Events

Tommaso Pacini, University of Torino

Kahler techniques beyond Kahler geometry: the case of pluripotential theory

Classical pluripotential theory was introduced into complex analysis in the 1940's, as an analogue of the theory of convex functions. In the early 2000's, Harvey and Lawson showed that both pluripotential theory and many of its analytic applications make sense in a much broader setting.

Starting with the work of Calabi in the 1950's, however, it has become clear that pluripotential theory is central also to Kahler geometry. In particular, it is closely related to the cohomology of Kahler manifolds via Hodge theory and the ddbar lemma, and it provides one of the main ingredients in proving the existence of canonical metrics.

Work in progress, joint with A. Raffero, shows how parts of this "second life" of pluripotential theory extend to other geometries, hinting towards new research directions in the field of calibrated geometry and manifolds with special holonomy.

The goal of this talk will be to present a non-technical overview of some of these topics, aimed at non-specialists.

MC 5501

Chi Hoi Yip, Georgia Institute of Technology

Inverse sieve problems

Many problems in number theory boil down to bounding the size of a set contained in a certain set of residue classes mod p for various sets of primes p; and then sieve methods are the primary tools for doing so. Motivated by the inverse Goldbach problem, Green–Harper, Helfgott–Venkatesh, Shao, and Walsh have explored the inverse sieve problem: if we let S \subseteq N be a maximal set of integers in this interval where the residue classes mod p occupied by S have some particular pattern for many primesp, what can one say about the structure of the set S beyond just its size? In this talk, I will give a gentle introduction to inverse sieve problems, and present some progress we made when S mod p has rich additive structure for many primes p. In particular, in this setting, we provide several improvements on the larger sieve bound for |S|, parallel to the work of Green--Harper and Shao for improvements on the large sieve. Joint work with Ernie Croot and Junzhe Mao.

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Rahim Moosa, University of Waterloo

Definable groups in CCM

I will survey what is known about the structure of definable groups in both the standard and nonstandard models of CCM.

MC 5479

Faisal Romshoo, University of Waterloo

Deformations of calibrations

We will look at a criterion for unobstructedness for calibrations and see when the corresponding moduli spacesform smooth manifolds, following the approach by Goto in https://arxiv.org/abs/math/0112197

MC 5403

Evan Sundbo, University of Waterloo

Broken Toric Varieties and Balloon Animal Maps

We will see the definition and some examples of broken toric varieties and balloon animal maps between them. After an overview of some of the many different areas in which they appear, we look at how their geometry can be studied via complexes of sheaves on an associated complex of polytopes. This yields results such as a version of the Decomposition Theorem and some explicit formulas for dimensions of rational cohomology groups of broken toric varieties.

MC 5417

Open Mic

Come listen to or contribute a minitalk (no longer than 15 minutes). Anything (as long as it vaguely relates tomathematics and is reasonably accessible) goes!

MC 5479

(Refreshments will start at 16:30)

Matthew Young, Rutgers University

The shifted convolution problem for Siegel modular forms

The shifted convolution problem for Fourier coefficients of cusp forms has seen a lot of attention due to applications towards moments of L-functions and the subconvexity problem. However, the problem for higher rank automorphic forms (beyond GL_2) has been a notorious bottleneck towards progress on the sixth moment of the Riemann zeta function. In this talk, I will discuss recent progress on the problem for Siegel cusp forms on Sp_4. This is joint work with Wing Hong (Joseph) Leung.

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Damaris Schindler, University of Göttingen

Density of rational points near manifolds

Given a bounded submanifold M in R^n, how many rational points with common bounded denominator are there in a small thickening of M? How does this counting function behave if we let the size of the denominator go to infinity? The study of the density of rational points near manifolds has seen significant progress in the last couple of years. In this talk I will explain why we might be interested in this question, focusing on applications in Diophantine approximation and the (quantitative) arithmetic of projective varieties.

MC 5403

Damaris Schindler, University of Göttingen

Quantitative weak approximation and quantitative strong approximation for certain quadratic forms

In this talk we discuss recent results on optimal quantitative weak approximation for certain projective quadrics over the rational numbers as well as quantitative results on strong approximation for quaternary quadratic forms over the integers. We will also discuss results on the growth of integral points on the three-dimensional punctured affine cone and strong approximation with Brauer-Manin obstruction for this quasi-affine variety. This is joint work with Zhizhong Huang and Alec Shute.

MC 5479