Joey Lakerdas-Gayle, Department of Pure Mathematics, University of Waterloo
"Computable Structure Theory VI"
We will discuss generic enumerations of structures, following Antonio Montalbán's monograph.
MC 5479
Xiao Zhong, Department of Pure Mathematics, University of Waterloo
"Green's Functions on the Berkovich Projective Line"
We introduce the green's functions and explore their properties. After this, we are ready to introduce a Bilu-type equidistribution theory in the next talk which is one of the main motivation for going deeply into this subject. The materials in this presentation cover the later half of the chapter 7 in Baker-Rumely's monograph on "Potential Theory and Dynamics on the Berkovich Projective Line".
MC 5417
Amanda Petcu, Department of Pure Mathematics, University of Waterloo
"An Introduction to (Lagrangian) Mean Curvature Flow"
In this talk, we will introduce the Mean Curvature Flow and explore some initial examples of the flow. We will show that in the compact case, the flow always produces singularities. We will also introduce type I and type II singularities. Finally, if time permits, we will discuss the Lagrangian Mean Curvature Flow and demonstrate that a mean curvature flow starting from a Lagrangian remains Lagrangian.
MC 5403
Anne Johnson, Department of Pure Mathematics, University of Waterloo
"Functor of Points and Examples"
We define and describe the functor of points. We then give examples of reduced schemes over algebraically closed fields from section 2.1 of Eisenbud and Harris.
MC 5417
Ákos Nagy, BEIT Canada
"On the hyperbolic Bloch transform"
Motivated by recent theoretical and experimental developments in the physics of hyperbolic crystals, I will introduce the noncommutative Bloch transform for Fuchsian groups which I will call the hyperbolic Bloch transform (HBT). The HBT transforms wave functions on the hyperbolic plane to sections of irreducible, flat, Hermitian vector bundles over the orbit space and transforms the hyperbolic Laplacian into the covariant Laplacian. I will prove that the HBT is injective and “asymptotically unitary”. If time permits, I will talk about potential applications to hyperbolic band theory. This is a joint work with Steve Rayan (arXiv:2208.02749).
MC 5417
Matthijs Vernooij, TU Delft
"Derivations for symmetric quantum Markov semigroups"
Quantum Markov semigroups describe the time evolution of the operators in a von Neumann algebra corresponding to an open quantum system. Of particular interest are so-called symmetric semigroups. Given a faithful state, one can define the GNS- and KMS-inner product on the von Neumann algebra, and a semigroup is GNS- or KMS-symmetric if it is self-adjoint w.r.t. the inner product. GNS-symmetry implies KMS-symmetry, and both coincide if the state is a trace. It was shown in 2003 that the generator of a tracially symmetric quantum Markov semigroup can be written as the 'square' of a derivation, i.e. d* after d, where d is a derivation to a Hilbert bimodule. This result has proven to be very influential in many different directions. In this talk, we will look at this problem in the case that our state is not tracial. We will start by discussing how a computer can be used to decide whether such a derivation exists in finite dimensions, and work our way up to a general result on KMS-symmetric quantum Markov semigroups. This is joint work with Melchior Wirth.
This seminar will be held both online and in person:
Kieran Mastel, Department of Pure Mathematics, University of Waterloo
"An Aperiodic Monotile"
Last year, David Smith, Joseph Samuel Myers, Craig S. Kaplan, and Chaim Goodman-Strauss found the first example of an aperiodic monotile (or ‘einstein’), solving a longstanding open problem. We will look at the ‘hat’ tile they define and try to visually understand why it tiles the plane and why none of its tilings are periodic.
MC 5501
Peter Oberly, University of Rochester
"Some Bounds on the Arakelov-Zhang Pairing"
The Arakelov-Zhang pairing (also called the dynamical height pairing) is a kind of dynamical distance between two rational maps defined over a number field. This pairing has applications in arithmetic dynamics, especially as a tool to study the preperiodic points common to two rational maps. We will discuss some bounds on the Arakelov-Zhang pairing of f and g in terms of the coefficients of f and investigate some simple consequences of this result.
MC 5417
Joey Lakerdas-Gayle, Department of Pure Mathematics, University of Waterloo
"Computable Structure Theory VII"
We will discuss degree spectra of structures, following Antonio Montalbán's monograph.
MC 5479
Guillermo Gallego, Universidad Complutense de Madrid
"Multiplicative Higgs bundles, monopoles and involutions"
Multiplicative Higgs bundles are a natural analogue of Higgs bundles on Riemann surfaces, where the Higgs field now takes values on the adjoint group bundle, instead of the adjoint Lie algebra bundle. In the work of Charbonneau and Hurtubise, they have been related to singular monopoles over the product of a circle with the Riemann surface.
In this talk we study the natural action of an involution of the group on the moduli space of multiplicative Higgs bundles, also from the point of view of monopoles. This provides a "multiplicative analogue" of the theory of Higgs bundles for real groups.
MC 5403