Rachael Alvir, University of Waterloo
More Fundamentals of Computability Theory
We continue to present results from Soare's book on computability theory.
MC 5403
Alex Pawelko, University of Waterloo
Prequantum Line Bundles and Geometric Quantization
Prequantum line bundles are objects in symplectic geometry that play a somewhat analogous role to holomorphic line bundles in complex geometry. In this talk, we will discuss the existence of prequantum line bundles, examples of them, and their uses in symplectic geometry, most notably in geometric quantization.
MC 5479
Owen Patashnick, King's College London
Motive-ating formal periods via special values of L-functions
In this talk we will use special values of L-functions as a gateway drug to explore the motivic periods that underlie geometric content associated to these L-values. In particular, we will "motivate" an explicit construction of classes built out of algebraic cycles associated with the L-values L(Sym^n(E), n+m) and muse on the consequences. We will try to make the talk as accessible as possible, and hopefully keep discussion of machinery to a minimum.
MC 5479
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 talk, I will extend this correspondence by providing necessary and sufficient conditions for when a co-Higgs field induces an integrable 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. We will also analyze this correspondence for co-Higgs fields over curves of genus g greater than one. Finally, I will show how stability can be understood geometrically through the zeros of the induced Poisson structure, establishing connections between \Phi-invariant subbundles, Poisson subvarieties, and the spectral curve. As this talk is a preparation for my thesis defense, please ask me many questions!
MC 5403
Spiro Karigiannis, University of Waterloo
A tale of two Lie groups
The classical Lie group SO(4) is well-known to possess a very rich structure, relating in several ways to complex Euclidean spaces. This structure can be used to construct the classical twistor space Z over an oriented Riemannian 4-manifold M, which is a 6-dimensional almost Hermitian manifold. Special geometric properties of Z are then related to the curvature of M, an example of which is the celebrated Atiyah-Hitchin-Singer Theorem. The Lie group Spin(7) is a particular subgroup of SO(8) determined by a special 4-form. Intriguingly, Spin(7) has several properties relating to complex Euclidean spaces which are direct analogues of SO(4) properties, but sadly (or interestingly, depending on your point of view) not all of them. I will give a leisurely introduction to both groups in parallel, emphasizing the similarities and differences, and show how we can nevertheless at least partially succeed in constructing a "twistor space" over an 8-dimensional manifold equipped with a torsion-free Spin(7)-structure. (I will define what those are.) This is joint work with Michael Albanese, Lucia Martin-Merchan, and Aleksandar Milivojevic. The talk will be accessible to a broad audience.
MC 5479
Mingyang Li, UC Berkeley
On 4d Ricci-flat metrics with conformally Kahler geometry.
Ricci-flat metrics are fundamental in differential geometry, and they are easier to study when they have additional structures. I will introduce my recent work on 4d conformally Kahler but non-Kahler Ricci-flat metrics, which is a condition analogous to hyperkahler. This leads to a complete classification of asymptotic geometries of such metrics at infinity and a classification of such gravitational instantons.
MC 5417
Rachael Alvir, University of Waterloo
Conclusion of the Fundamentals of Computability Theory
We will finish presenting results from Soare's book. We will look at Low n and High n sets.
MC 5403
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.
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Meeting ID: 971 4907 1044
Passcode: 776121
Ruiran Sun, University of Toronto
Rigidity problems on moduli spaces of polarized manifolds.
Motivated by Shafarevich’s conjecture, Arakelov and Parshin established a significant finiteness result: for any curve C, the set of isomorphism classes of non-constant morphisms C → M_g is finite for g≥2. However, for moduli stacks parametrizing higher-dimensional varieties, the Arakelov-Parshin finiteness theorem fails due to the presence of non-rigid families. In this talk, I will review recent advances in rigidity problems for moduli spaces of polarized manifolds, focusing on two main topics: an "one-pointed" version of Shafarevich’s finiteness theorem and the distribution of non-rigid families within moduli spaces.
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Thomas Bray, University of Waterloo
To b or Not to b: The Art of Generalizing Metric Spaces
Metric spaces form the foundation of many areas in mathematics, offering a rigorous framework for understanding distance and convergence. But what happens when we relax the triangle inequality? Enter b-metric spaces, where the triangle inequality is replaced by a more flexible inequality scaled by a constant. In this talk, we will explore how this generalization leads to surprising results and broadens the scope of classical fixed-point theory, topology, and functional analysis. Join me as we delve into the rich structure of b-metric spaces and uncover their role in contemporary mathematical research.
Snacks will be available from 4:00pm
MC 5417