Events

Diophantine Results for Shimura Varieties

Welcoming remarks: Dean Mark Giesbrecht (University of Waterloo)

Laudation: Professor Peter Sarnak (Institute for Advanced Study and Princeton University)

Ostrowski Lecture: Professor Jacob Tsimerman (University of Toronto)

Shimura Varieties are higher dimensional analogues of modular curves, and they play a foundational role in modern number theory. The most familiar Shimura varieties are the moduli spaces of Abelian varieties, and in this context we have a wealth of diophantine results, both in the number field and function field setting: Finiteness of S-rational points, the Tate conjecture, the Shafarevich conjecture, semisimplicity of Galois representations, and others. These results constitute a blueprint for what we expect to be true in other settings but is largely out of reach.

DC1302

Sunil Naik, Queen's University

On a question of Christensen, Gipson and Kulosman

The study of irreducible polynomials in various polynomial rings is an important topic in mathematics. In this context, polynomials with restricted exponents have become the focus of considerable attention in recent years. Motivated by these considerations, Matsuda introduced the ring $F[X;M]$ of polynomials with coefficients in a field $F$ and exponents in a commutative, torsion-free, cancellative (additive) monoid $M$ and began an inquiry into the irreducibility of various polynomials in these rings. For any prime $\ell$, we say that $M$ is a Matsuda monoid of type $\ell$ if for each indivisible $\alpha$ in $M$, the polynomial $X^{\alpha}-1$ is irreducible in $F[X;M]$ for any field $F$ of characteristic $\ell$.

Let $M$ be the additive submonoid of non-negative integers generated by 2 and 3. In a recent work, Christensen, Gipson, and Kulosman proved that $M$ is not a Matsuda monoid of type 2 and type 3 and they have raised the question of whether $M$ is a Matsuda monoid of type $\ell$ for any prime $\ell$. Assuming the Generalized Riemann Hypothesis (GRH), Daileda showed that $M$ is not a Matsuda monoid of any positive type. In this talk, we will discuss an unconditional proof of the above result using its connection with Artin’s primitive root conjecture.

Anne Johnson, University of Waterloo

The Arc Space of the Grassmannian

We give a brief description of the arc space of a scheme and discuss a decomposition of the arc space of the Grassmanian given by Decampo and Nigro in 2016. To do so, we give just enough detail on Schubert calculus as is necessary to make sense of the decomposition. We present some of their related results on plane partitions and irreducibility and then discuss extensions of this work to flag varieties.

MC 5403

Facundo Camano, University of Waterloo

Gromov-Hausdorff Convergence

I will introduce Hausdorff and Gromov-Hausdorff distances on metric spaces. We will look at examples of calculating distances and convergent sequences of metric spaces. We will end off with proving Gromov’s precompactness theorem and a few pathological examples of convergence stemming from the result.

MC 5479

Tobias Shin, University of Chicago

Almost complex manifolds are (almost) complex

What is the difference topologically between an almost complex manifold and a complex manifold? Are there examples of almost complex manifolds in higher dimensions (complex dimension 3 and greater) which admit no integrable complex structure? We will discuss these two questions with the aid of a deep theorem of Demailly and Gaussier, where they construct a universal space that induces almost complex structures for a given dimension. A careful analysis of this space shows the question of integrability of complex structures can be phrased in the framework of Gromov's h-principle. If time permits, we will conclude with some examples of almost complex manifolds that admit a family of Nijenhuis tensors whose sup norms tend to 0, despite having no integrable complex structure (joint with L. Fernandez and S. Wilson).

MC 5417

Rizwanur Khan, University of Texas at Dallas

Eisenstein series and the Random Wave Conjecture

What do automorphic forms "look" like when plotted on the modular surface? Quantum chaos predictions say that they should tend to look more and more like random waves. We'll discuss the relevant conjecture and report on progress for a fundamental type of automorphic form - the Eisenstein series. This is joint work with Goran Djanković.

Jesse Huang, University of Waterloo

Cohen-Macaulay Modules

Cohen-Macaulay modules are central objects of study in commutative algebra, with deep connections to algebraic geometry, singularity theory, and homological algebra. In this talk, we give a brief overview of the connection between Cohen-Macaulay modules and geometric objects, particularly how these modules can be used to study the local behavior of varieties at singular points. Several classical examples, including modules over regular local rings and isolated singularities, will illustrate the practical utility of Cohen-Macaulay theory in understanding algebraic structures. We will also touch on Cohen-Macaulay modules over toric Gorenstein rings and the role of mirror symmetry in the study of these modules.

MC 5403

Chris Schulz, University of Waterloo

Toward a characterization of k-automatic structures

We consider structures over Presburger arithmetic that include k-automatic sets, that is to say, sets recognized by a base-k finite automaton. The question of how many such structures exist up to interdefinability is a complex one, with a deceptively simple conjectured answer. We give a proof of this conjecture in the restricted case of expansion by a single unary set, and we discuss potential strategies for handling the multivariate case. This talk is based on joint work with Jason Bell and Alexi Block Gorman.

MC 5403

Francisco Villacis, University of Waterloo

Convexity of Toric Moment Maps

Toric moment maps are arguably the nicest family of moment maps in symplectic geometry. A classical theorem from the 80s state that the images of these moment maps are convex polytopes, which was proven independently by Atiyah, and Guillemin and Sternberg. In this talk I will go through Atiyah's slick proof of the convexity theorem using Morse theory, and if time permits I will talk about other results in this area.

MC 5479

Yasuyuki Kawahigashi, University of Tokyo

Subfactors, quantum 6j-symbols and alpha-induction

Tensor categories have found many applications in physics and mathematics, particularly quantum field theory and condensed matter physics in recent years, as a new type of symmetry generalizing a classical notion of a group. Operator algebras give useful and efficient tools to study tensor categories. A fusion category, a tensor category with certain finiteness condition, is characterized by a finite set of complex numbers satisfying certain compatibility condition, called quantum 6j-symbols. Its variant, called bi-unitary connections, has played an important role in the Jones theory of subfactors in operator algebras. We have a tensor functor called alpha-induction for a braided fusion category, as a quantum version of a classical machinery of an induced representation for a subgroup. We describe alpha-induction in the framework of quantum 6j-symbols from a viewpoint of being of a canonical form.

MC 4042 or join on Zoom