Rahim Moosa, University of Waterloo
Organisational meeting
We will meet to discuss the seminars taking place in the Winter term.
MC 5403
Dror Varolin, Stony Brook University
Extending sections of Holomorphic Vector Bundles
In 1987 Ohsawa and Takegoshi published their fundamental result on L2 extension of holomorphic functions. It did not take long for this result to be generalized to sections of holomorphic line bundles, and a spectacular array of applications appeared in a number of areas of complex analytic and algebraic geometry. By contrast, the L2 Extension of sections of holomorphic vector bundles has been much less considered. In particular, until recently optimal positivity conditions were not totally understood. In this talk I will present a result about L2 Extension in the higher rank case, and also an example showing that this type of positivity is optimal. I will also discuss the relevance to a question about deformation of spaces of holomorphic sections.
MC 5417
Rachael Alvir, University of Waterloo
Introduction to Continuous Logic I
We begin with an introduction to continuous logic and metric structures following the manuscript "Model Theory for Metric Structures" by Berenstein, Ben Yaacov, Henson, and Iovino.
MC 5403
Almut Burchard, University of Toronto
On spatial monotonicity of heat kernels
The heat kernel on a manifold contains a wealth of global geometric information about the underlying space. It is of central importance for partial differential equations (describing diffusion of a unit of heat released from a point through the space) and for probability (giving the transition densities for Brownian motion).
On flat n-dimensional space, the heat kernel K_t(x,y) decreases with the distance between the points x and y (that is, temperature decreases as we move away from the heat source); the same is true on the sphere. Does the heat kernel on different Riemannian manifolds have similar properties? In general, the answer is "No!" ... except sometimes ...
MC 5501
Refreshments available at 3:30pm
Aristomenis Papadopoulos, University of Maryland
Zarankiewicz's Problem and Model Theory
"A shower thought that anyone interested in graph theory must have had at some point in their lives is the following: 'How ""sparse"" must a given graph be, if I know that it has no ""dense"" subgraphs?'. This curiosity definitely crossed the mind of Polish mathematician K. Zarankiewicz, who asked a version of this question formally in 1951. In the years that followed, many central figures in the development of extremal combinatorics contemplated this problem, giving various kinds of answers. Some of these will be surveyed in the first part of my talk.
So far so good, but this is a logic seminar and the title says the words ""Model Theory""… In the second part of my talk, I will discuss how the celebrated Szemerédi-Trotter theorem gave a starting point to the study of Zarankiewicz's problem in ""geometric"" contexts, and how the language of model theory has been able to capture exactly what these contexts are. I will then ramble about improvements to the classical answers to Zarankiewicz's problem, when we restrict our attention to semilinear/semibounded o-minimal structures, Presburger arithmetic, and various kinds of Hrushovski constructions.
The new results that will appear in the talk were obtained jointly with Pantelis Eleftheriou."
MC 5479
Sourabh Das, University of Waterloo
Tools in Analytic Number Theory
Analytic number theory provides several classical tools to explore the distribution of arithmetic functions, including the rearrangement of sums, generating series, and counting arguments. In this talk, we will use these techniques to investigate the distribution of h-free and h-full numbers, which play an important role in understanding the structure of integers. If time allows, we will extend our discussion to prove results about the distribution of the prime-counting ω-function over these subsets of integers. Designed as an introductory talk, this presentation will be accessible to anyone with a basic background in number theory.
MC 5403
Spiro Karigiannis, University of Waterloo
Infinitesimal deformations of G-structures
I will introduce the setting of G-structures on an oriented Riemannian n-manifold, where G is a closed Lie subgroup of SO(n). These can be understood in terms of global sections of the SO(n) bundle which is the quotient of the SO(n)-prinicipal bundle of oriented orthonormal frames by the free action of G. We will define the intrinsic torsion of a G-structure, and explain how to describe infinitesimal deformations of G-structures. If time permits, we will discuss a Dirichlet energy type of functional on the space of G-structures, whose critical points are called harmonic G-structures. This condition includes the torsion-free G-structures but is more general. These ideas were developed recently by Fowdar, Loubeau, Moreno, Sa Earp building on earlier work in the G2 and Spin(7) cases by myself from 2006-2007.
MC 5479
Rahim Moosa, University of Waterloo
Curve excluding fields I
Recently, Johnson and Ye have proved an attractive and somewhat surprising result: Suppose C is an algebraic curve of genus at least two having no rational points. Then the class of fields over which C has no rational points, has a model companion. This model companion, they call it CXF, turns out to answer several old questions.
I will start presenting the results of the paper.
MC 5403
Daniel Gromada, Czech Technical University
A brief introduction to quantum symmetries
In this talk, I would like to explain the concept of a quantum symmetry. We will focus on symmetries of simple combinatorial objects like finite sets and graphs. This can be approached either from the viewpoint of quantum groups or via diagrammatic categories. I will try to explain how drawing simple string diagrams can reveal interesting findings about quantum symmetries of certain objects.
MC 5501
Krishnarjun Krishnamoorthy, BIMSA
Moments of non-normal number fields
Let K be a number field and a_K(m) be the number of integral ideals in K of norm equal to m. We asymptotically evaluate the sum \sum_{m\leqslant X} a_K^l(m) as X grows to infinity. We also consider the continuous moments of the associated Dedekind zeta function and prove lower bounds of the expected order of magnitude.