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William Dan, University of Waterloo

Random Left C.E. Reals and Solovay Reducibility

In the last seminar we discussed how the halting probability of a universal prefix-free machine is left c.e. andrandom, and asked if the converse would hold. We then studied Solovay reducibility and the resulting concept ofSolovay completeness, which turns out to be key in proving the converse. In this seminar, we will use thisconcept to prove the two theorems giving the converse, a theorem from Calude et al. and the Kucera-Slamantheorem. Then, we will go back to expand further on the properties of Solovay reducibility and how it connectsto relative randomness, and relate this connection back to the theorems we proved. This seminar follows sections9.1 and 9.2 from the Downey and Hirschfeldt book.

MC 5403

Elisabeth Werner, Case Western Reserve University

The $L_p$-Floating Area and Isoperimetric Inequalities on the Sphere

Euclidean convex bodies in spaces of constant positive curvature. We introduce the family of $L_p$-floatingareas for spherical convex bodies, as an analog to $L_p$-affine surface area measures from Euclidean geometry.We investigate a duality formula, monotonicity and isoperimetric inequalities for this new family of curvaturemeasures on spherical convex bodies. Based on joint works with Florian Besau.

MC 5417

Facundo Camano, University of Waterloo

Moduli Space Degeneration via Monopole Deformation

In this talk, I will discuss the theory behind the deformation of monopoles. I will then apply the theory to show monopole moduli spaces degenerate as a singularity is sent off towards infinity.

MC 5403

Rahim Moosa, University of Waterloo

Definable groups in CCM

I will continue the compactification argument for complex manifolds that are compactifiable outside one point.

MC 5479

Mathilde Gerbelli-Gauthier, University of Toronto

Equidistribution of Root Numbers

The root number of an L-function captures important arithmetic information, such as, conjecturally, the parity of the rank in the case of elliptic curves. As such, statistics of root numbers can tell us about the typical behavior of arithmetic objects. In joint work with Rahul Dalal, we prove an equidistribution result for root numbers of self-dual automorphic representations of GL_N as the weight varies. This is done in the framework of endoscopy and the stable trace formula.

MC 5479

Xiao Zhong, University of Waterloo

Bounds on the Greatest Common Divisors and a Dynamical Analogy

In this talk, I will discuss a dynamical analogue of a classical number-theoretic question concerning bounds on the greatest common divisors of two integer sequences. I will present some recent progress on this problem and highlight several open directions for future research. Finally, I will explain how this question relates to the Dynamical Mordell–Lang Conjecture, a central topic in algebraic and arithmetic dynamics.

MC 5501

(with snacks afterward)

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)

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

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

William Dan, University of Waterloo

A Characterization of Random, Left C.E. Reals

An immediate property of the halting probability of a prefix-free machine is that it is a left c.e. real. An easycorollary of the Kraft-Chaitin theorem is that the converse is true: any left c.e. real is the halting probability ofsome prefix-free machine. The most common example of a random real is Chaitin's omega, the haltingprobability of a universal prefix-free machine. In fact, it is a random left c.e. real. It is natural then to ask if theconverse holds in this case as well: that any random left c.e. real is the halting probability of some universalprefix-free machine. As it turns out, this is the case, and in this talk I will explain the concept used to solve thisquestion, Solovay reducibility, then prove the theorems demonstrating the converse. This talk follows sections9.1 and 9.2 of the Downey and Hirschfeldt book.

MC 5403