Mathematics

A vanishing sum of roots of unity (VSRU) is a finite list $z_1,\ldots,z_K$ of $N$-th complex roots of unity whose sum is zero. While there are many simple examples—including the famous "beautiful equation" of Euler, $e^{i \pi} + 1 = 0$—such sums become extremely complex as the parameter $N$ attains more complex prime power divisors (and we will see several classical examples illustrating this idea, as well as new examples from my work).

One fruitful line of inquiry is to seek a quantitative relationship between the prime divisors of $N$, their associated exponents, and the cardinality parameter $K$. A theorem of T.Y. Lam and K.H. Leung from the early '90's states: $K$ must always be (at least) as large as the smallest prime dividing $N$. This generalizes the well known observation that that sum of all $p$-th roots of unity (where $p$ is any prime number) must vanish; and, one notices that Euler's equation is one example of this fact.

In this talk, we will discuss two significant strengthenings of this result (one due to myself and I. Łaba, another due to myself, G. Kiss, I. Łaba and G. Somlai), which are derived from complexity measurements for polynomials with integer coefficients which have many cyclotomic polynomial divisors. As applications, we give connections in two other areas of mathematics. The first is in the study of integer tilings: additive decompositions of the integers $Z = A+B$ as a sum set, where each integer is represented uniquely. The second application is to the Favard length problem in fractal geometry, which asks for bounds upon the average length of the projections of certain dynamically-defined fractals onto lines.

This talk is based upon my individual work, as well as my joint work with I. Łaba, as well as my joint work with G. Kiss, I. Łaba and G. Somlai. All are welcome, and the first 15-20 minutes will include introductory ideas and examples for all results discussed in the latter portion of the talk.

Lagrangian submanifolds are a central object of study in symplectic topology. Their rigidity properties have been uncovered via Floer theory since the early ’90’s. The talk will briefly review the subject, in particular how triangulated category structures naturally arise in this context through work of Donaldson, Kontsevich, Fukaya, and others. Further, will be discussed the more recent, natural role of persistence theory, in the sense common in data science. Finally, we will outline how complexity measurements based on persistence methods reflect topological and dynamical invariants, such as topological entropy.

In this talk, we first introduce the Brumer-Rosen-Zantema exact sequence (BRZ), a four-term sequence related to strongly ambiguous ideal classes in finite Galois extensions of number fields. Then, using BRZ, we obtain some known cohomological results in the literature concerning Hilbert's Theorem 94, the capitulation map, and the Principal Ideal Theorem. This is a joint work with Ali Rajaei (Tarbiat Modares University) and Ehsan Shahoseini (Institute for Research in Fundamental Sciences).

The rainbow and the brain have in common that frequencies are produced. In both cases there is a function of frequency, f, called the power spectral density (PSD). In both cases invasive investigation spoils the investigated object. This talk will describe using noninvasive electroencephalography (EEG) to evaluate the PSD of the brain, via stochastic modelling of associated brain structure. We explore the popular question: does the human brain manifest the mysterious property called "1/f"? Is the PSD of the brain proportional to the function "f to the power -a", for some a > 0, and hence scale-free? What would that mean about the brain? Independent of these fascinating questions, the exponent, a, has many successful applications as a diagnostic of brain disorders and treatments.

The heat kernel is the fundamental solution to a parabolic partial differential equation. From a probabilistic perspective, the heat kernel is the transition probability density of a stochastic process. Harnack inequalities and functional inequalities such as Poincare and Sobolev inequalities provide tools to understand the relationship between the behavior of the heat kernel and the geometry of the underlying space. An important feature of the approach using functional inequalities is its robustness under perturbations.

The study of the heat kernel and its estimates has produced fruitful interactions between the fields of Analysis, Geometry, and Probability. One of the goals of this course is to illustrate these interactions of heat kernel estimates with functional inequalities, boundary trace processes, quasisymmetric maps, circle packings, the time change of Markov processes, Doob's h-transform, and estimates of harmonic measure or exit distribution.

The setting for this course is a symmetric Markov process which is equivalentlydescribed using a Dirichlet form. This course will contain an introduction to the theory of Dirichlet forms. This theory will be used to construct and analyze Markov processes. This course will survey both classical results and recent progress in our understanding of heat kernel estimates and Harnack inequalities.

• Lecture 1
• Lecture 2
• Lecture 3
• Lecture 4
• Lecture 5
• Lecture 6
• Lecture 7
• Lecture 8
• Lecture 9
• Lecture 10
• Lecture 11
• Lecture 12
• Lecture 13
• Lecture 14
• Lecture 15
• Lecture 16