Aleksandar Milivojevic, Department of Pure Mathematics, University of Waterloo
"Formality and non-zero degree maps"
Natasha Dobrinen, University of Notre Dame
"Cofinal types of ultrafilters on measurable cardinals"
Nicole Kitt, Department of Pure Mathematics, University of Waterloo
"Characterization of Cofree Representations of SL_n\times SL_m"
Ty Ghaswala, Centre for Education in Mathematics and Computing, University of Waterloo
"Kirby diagrams -- because 3-dimensions just isn't enough"
I will introduce Kirby diagrams, which are a way of denoting 4-dimensional handles. The talk will almost entirely consist of staring at pictures of knots and balls and convincing ourselves that these are actually 4-dimensional handlebodies!
MC 5403
Rahim Moosa, Department of Pure Mathematics, University of Waterloo
"NIP"
This learning seminar will be reading through parts of Pierre Simon’s book “A Guide to NIP Theories”. In this talk I will give a non-technical introduction to the subject, explaining some of the history and motivation behind considering these theories.
MC 5403
Kaleb Ruscitti, Department of Pure Mathematics, University of Waterloo
"Intersection Theory III: Finishing the introduction and first examples"
This week, we will finish up the theoretical aspects in Chapter 1: Push-pull formula, Chern classes of line bundles, and the adjunction formula. Then we will spend the remainder of our time going through as many examples from Chapter 2 as time permits: Veronese embeddings, Segre embeddings, graphs of functions, etc.
This seminar will be held both online and in person:
Pierre Popoli, University of Liège
"On the binary digits of $n$ and $n^2$"
Let $s(n)$ denote the sum of digits in the binary expansion of the integer $n.$ Hare, Laishram and Stoll (2011) studied the number of odd integers such that $s(n)=s(n^2)=k,$ for a given positive integer $k.$ The remaining cases that could not be treated by theses authors were $k=9,$ 10, 11, 14 or 15. In this talk, I will present the results of our article on the cases $k=9,$10 and 11 and the difficulties to settle for the two remaining cases $k=14$ and 15.
A related problem is to study perfect squares of odd integers with four binary digits. Bennett, Bugeaud and Mignotte (2012) proved that there are only finitely many solutions and conjectured that the set of solutions is composed of 13, 15, 47 and 111. In the same paper, we give an algorithm to find all solutions with fixed sum of digits value, supporting this conjecture, as well as show related results for perfect squares of odd integers with five binary digits.
This is joint work with Aloui, Jamet, Kaneko, Kopecki and Stoll.
Zoom link: https://uwaterloo.zoom.us/j/94730210787?pwd=NUEwNk8rZlFqUm9vSDZWY1lCL1IyZz09
Chatchai Noytaptim, Department of Pure Mathematics, University of Waterloo
"Equidistribution of small points and GCD of iterated polynomials"
In 2006, Favre-Rivera-Letelier, Baker-Rumely, and Chambert-Loir independently proved equidistribution theorems of points of small height. These equidistribution results have many applications in dynamical unlikely intersection problems. In this (expository) talk, I discuss a result of Hsia and Tucker in which they show that, under nice conditions, there are finitely many irreducible factors of the greatest common divisor of iterates of polynomials. A crucial step in the proof—in the case the degree of both polynomials is greater than 1—is to apply the equidistribution theorems to obtain dynamical relations of two polynomials (namely, Call-Silverman canonical height and Julia set).
MC 4063
Joey Lakerdas-Gayle, Department of Pure Mathematics, University of Waterloo
"Effective Descriptive Set Theory 2"
We will continue to introduce effective descriptive set theory following Andrew Marks's notes.
MC 5479
Amanda Petcu, Department of Pure Mathematics, University of Waterloo
"Partial progress on a conjecture of Donaldson by Fine and Yao (Part 2)"
Given a compact hypersymplectic manifold $X^4$, Donaldson conjectured that the hypersymplectic structure can be deformed through cohomologous hypersymplectic structures to a hyperk\"{a}hler structure. Fine and Yao consider a manifold with closed $G_2$-structure that is set up as $\mathbb{T}^3 \times X^4$. They examine the $G_2$-Laplacian flow under in this setting and give a flow of hypersymplectic structures which evolve according to the equation
\[\partial_t \underline{\omega} = d(Q d^*(Q^{-1} \underline{\omega}))\]
where $\underline{\omega}$ is the triple that gives the hypersymplectic structure and $Q$ is a $3 \times 3$ symmetric matrix that relates the symplectic forms $\omega_i$ to one another. Lotay-Wei have established long time existence of the $G_2$-Laplacian flow provided the velocity of the flow remains bounded. Fine-Yao use this extension theorem in their setup and manage to improve it by proving long time existence of the hypersymplectic flow provided the torsion tensor $T$ remains bounded. Furthermore, one can relate the scalar curvature and torsion tensor of manifold with closed $G_2$-structure and thus they conclude long time existence for the hypersymplectic flow provided the scalar curvature remains bounded. In this talk we will go over some details from this paper by Fine-Yao.
MC 5403