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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

Alina Stancu, Concordia University

"On the fundamental gap of convex domains in hyperbolic space"

The difference between the first two eigenvalues of the Dirichlet Laplacian on convex domains of R^n and, respectively S^n, satisfies the same strictly positive lower bound depending on the diameter of the domain. In work with collaborators, we have found that the gap of the hyperbolic space on convex sets behaves strikingly different even if a stronger notion of convexity is employed. This is very interesting as many other features of first two eigenvalues behave in the same way on all three spaces of constant sectional curvature. 

MC 5501

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

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

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

Noah Snyder, Indiana University

"Tensor categories, string diagrams, and the Quantum Exceptional Series"

A representation of a group is a vector space on which the group acts linearly, and the collection of all finite dimensional representations of a group forms a structure called a tensor category. Unlike ordinary algebra which is written on a line (you can multiply on the left or on the right), tensor categories are better understood by doing calculations using diagrams in higher dimensions! In particular, "braided" tensor categories have 3-dimensional diagrams which are closely connected to knot polynomials like the Jones Polynomial, the Kauffman Polynomial, and the HOMFLY-PT polynomial. I will explain how the Kauffman polynomial is related to the family of orthogonal groups O(n), and at the end of the talk I will introduce a new conjectural knot polynomial related to the Exceptional Lie groups (from work joint with Thurston and joint in part with Morrison arxiv:2402.03637).

MC 5501

Andrés Chirre, Pontifical Catholic University of Peru

"Remarks on a formula of Ramanujan"

In this talk, we will discuss a well-known formula of Ramanujan and its relationship with the partial sums of the Möbius function. Under some conjectures, we analyze a finer structure of the involved terms. It is a joint work with Steven M. Gonek.

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

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:

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

Timothy Ponepal, Wilfrid Laurier University

"The flow of the horizontal lift of a vector field"

Let $E$ be a vector bundle over a manifold $M$, and let $\nabla$ be a connection on $E$. Given a vector field $X$ on $M$, the connection determines its horizontal lift $X^h$, which is a vector field on the total space of $E$. We will show that the flow of $X^h$ is related to parallel transport with respect to $\nabla$. If time permits, we will show that in the special case when $E$ is a rank 3 oriented real vector bundle with fibre metric, the flow of $X^h$ preserves the cross product on the fibres.

MC 5403