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Roberto Hernandez Palomares, Department of Pure Mathematics, University of Waterloo

"C* Quantum Dynamics"

A subfactor is a unital inclusion of simple von Neumann algebras, which can be presented as a non-commutative dynamical system governed by a tensor category. Popa established that in ideal scenarios, dynamical data is a strong invariant for hyperfinite subfactors. These reconstruction results in a way give an equivariant version of Connes' classification for amenable factors. On the topological side, after the recent culmination of the classification program for amenable C*-algebras, whether there is an analogue of Popa's Reconstruction results is not clear. In this talk, I will describe the transfer of subfactor techniques to C*-algebras, introducing the largest class of inclusions of C*-algebras admitting a quantum dynamical invariant akin to subfactors. Examples include the cores of Cuntz algebras, certain semicircular systems, and crossed products by actions of tensor categories. Time allowing, we will discuss some interactions with the C* classification program. This is based on joint work with Brent Nelson.

This seminar will be held both online and in person:

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

Gian Cordana Sanjaya, Department of Pure Mathematics, University of Waterloo

"Even More Examples of Schemes"

Last time, we looked at reduced schemes over algebraically closed fields. Now we remove the algebraically closed condition, and look at even more interesting schemes.

MC 5417

Chatchai Noytaptim, Department of Pure Mathematics, University of Waterloo

"Adelic equidistribution theorem for points of small height"

Bilu’s celebrated equidistribution theorem asserts that if there is an infinite sequence of distinct algebraic numbers with low  arithmetic complexity, then its Galois orbit is equidistributed with respect to the uniform probability measure on the complex unit circle. We present the proof of an adelic version of Bilu-type equidistribution theorem in dynamical setting. The material in this presentation covers section 7.9 in Baker-Rumely’s monograph on “Potential Theory and Dynamics on the Berkovich Projective Line”.

MC 5417

Joey Lakerdas-Gayle, Department of Pure Mathematics, University of Waterloo

"Isomorphism Spectra and Computably Composite Structures"

If $\mathcal{A}$ and $\mathcal{B}$ are two computable copies of a structure, their isomorphism spectrum is the set of Turing degrees that compute an isomorphism from $\mathcal{A}$ to $\mathcal{B}$. We introduce a framework for constructing computable structures with the property that the isomorphisms between arbitrary computable copies of these structures are constructed from isomorphisms between computable copies of their component structures. We call these \emph{computably composite structures}. We show that given any uniformly computable collection of isomorphism spectra, there exists a pair of computably composite structures whose isomorphism spectrum is the union of the original isomorphism spectra. We use this to construct examples of isomorphism spectra that are not equal to the upward closure of any finite set of Turing degrees.

MC 5479

Charles Cifarelli, CIRGET & Stony Brook

"Steady gradient Kähler-Ricci solitons and Calabi-Yau metrics on C^n"

I will present recent joint work with V. Apostolov on a new construction of complete steady gradient Kähler-Ricci solitons on C^n, using the theory of hamiltonian 2 forms, introduced by Apostolov-Calderbank-Gauduchon-Tønnesen-Friedman, as an Ansatz. The metrics come in families of two types with distinct geometric behavior, which we call Cao type and Taub-NUT type. In particular, the Cao type and Taub-NUT type families have a volume growth rate of r^n and r^{2n-1}, respectively. Moreover, each Taub-NUT type family contains a codimension 1 subfamily of complete Ricci-flat metrics.

MC 5417

Konstantin Tikhomirov, Carnegie Mellon University

"On the width of random polyhedra"

We consider the problem of estimating the width of a polyhedron defined as the intersection of m i.i.d random affine subspaces of n-dimensional space. Such polyhedra naturally appear in probabilistic analysis of linear programs, as well as in convex geometric analysis as extremizers of various quantities associated with convex sets. For a wide range of parameters m, n, we obtain sharp estimates of the width of the polyhedron in any given direction.

MC 5501

Amir Akbary, University of Lethbridge

"eta-Quotients whose Derivatives are eta-Quotients"

The Dedekind eta function is defined by the infinite product
\[
\eta(z) = e^{\pi i z/12}\prod_{n=1}^\infty (1 - e^{2 \pi i z}) = q^{1/24}\prod_{n=1}^\infty (1 - q^n).
\]
and
\[
f(z) = \prod_{t\mid N} \eta^{r_t}(tz),
\]
where the exponent r_t are integers. Let k be an even positive integer, p be a prime, and m be a nonnegative integer. We find an upper bound for orders of zeros (at cusps) of a linear combination of classical Eisenstein series of weight k and level p^m. As an immediate consequence, we find the set of all eta quotients that are linear combinations of these Eisenstein series and, hence, the set of all eta quotients of level p^m whose derivatives are also eta quotients.

This is joint work with Zafer Selcuk Aygin (Northwestern Polytechnic).

MC 5417

Gian Cordana Sanjaya, Department of Pure Mathematics, University of Waterloo

"More Examples of Schemes"

We describe a number of examples of schemes and how to make sense of them.

MC 5417

Chris Schulz, Department of Pure Mathematics, University of Waterloo

"Automatic structure on Z[F]-modules"

The structure on the integers induced by the base-k representation has been well-studied using finite automata, by Büchi and others. Less well-explored are the extensions of these results to underlying groups other than Z. We will discuss a recent preprint of Francoise Point, in which the author uses the F-sets defined by Moosa and Scanlon in order to generalize Büchi's work. The end result is an expansion of a finitely generated Z[F]-module that has IP but maintains decidability.

MC 5479

Christine Eagles, Department of Pure Mathematics, University of Waterloo

"Splitting the differential logarithm map using Galois theory"

An ordinary algebraic differential equation is said to be internal to the constants if its general solution is obtained as a rational function of finitely many of its solutions and finitely many constant terms. Such equations give rise to algebraic groups behaving as Galois groups. In this talk I give a characterisation of when the pullback of the differential logarithm of an equation is internal to the constants when the Galois group is unipotent or a torus. This is joint work in progress with Leo Jimenez.

MC 5479