Rachael Alvir, University of Waterloo
Effective Algebra 3
We will begin learning about Higman's Theorem.
MC 5417
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
Jack Jia, University of Waterloo
Noncommutative Grothendieck Duality
Grothendieck duality states, up to adding some correct adjectives, that the derived functor of a sufficiently nice morphism of schemes will admit a right adjoint. We will translate this into the language of dualizing complexes and give a noncommutative analog.
MC 5403
Spiro Karigiannis, University of Waterloo
Unique continuation in geometry
I will introduce the notion of unique continuation in geometry, closely following a survey article by Jerry Kazdan (CPAM 1988). Not all elliptic PDE exhibit the phenomenon of unique continuation, but most important elliptic PDE arising in geometry do, such as the Laplace equation, the Cauchy-Riemann equation, and the harmonic map equation.
MC 5403
Isabella Wang, University of Waterloo
The Partite Construction
Using the Hales-Jewett theorem, we use a technique of Nesetril and Rodl to show that the class of finite ordered graphs has the Ramsey property.
MC 5417
Justin Fus, University of Waterloo
The Geometry of the Based Loop Group and Moment Maps
Given a compact Lie group, we will explore a symplectic structure on the infinite-dimensional based loop group consisting of smooth maps from the circle to the Lie group with the identity as a basepoint. The maximal torus of the Lie group and the circle group together generate a Hamiltonian torus action on the loop group. Results on connectedness of level sets and convexity of the moment map, which are attempts to generalize those for finite-dimensional compact symplectic manifolds, will be previewed.
MC 5403
Spiro Karigiannis, University of Waterloo
Unique continuation in geometry (conclusion)
I will finish discussing the paper by Jerry Kazdan on unique continuation in geometry. I will try to make this second talk self-contained, by stating the various estimates which we derived in my first talk, and continuing the proof from there.
MC 5403
Nicole Kitt, University of Waterloo
Characterizing Cofree Representations of SL_n x SL_m
The study, and in particular classification, of cofree representations has been an interest of research for over 70 years. The Chevalley-Shepard Todd Theorem provides a beautiful intrinsic characterization for cofree representations of finite groups. Specifically, this theorem says that a representation V of a finite group G is cofree if and only if G is generated by pseudoreflections. Up until the late 1900s, with the exception of finite groups, all of the existing classifications of cofree representations of a particular group consist of an explicit list, as opposed to an intrinsic group-theoretic characterization. However, in 2019, Edidin, Satriano, and Whitehead formulated a conjecture which intrinsically characterizes stable irreducible cofree representations of connected reductive groups and verified their conjecture for simple Lie groups. The conjecture states that for a stable irreducible representation V of a connected reductive group G, V is cofree if and only if V is pure. In comparison to the classifications comprised of a list of cofree representations, this conjecture can be viewed as an analogue of the Chevalley–Shepard–Todd Theorem for actions of connected reductive groups. The aim of this thesis is to further expand upon the techniques formulated by Edidin, Satriano, and Whitehead as a means to work towards the verification of the conjecture for all connected semisimple Lie groups. The main result of this thesis is the verification of the conjecture for stable irreducible representations V\otimes W of SL_n x SL_m satisfying dim V>=n^2 and dim W>=m^2.
Gian Cordana Sanjaya, University of Waterloo
Density of Special Classes of Polynomials with Squarefree Discriminant
In this talk, we discuss the problem of determining the local densities of monic integer polynomials of fixed degree with squarefree discriminant, with some conditions on the coefficients. Previously, Yamamura solved the case where no extra conditions are imposed, and Bhargava, Shankar, and Wang proved that the global density of general monic integer polynomials (of fixed degree) with squarefree discriminant is equal to the product of the local densities.
Our main case of interest is when the first two non-leading coefficients are fixed. However, we also obtain results on other cases, such as the case where the constant coefficient is fixed. Moreover, the technique used here allows us to compute the local densities exactly.
This talk is based on the recent preprint: https://arxiv.org/abs/2505.06820
MC 5417
Andy Zucker, University of Waterloo
Dynamical Partite Construction
We revisit the partite construction using some of the dynamical ideas we have developed.
MC 5417