Joey Lakerdas-Gayle, University of Waterloo
Effective Algebra 2
We will continue learning about recursively presented groups.
MC 5417
Francisco Villacis, University of Waterloo
Algebraic Geometry Working Seminar
In this talk, we will explore the "marriage of matroid theory and convex set theory" initiated by I.M. Gelfand and R. MacPherson back in the 80s. In their seminal work, they construct a bijection between projective configurations of n points in P^(k-1) and C*^n-orbits of the Grassmannian of n-k-planes in C^n. This gives a one-to-one correspondence between representable matroids over C and certain convex polyhedra, which in turn gives two equivalent decompositions of the Grassmannian into strata. This decomposition is also equivalent to the decomposition of the Grassmannian into intersections of translates of certain Shubert cells, as shown by Gelfand, Goresky, MacPherson and Serganova a few years later. We will explore these three decompositions and talk about related results.
MC 5403
Eason Li, University of Waterloo
The Hales-Jewett Theorem
We discuss the Hales-Jewett theorem, time permitting giving a full proof.
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
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