Events

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

Akash Sengupta, Department of Pure Mathematics, University of Waterloo

"Approximation of rational points and a characterization of projective space"

Given a real number x, how well can we approximate it using rational numbers? This question has been classically studied by Dirichlet, Liouville, Roth et al, and the approximation exponent of a real number x measures how well we can approximate x. Similarly, given an algebraic variety X over a number field k and a point x in X, we can ask how well can we approximate x using k-rational points? McKinnon and Roth generalized the approximation exponent to this setting and showed that several classical results also generalize to rational points algebraic varieties.

In this talk, we will define a new variant of the approximation constant which also captures the geometric properties of the variety X. We will see that this geometric approximation constant is closely related to the behavior of rational curves on X. In particular, I’ll talk about a result showing that if the approximation constant is larger than the dimension of X, then X must be isomorphic to projective space. This talk is based on joint work with David McKinnon.

MC 5417

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

"Computable Structure Theory VIII"

We will discuss effective embeddings and interpretability, following Antonio Montalbán's monograph.

MC 5479

AJ Fong, Department of Pure Mathematics, University of Waterloo

"Non-reduced schemes"

Last time, we looked at the case where the ground field is not algebraically closed. Now we will drop the hypothesis that the ring of regular functions is not an integral domain and explain what the simplest schemes of this sort look like. We will also introduce the central concepts of limits and flatness and begin to discuss them in detail.

MC 5417

Jiahui Huang, Department of Pure Mathematics, University of Waterloo

"Arc-Floer conjecture"

For a hypersurface singularity, the arc-Floer conjecture states an isomorphism between the compactly supported cohomology of $X_m$, the m-th restricted contact locus (of algebraic nature), and the Floer homology of $\varphi^m$, the m-th iterate of the monodromy on the Milnor fiber (of topological nature). In particular, this gives the Floer homology a mixed Hodge structure.

It was known by a result of Denef and Loeser that the Euler characteristic of $X_m$ agrees with the Lefschetz number of $\varphi^m$, which is given by the Euler characteristic of its Floer homology. The conjecture predicts an equivalence at the level of cohomology. It has been proven for plane curves by de la Bodega and de Lorenzo Poza. We shall look at the case where the singularity is the affine cone of a smooth projective hypersurface.

MC 5417

Christoph Kesting, McMaster University

"The Klein j-Function is not Pfaffian over the Real Exponential Field"

James Freitag showed that the Klein j-function is not pfaffian over the complex numbers. In this talk, I will give a brief introduction to pfaffian functions, their current place in model theory and Freitag's result. Then I will discuss recent work expanding Freitag's result to a restriction of the j-function to the imaginary interval (0, i) not being pfaffian over the real exponential field.

MC 5479

François Greer, Michigan State University

"Finiteness of monodromy for fibered Calabi-Yau threefolds"

An old question going back to S.T. Yau asks whether there are finitely many diffeomorphism types for smooth projective Calabi-Yau manifolds of a given dimension. The answer is affirmative for dimensions one and two (elliptic curves and K3 surfaces). It has recently been settled for Calabi-Yau threefolds admitting elliptic fibrations. We discuss the case of CY3’s admitting abelian surface or K3 fibrations. 

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

Micah Milinovich, University of Mississippi

"Fourier optimization, prime gaps, and the least quadratic non-residue"

There are many situations where one imposes certain conditions on a function and its Fourier transform and then wants to optimize a certain quantity. I will describe two such Fourier optimization frameworks that can be used to study classical problems in number theory: bounding the maximum gap between consecutive primes assuming the Riemann hypothesis and bounding for the size of the least quadratic non-residue modulo a prime assuming the generalized Riemann hypothesis (GRH) for Dirichlet L-functions. The resulting extremal problems can be stated in accessible terms, but finding the exact answer appears to be rather subtle. Instead, we experimentally find upper and lower bounds for our desired quantity that are numerically close. If time allows, I will discuss how a similar Fourier optimization framework can be used to bound the size of the least prime in an arithmetic progression on GRH. This is based upon joint works with E. Carneiro (ICTP), E. Quesada-Herrera (TU Graz), A. Ramos (SISSA), and K. Soundararajan (Stanford). 

MC 5417

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

"Computable Structure Theory IX"

We will discuss effective interpretability of graphs, following Antonio Montalbán's monograph.

MC 5479