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Sourabhashis Das, University of Waterloo

On the distributions of prime divisor counting function

In 1917, Hardy and Ramanujan established that $\omega(n)$, the number of distinct prime factors of a natural number $n$, and $\Omega(n)$, the total number of prime factors of $n$ have normal order $\log \log n$. In 1940, Erdős and Kac refined this understanding by proving that $\omega(n)$ follows a Gaussian distribution over the natural numbers.

In this talk, we extend these classical results to the subsets of $h$-free and $h$-full numbers. We show that $\omega_1(n)$, the number of distinct prime factors of $n$ with multiplicity exactly $1$, has normal order $\log \log n$ over $h$-free numbers. Similarly, $\omega_h(n)$, the number of distinct prime factors with multiplicity exactly $h$, has normal order $\log \log n$ over $h$-full numbers. However, for $1 < k < h$, we prove that $\omega_k(n)$ does not have a normal order over $h$-free numbers, and for $k > h$, $\omega_k(n)$ does not have a normal order over $h$-full numbers.

Furthermore, we establish that $\omega_1(n)$ satisfies the Erdős-Kac theorem over $h$-free numbers, while $\omega_h(n)$ does so over $h$-full numbers. These results provide a deeper insight into the distribution of prime factors within structured subsets of natural numbers, revealing intriguing asymptotic behavior in these settings.

MC 5417

Micah Milinovich, University of Mississippi

Hilbert spaces and low-lying zeros of L-functions

Given a family of L-functions, there has been a great deal of interest in estimating the proportion of the family that does not vanish at special points on the critical line. Conjecturally, there is a symmetry type associated to each family which governs the distribution of low-lying zeros (zeros near the real axis). Generalizing a problem of Iwaniec, Luo, and Sarnak (2000), we address the problem of estimating the proportion of non-vanishing in a family of L-functions at a low-lying height on the critical line (measured by the analytic conductor). We solve the Fourier optimization problems that arise using the theory of reproducing kernel Hilbert spaces of entire functions (there is one such space associated to each symmetry type), and we can explicitly construct the associated reproducing kernels. If time allows, we will also address the problem of estimating the height of the "lowest" low-lying zero in a family for all symmetry types. These results are based on joint work with Emanuel Carneiro and Andrés Chirre.

MC 5417

Eason Li, University of Waterloo

The Hales-Jewett Theorem

We discuss the Hales-Jewett theorem, time permitting giving a full proof.

MC 5417

Joey Lakerdas-Gayle, University of Waterloo

Effective Algebra 2

We will continue learning about recursively presented groups.

MC 5417

Paul Pollack, University of Georgia

How nonunique is your factorization?

Number theorists learn early on not to take unique factorization for granted. In 1980, R.J. Valenza introduced the "elasticity” of an integral domain as a means of measuring how far away the domain is from possessing unique factorization. I will survey what is known about elasticity of rings of number theoretic interest. Particular attention will be paid to recent work of the speaker on elasticity of orders in quadratic fields. Much of this is joint with Steve Fan (UGA) and Enrique Trevino (Lake Forest).

MC 5417

Jeremy Champagne, University of Waterloo 

On the generalisation of a theorem of Watson

This talk is a continuation of the one I gave in March. In essence, we are discussing the set of real valued functions f(n) such that gcd(n,[f(n)])=1 happens with probability 1/zeta(2) (in the sense of natural densities), and related problems. I will give a general gameplan to establishing such results, and I will prove that gcd(n, [alpha_1n], [alpha_2n^2],...,[alpha_kn^k])=1 happens with probability 1/zeta(k+1) foralpha_1,...,alpha_k irrational.

MC 5403

Larissa Kroell, University of Waterloo

Partial C*-Dynamical Systems: Injective Envelopes and the Ideal Intersection Property

Partial C*-dynamical systems are a generalization of ordinary C*-dynamical systems and were first introduced by Ruy Exel (1994) to express certain C*-algebras as crossed products by a single partial automorphism. In this talk, we will show the existence of an equivariant injective envelope for partial C*-dynamical systems — a concept which has been a driving force in many recent results regarding the ideal structure of ordinary C*-dynamical systems. We will motivate our construction by highlighting its connection to ordinary C*-dynamical systems via enveloping actions. Furthermore, we provide a characterization of the ideal intersection property for partial C*-dynamical systems as an application of equivariant injective envelopes in this setting. This is based on joint work with Matthew Kennedy and Camila Sehnem.

This seminar is held jointly with the Canadian Operator Symposium.

M3 1006

Dashen Yan, Stony Brook University

Non-degenerate Z_2 harmonic 1-forms with shrinking branching sets

In this talk, we will explain the technical aspects of the gluing construction presented in Thursday’s talk. Specifically, we adapt Donaldson’s framework for deforming multivalued harmonic functions to our gluing setting and establish a weighted version of the Hamilton–Nash–Moser–Zehnder implicit function theorem to prove the gluing result.

MC 5417

Dashen Yan, Stony Brook University

Non-degenerate Z_2 harmonic 1-forms on R^n and their geometric applications

The Z_2 harmonic 1-form arises in various compactification problems in gauge theory, including those involving PSL(2,C) connections and Fueter sections. In this talk, we will describe a recent construction of non-degenerate Z_2 harmonic 1-forms on R^n for n >(=) 3 , and explore their relation to Lawlor’s necks—a family of special Lagrangian submanifolds in C^n.

We will also discuss a gluing construction in which these examples are glued to a regular zero of a harmonic 1-form on a compact manifold. This yields a sequence of non-degenerate Z_2 harmonic 1-forms whose branching sets shrink to points. As a result, we obtain many new examples of non-degenerate Z_2 harmonic 1-forms on compact manifolds.

STC 0010

Jashan Bal, University of Waterloo

More on Ramsey degrees

We continue discussing various dynamical reformulations of having finite Ramsey degree.

MC 5417