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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

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

Organisational meeting

We will meet to discuss the seminars taking place in the Winter term.

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

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

Spiro Karigiannis, University of Waterloo

Organizational Meeting

We will meet to plan out the Differential Geometry Working Seminar for the Winter 2025 term.

MC 5479

Faisal Romshoo, University of Waterloo

Constructing associatives in 7-manifolds

We will revisit the classical examples of Special Lagrangians invariant under some group G in SU(n) using a new method and check if we can use the same method to construct associative submanifolds, which are a type of calibrated 3-submanifolds in 7-manifolds, in R^7.

MC 5479

Zev Friedman, University of Waterloo

4-dimensional U(m) structures

We will define the modified deRham operator D on U(m) structures, and prove that D^2=0 is equivalent to d \omega=0 in the 4-dimensional case.

MC 5479

Ryoya Arimoto, Kyoto University

Simplicity of crossed products of the actions of totally disconnected locally compact groups on their boundaries

Results of Archbold and Spielberg, and Kalantar and Kennedy assert that a discrete group admits a topologically free boundary if and only if the reduced crossed product of continuous functions on its Furstenberg boundary by the group is simple. In this talk, I will show a similar result for totally disconnected locally compact groups.

MC 5417 or Join on Zoom

Brady Ali Medina, University of Waterloo

Co-Higgs Bundles and Poisson Structures.

There is a correspondence between co-Higgs fields and holomorphic Poisson structures on P(V) established by Polishchuk in the rank 2 case and by Matviichuk in the case where the co-Higgs field is diagonalizable. In this thesis, we extend this correspondence by providing necessary and sufficient conditions for when a co-Higgs field induces a Poisson structure on V and P(V), showing that the co-Higgs field must either be a function multiple of a constant matrix or have only one non-zero column. Furthermore, we analyze this correspondence for co-Higgs fields over curves of genus greater or equal to one.  Finally, we analyze how stability can be interpreted geometrically through the zeros of the induced Poisson structure, establishing connections between \Phi -invariant subbundles, Poisson subvarieties, and the spectral curve.

Join on Zoom

Meeting ID: 971 4907 1044

Passcode: 776121