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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

Paul Cusson, University of Waterloo

Various moduli spaces of monopoles

We will go over, in increasing generality, results classifying various classes of Euclidean $SU(n)$-monopoles, starting with the $n=2$ case. We will see that these moduli spaces are described as spaces of rational maps from the projective line to flag varieties.

MC 5403

Cole Wyeth, University of Waterloo

ntroduction to Algorithmic Probability and the Coding Theorem

Building on the prefix-free Kolmogorov complexity discussed at our last meeting, I will introduce the basic objects of algorithmic probability. In particular, with a theory of effective explanations in hand, it is natural to ask which strings are more probable a priori? After all, it is harder to predict the data before you have seen it! The distributions generated by probabilistic Turing machines can be fully characterized as the (normalized) "lower semicomputable semimeasures," which naturally leads to the so-called "discrete universal distribution" m by simply mixing them all together. I will sketch a proof of Leonid A. Levin's coding theorem, which tells us that -lg m(x) = K(x) up to constants, meaning that all of our work was, in the most satisfying possible sense, for nothing: we can take only the shortest algorithmic explanation without losing anything. However, this is all just a warm-up: we will find that the situation is much more intricate when we turn to the prediction of infinite sequences, which I hope to gesture at, time permitting. 

MC 5403

Eduardo de Lorenzo Poza, KU Leuven

Algebraic Geometry Seminar:Singularities via arc spaces and Floer homology

Given an isolated hypersurface singularity, the arc-Floer conjecture relates the cohomology of the associated contact loci with the Floer homology of the monodromy iterates. In this talk we will explain the origin of this conjecture and what is known about it, and we will explore the key ingredients of the proof of the conjecture in the cases of plane curve singularities and semihomogeneous singularities. This is joint work with Javier de la Bodega and Jiahui Huang.

Join on Zoom

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

Roberto Hernandez Palomares, University of Waterloo

Quantum graphs and spin models

Spin models for singly-generated Yang-Baxter planar algebras are known to be determined by certain highly-regular classical graphs such as the pentagon or the Higman-Sims graph. Examples of spin models include the Jones and Kauffman polynomials, as well as certain fiber functors. We will explore the notion of higher-regularity for quantum graphs as well as their potential to encode spin models. Time allowing, we will give examples of non-classical graphs with these properties.

QNC 1507 or Join on Zoom