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Paul Cusson, University of Waterloo

Holomorphic vector bundles over an elliptic curve

We'll go over the classification of holomorphic vector bundles over an elliptic curve, with a focus on the rank 1 and 2 cases. For the case of line bundles, we'll show that the space of degree 0 line bundles is isomorphic to the elliptic curve itself. The classification of rank 2 bundles rests on the existence of two special indecomposable 2-bundles of degree 0 and 1, which we will describe in detail. The general case for higher ranks would then follow essentially inductively

MC 5479

William Verreault, University of Toronto

On the minimal length of addition chains

An addition chain is a sequence of increasing numbers, starting with 1 and ending with n, such that each number is the sum of two previous ones in the sequence. A challenging problem is, given a positive integer n, to find the minimal length of an addition chain leading to n. I will present bounds on the distribution function of this minimal length, which are sharp up to a small constant. This is joint work with Jean-Marie De Koninck and Nicolas Doyon.

MC 2034

AJ Fong, University of Waterloo

Finite automorphism groups of fans (with some adjectives)

A fan (with some aformentioned adjectives) is a subdivision of n-space into polyhedral cones from the origin subject to some conditions. I will make this precise and describe a classification of the finite automorphism groups when n=2.

MC 5403

Rahim Moosa, University of Waterloo

Zilber dichotomy in DCF_m

We will start reading Omar Leon Sanchez' recent paper by that name.

MC 5403

Facundo Camano, University of Waterloo

Bows to Singular Monopoles

We will discuss an approach to constructing singular monopoles on R^3 by Sergey Cherkis. We begin with the background and the traditional approach to constructing singular monopoles via the Nahm equations. We then talk about bows and their representations, along with the resulting moduli spaces and their self-dual instantons. We apply the bow approach to construct self-dual instantons on the multi-Taub-NUT space and then exploit Kronheimer's correspondence to obtain singular monopoles.

MC 5479

Pavlos Kalantzopoulos, UC Irvine

Analysis Seminar: A multiversion of real and complex hypercontractivity

We establish a multiversion of real and complex Gaussian hypercontractivity. More precisely, our result generalizes Nelson’s hypercontractivity in the real setting and the works of Beckner, Weissler, Janson, and Epperson in the complex setting to several functions. The proof relies on heat semigroup methods, where we construct an interpolation map that connects the inequality at the endpoints. As a consequence, we derive sharp multidimensional versions of the Hausdorff-Young inequality, a Noisy Gaussian-Jensen inequality, and the log-Sobolev inequality. This is joint work with Paata Ivanisvili.

MC 5417 or Join on Zoom

Clement Yung, University of Toronto

Weak A2 spaces, the Kastanas game and strategically Ramsey sets

In the past century, the insight behind the original Ramsey's theorem proved to be applicable to a wide range of mathematics, such as number theory, functional analysis and topology. This spurred two particular directions of Ramsey theory: The first one is known as topological Ramsey theory, a general procedure developed by Todorcevic to prove many seemingly unrelated Ramsey's theorem-like results. The second one is the Ramsey theory of Banach spaces, kickstarted by Gowers' shocking application of Ramsey theory to resolve a long-standing open problem in Banach space theory. In this talk, I introduce the theory of weak A2 spaces, which serves as a possible intersection between these two Ramsey theories and discuss how several infinite games that appeared in these Ramsey theories (the Kastanas game, the Gowers game and the asymptotic game) are closely related.

MC 5479

Elisabeth Werner, Case Western Reserve University

Affine invariants in convex geometry

In analogy to the classical surface area, a notion of affine surface area (invariant under affine transformations) has been defined. The isoperimetric inequality states that the usual surface area is minimized for a ball. Affine isoperimetric inequality states that affine surface area is maximized for ellipsoids. Due to this inequality and its many other remarkable properties, the affine surface area finds applications in many areas of mathematics and applied mathematics. This has led to intense research in recent years and numerous new directions have been developed. We will discuss some of them and we will show how affine surface area is related to a geometric object, that is interesting in its own right, the floating body.

MC 5501

Miao Gu, University of Michigan

On Triple Product L-functions

The Poisson summation conjecture of Braverman-Kazhdan, Lafforgue, Ngo and Sakellaridis is an ambitious proposal to prove analytic properties of quite general Langlands L-functions using vast generalizations of the Poisson summation formula. In this talk, we present the construction of a generalized Whittaker induction such that the associated L-function is the product of the triple product L-function and L-functions whose analytic properties are understood. We then formulate an extension of the Poisson summation conjecture and prove that it implies the expected analytic properties of triple product L-functions. Finally, we use the fiber bundle method to reduce this extended Poisson summation conjecture to a case of the Poisson summation conjecture in which spectral methods can be employed together with certain local compatibility statements. This is joint work with Jayce Getz, Chun-Hsien Hsu, and Spencer Leslie.

MC 5479

Becky Armstrong, Victoria University of Wellington

Analysis Seminar: Twisted groupoids that are not induced by continuous 2-cocycles

Twisted groupoids are generalisations of group extensions that play an important role in C*-algebraic theory: every classifiable C*-algebra has an underlying twisted groupoid model. It is well known that group extensions are in one-to-one correspondence with group 2-cocycles. Analogously, every groupoid 2-cocycle gives rise to a twisted groupoid. However, an example due to Kumjian shows that the converse is not true. Kumjian’s counterexample is a twisted groupoid consisting entirely of isotropy, but in this talk I will present a new example of a twisted groupoid that is not all isotropy, such that the twisted isotropy subgroupoid is not induced by a 2-cocycle. (This is joint work with Abraham C.S. Ng, Aidan Sims, and Yumiao Zhou.)

MC 5417