Events

Nicolas Banks, University of Waterloo

Classification Results for Intersective Polynomials With No Integral Roots

In this thesis defence, we introduce strongly intersective polynomials - polynomials with no integer roots but with a root modulo every positive integer - of degree 5-10. These are fascinating objects which make contact with many areas of math, including permutation group theory, splitting behaviour of prime ideals in number fields, and Frobenius elements from class field theory.

In particular, we explain the computation of a list of possible Galois groups of such polynomials. We also discuss constraints on the splitting behaviour of ramified primes; in the process, we argue that intersectivity can be thought of as a property of a Galois number field, together with its set of subfields of specified degrees.

MC 5417 or Join on Teams

Faisal Romshoo, University of Waterloo

Constructing examples of Smith maps

I will start off by giving some background on Smith maps, which are special k-harmonic maps between two Riemannian manifolds. Smith maps have deep connections with both calibrated geometry and conformal geometry. I will then discuss my current work, where I am trying to construct explicit examples of Smith immersions.

MC 5403

AJ Fong, University of Waterloo

Galois actions on torus-invariant curves of toric surfaces

We are interested in studying the actions of the absolute Galois group on curves on varieties. Toric surfaces form an instructive class of examples since they are geometrically well-understood, and it is natural to study torus-invariant curves here. In this talk I will describe upper bounds on the Galois actions, and ongoing work regarding the inverse problem.

MC 5479

Barbara Csima, University of Waterloo

Measuring complexity of structures via their Scott Sentences

Scott’s Isomorphism Theorem shows that each countable structure can be uniquely defined, up to isomorphism, by a sentence of infinitary logic, now known as the Scott Sentence of the structure. The complexity of a structure’s Scott Sentence can then be viewed as a measure of complexity of the structure. In this talk, we will discuss the relationship of Scott complexity with other measures of complexity, as well as discuss the Scott Complexity of certain structures.

MC 5403

Kateryna Tatarko, University of Waterloo

Isoperimetric problem: from classical to reverse

The well-known classical isoperimetric problem states that the Euclidean ball has the largest volume among all convex bodies in R^n of a fixed surface area. We will discuss the question of reversing this result for the special class of convex bodies which are intersections of (finitely or infinitely many) balls of radius 1/lambda for some lambda>0. In particular, we will discuss the problem of determining which bodies in this class minimize the volume for a prescribed surface area and completely resolve it in R^3.

MC 5501

Alex Cowan, University of Waterloo

Murmurations from functional equations

Unexpected and striking oscillations in the average a_p values of sets of elliptic curves, dubbed murmurations, were recently discovered using techniques from data science. Since then, similar patterns have been discovered for many other types of arithmetic objects. In this talk we present a new approach for studying murmurations, revolving around mean values of L-functions in the critical strip and guided by random matrix theory. With our approach, we prove murmurations in many cases conditional on standard conjectures, and unconditionally for all GL_1 automorphic representations. To handle the case of elliptic curves knowledge is needed of the distribution of conductors of elliptic curves when ordered by height, which is of independent interest.

MC 5417

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

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

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