study definable groups in the standard model of CCM. We will look at the proof of a Chevalleystructure type theorem for meromorphic groups (that is, meromorphic groups are regular in the sense of Fujiki)from the article of Pillay and Scanlon.
MC 5479