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Kunjakanan Nath, University of Illinois, Urbana-Champaign

"Circle method and binary correlation problems"

One of the key problems in number theory is to understand the correlation between two arithmetic functions. In general, it is an extremely difficult question and often leads to famous open problems like the Twin Prime Conjecture, the Goldbach Conjecture, and the Chowla Conjecture, to name a few. In this talk, we will discuss a few binary correlation problems involving primes, square-free integers, and integers with restricted digits. The objective is to demonstrate the application of Fourier analysis (aka the circle method) in conjunction with the arithmetic structure of the given sequence and the bilinear form method to solve these problems.

Zoom link: https://uwaterloo.zoom.us/j/98937322498?pwd=a3RpZUhxTkd6LzFXTmcwdTBCMWs0QT09

Jiahui Huang, Department of Pure Mathematics, University of Waterloo

"Deformations and Lines on Cubics"

We continue studying the problem of lines on cubic surfaces. By considering first order deformations on the Hilbert scheme, we show that there are 27 distinct lines on any smooth surfaces.

MC 5501

Faisal Romshoo, Department of Pure Mathematics, University of Waterloo

"Some computations with gauge transformations on a $G_2$ manifold"

Given a (torsion-free) $G_2$-manifold $(M, \varphi, g)$ and a gauge transformation $P: TM \rightarrow TM$, we want to look at the $G_2$ structures $\Tilde{\varphi} = P^*g$ and explore the conditions for it to be torsion-free. In this talk, we will start in a more general setting with a Riemannian manifold $(M, g)$ and obtain an expression for the tensor $B(X, Y) = \tilde{\nabla}_X Y -\nabla_X Y$ before moving on to computing the full torsion tensor $\tilde{T}_{pq}$ in the case when $M$ is a $G_2$ manifold.

MC 4058

Rachael Alvir, Department of Pure Mathematics, University of Waterloo

"Computable Structure Theory III"

We will continue our discussion on forcing and take a look at nontrivial structures. 

MC 5479

Chatchai Noytaptim, Department of Pure Mathematics, University of Waterloo

"Potential function, transfinite diameter, and Fekete-Szego theorem"

We introduce the potential function attached to a probability measure and the transfinite diameter of any compact set in the Berkovich projective line. Time permitting, we briefly discuss a Berkovich version of the adelic Fekete-Szego theorem. The materials in this presentation cover sections 6.3-6.5 in Baker-Rumely’s monograph on “Potential Theory and Dynamics on the Berkovich Projective Line”.

MC 5417

Elliot Kaplan, McMaster University

"Generic derivations on o-minimal structures"

Let T be a model complete o-minimal theory that extends the theory of real closed ordered fields (RCF). We introduce T-derivations: derivations on models of T which cooperate with T-definable functions. The theory of models of T expanded by a T-derivation has a model completion, in which the derivation acts "generically." If T = RCF, then this model completion is the theory of closed ordered differential fields (CODF) as introduced by Singer. We can recover many of the known facts about CODF (open core, distality) in our setting. We can also describe thorn-rank for models of T with a generic T-derivation. This is joint work with Antongiulio Fornasiero.

MC 5479

AJ Fong, Department of Pure Mathematics, University of Waterloo

"Schemes in general"

We will introduce general schemes, and define important notions such as subschemes, local rings of schemes at a point, and morphisms. If time permits, we will also describe the gluing of arbitrary schemes by open subsets. This talk closely follows section I.2 of Eisenbud-Harris (with some necessary sheaf theory from I.1 which was omitted from the last talk for time).

MC 5417

Yash Singh, Department of Pure Mathematics, University of Waterloo

"Chern classes of vector bundles and applications"

We study Chern classes of vector bundles and their connection to grassmanians. We also study the problem of 27 lines on a cubic surface if time permits.

MC 5501

Eric Culf, Department of Applied Mathematics, University of Waterloo

"Approximation algorithms for noncommutative constraint satisfaction problems"

Constraint satisfaction problems (CSPs) are an important topic of investigation in computer science. For example, the problem of finding optimal k-colourings of graphs, Max-Cut(k), is NP-hard, but it is easy to approximate in the sense that it is possible to find a colouring that satisfies a large fraction of the constraints of an optimal one. We study a noncommutative variant of CSPs that is central in quantum information, where the variables are replaced by operators. In this context, even approximating general CSPs is known to be much harder than the classical case, in fact uncomputably hard. Nevertheless, Max-Cut(2) becomes efficiently solvable. We introduce a framework for designing approximation algorithms for noncommutative CSPs, which allows us to find classes of CSPs that are efficiently approximable but not efficiently solvable. To determine the quality of our approximation algorithm, we make use of results from free probability to characterise a distribution arising from random matrices. This talk is based on work with Hamoon Mousavi and Taro Spirig (arxiv.org/abs/2312.16765).

This seminar will be held both online and in person:

• Room: MC 5479
• Zoom link: https://uwaterloo.zoom.us/j/94186354814?pwd=NGpLM3B4eWNZckd1aTROcmRreW96QT09

Kieran Mastel, Department of Pure Mathematics, University of Waterloo

"Introduction to the PCP theorem"

This is the first meeting of a working seminar that will take place every Wednesday 1:30PM-3:00PM in QNC 1201. The PCP theorem from computational complexity theory was a important part of the groundbreaking MIP*=RE result, and has very strong connections to stability problems for representations of algebras and groups. We plan to study this theorem, the quantum PCP conjecture, and their connections to (approximate) representations and stability.  The first meeting will be organizational and will include an introductory talk by Kieran Mastel on the PCP theorem.

QNC 1201