Scientific

We talk about how some simple sounding problems about straight line paths on surfaces require many different kinds of mathematical thinking to solve, focusing on the example of understanding straight line paths on Platonic solids. We'll use this to start a discussion of how we can emphasize teaching different ways of thinking, and why geometry is an important resource for students. There will be lots and lots of fun pictures and hopefully interesting and provocative ideas!

The following resources are referenced during this talk:

• http://userhome.brooklyn.cuny.edu/aulicino/dodecahedron/

The sign test (Arbuthnott, 1710) and the Wilcoxon signed-rank test (Wilcoxon, 1945) are among the first examples of a nonparametric test. These procedures — based on signs, (absolute) ranks and signed-ranks — yield distribution-free tests for symmetry in one-dimension. However, multivariate distribution-free generalizations of these tests are not known in the literature. In this talk we propose a novel framework — based on the theory of optimal transport — which leads to distribution-free generalized multivariate signs, ranks and signed-ranks, and, as a consequence to analogues of the sign and Wilcoxon signed-rank tests that share many of the appealing properties of their one-dimensional counterparts. In particular, the proposed tests are exactly distribution-free in finite samples, with an asymptotic normal distribution, and adapt to various notions of multivariate symmetry such as central symmetry, sign symmetry, and spherical symmetry. We study the consistency of the proposed tests and their behaviors under local alternatives, and show that the proposed generalized Wilcoxon signed-rank test is particularly powerful against location shift alternatives. We show that in a large class of models, our generalized Wilcoxon signed-rank test suffers from no loss in (asymptotic) efficiency, when compared to the Hotelling’s T^2 test, despite being nonparametric and exactly distribution-free. These ideas can also be used to construct distribution-free confidence sets for the location parameter for multivariate distributions.

This is joint work with Zhen Huang at Columbia University.

Let the triple $(a, b, c)$ of integers be such that $\gcd{\left(a, b, c\right)} = 1$ and $a + b = c$. We call such a triple an ABC-solution. The quality of an ABC-solution $(a, b, c)$ is defined as
$$
q(a, b, c) = \frac{\max\left\{\log |a|, \log |b|, \log |c|\right\}}{\log \operatorname{rad}\left({|abc|}\right)}
$$
where $\operatorname{rad} (|abc|)$ is the product of distinct prime factors of $|abc|$. The ABC-conjecture states that given $\epsilon >0$ the number of the ABC-solutions $(a, b, c)$ with $q(a, b, c)\geq 1 + \epsilon$ is finite. In the first part of this talk, under the ABC-conjecture, we explore the quality of certain families of the ABC-solutions formed by terms in Lucas and associated Lucas sequences. We also introduce, unconditionally, a new family of ABC-solutions that has quality $> 1$. In the remainder of the talk, we provide a result on a conjecture of Erdõs on the solutions of the Brocard-Ramanujan equation
$$
n! + 1 = m^2
$$
by assuming an explicit version of the ABC-conjecture proposed by Baker.

The world is going through major changes with the climate crisis and the dual revolutions of artificial intelligence and bio-technology. These changes require us to rethink our systems and build new ones. In this talk, I will discuss why mathematics as the common language for sciences is the most important factor in building this new world and how diverse perspectives can help us solve these problems better.

The classical notion of symmetry can be formalized by actions of groups. Quantum symmetry is a generalization of the notion of symmetry to the quantum setting, where symmetries can no longer be completely described by the actions of groups. In this setting, quantum symmetries are given by Hopf actions of quantum groups on algebras. I will start with background on quantum groups and Hopf actions and then give examples of quantum symmetries of quiver path algebras. Path algebras can be described in terms of directed graphs and play an important role in the representation theory of finite-dimensional algebras. While quantum symmetries are not straightforward to visualize, path algebras give us a nice tool for doing so. Then, I will discuss a tensor categorical perspective for understanding quantum symmetry and how this perspective can be applied to quantum symmetries of path algebras and finite-dimensional algebras.