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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

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

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

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

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