MATLAB is generally considered to be the leading software package for scientific computing. In this talk we consider a number of computational examples where MATLAB gives or appears to give wrong answers. These examples are useful to help better understand the inner workings, evolution, limits and tradeoffs of a software package such as MATLAB. The author has collected and developed these examples over the years and has used them in his classes. The examples help students gain greater insight into the fundamentals of matrix computations and also into the basics of finite precision arithmetic and related concepts such as round off error, machine precision, numerical stability, and conditioning.

The Lucas sequences is a sequence of polynomials in $s,t$ defined recursively by $\{0\} = 0, \{1\} = 1,$ and
$\{n\} = s \{n-1\} + t \{n-2\}$ for $n \geq 2$. On specialization of $s$ and $t$ one can recover the Fibonacci numbers, the nonnegative integers, and the $q$-integers $[n]_q$. Given a quantity which is expressed in terms of products and quotients of positive integers, one obtains a Lucas analogue by replacing each factor of $n$ in the expression with $\{n\}$. It is then natural to ask if the resulting rational function is actually a polynomial in $s$ and $t$ and, if so, what it counts. Using lattice paths, we give a combinatorial model for the Lucas analogue of the binomial coefficients. This is joint work with Curtis Bennett, Juan Carrillo, and John Machacek. We then give an algebraic method for proving polynomiality using a connection with cyclotomic polynomials. This part of the talk is joint work with Jordan Tirrell and based on an idea of Richard Stanley. Finally, we also consider Catalan numbers and their relatives, such as those for finite Coxeter groups.

We show how heat kernel bounds can be used to establish lower curvature bounds for the Ricci flow. The method is fairly robust and even can be applied even if initially one only has integral lower curvature bounds.

We use a simple method from harmonic maps theory to investigate singularities
of period mappings from a surface with punctures. More precisely, we derive a harmonic map version
of Schmid’s nilpotent orbit theorem. This is a joint work with J. Jost and K. Zuo.

We consider the problem of minimizing the sum of two convex functions: one is differentiable and relatively smooth with respect to a reference convex function, and the other can be nondifferentiable but simple to optimize. The relatively smooth condition is much weaker than the standard assumption of uniform Lipschitz continuity of the gradients, thus significantly increases the scope of potential applications. We present accelerated Bregman proximal gradient (ABPG) methods that employ the Bregman distance of the reference function as the proximity measure. These methods attain an $O(k^{-\gamma})$ convergence rate in the relatively smooth setting, where $\gamma\in [1, 2]$ is determined by a triangle scaling property of the Bregman distance. We develop adaptive variants of the ABPG method that automatically ensure the best possible rate of convergence and argue that the $O(k^{-2})$ rate is attainable in most cases. We present numerical experiments with three applications: D-optimal experiment design, Poisson linear inverse problem, and relative-entropy nonnegative regression. In all experiments, we obtain numerical certificates showing that these methods do converge with the $O(k^{-2})$ rate. This is joint work with Filip Hanzely and Peter Richtarik.

Nonstandard analysis, a powerful machinery derived from mathematical logic,
has had many applications in probability theory as well as stochastic processes.
Nonstandard analysis allows construction of a single object - a hyperfinite probability
space - which satisfies all the first order logical properties of a finite probability space,
but which can be simultaneously viewed as a measure-theoretical probability space
via the Loeb construction. As a consequence, the hyperfinite/measure duality has
proven to be particularly useful in porting discrete results into their continuous settings.

In this talk, for every general-state-space continuous-time Markov process satisfying appropriate
conditions, we construct a hyperfinite Markov process which has all the basic order logical properties of a finite Markov process to represent it. We show that the mixing time and the hitting time agree with
each other up to some multiplicative constants for discrete-time general-state-space reversible Markov
processes satisfying certain condition. Finally, we show that our result is applicable
to a large class of Gibbs samplers and Metropolis-Hasting algorithms.

In this talk we will discuss a new approach to understanding
eigenfunction concentration. We characterize the features that cause an
eigenfunction to saturate the standard supremum bounds in terms of the
distribution of $L^2$ mass along geodesic tubes emanating from a point. We
also show that the phenomena behind extreme supremum norm growth is
identical to that underlying extreme growth of eigenfunctions when
averaged along submanifolds. Finally, we use these ideas to understand a
variety of measures of concentration including high $L^p$ norms and Weyl
laws; in each case obtaining quantitative improvements over the known
bounds.

Reducing the complexity of system models by averaging fast subsystems has a long history in applied mathematics in general and for stochastic models in particular. The fast components of the model determine an occupation measure, and the averaging argument seeks to replace this occupation measure by a simpler measure. Classically, the simpler measure has been identified as a limit of the occupation measure as some parameter in the model goes to infinity. This averaging argument will be discussed along with a recent approach by Cotter and collaborators that identifies an averaging measure that appears to give a more accurate approximation than the classical limiting argument.

Given an $r$-uniform hypergraph $\mathcal{A} \subset X^{(r)}$, the (lower) shadow of $\mathcal{A}$, denoted $\delta(\mathcal{A})$ is defined as $\delta(\mathcal{A}):= \{ B \in X^{(r-1)} : B \subset A \text{ for some } A \in \mathcal{A} \}$. In this talk, we will explore the classical Kruskal-Katona theorem which gives a lower bound on $|\delta(\mathcal{A})|$ and describe related notions of colex order and compression operators on set families.