Activities This Week

I will survey the geopolitical situation (at least as reflected from some recent ArXiv preprints)

A multivariable dynamical system (MDS) consists of a compact Hausdorff space X together with a finite family of continuous self-maps σᵢ: X → X. To each such system one naturally associates a non-selfadjoint operator algebra known as the tensor algebra, denoted by A(X, σ), which encodes the dynamical information algebraically. In the single-variable case (n = 1), Davidson and Katsoulis established that two tensor algebras are isomorphic if and only if the corresponding dynamical systems are conjugate. For n ≥ 2, however, conjugacy is too strong to capture algebraic isomorphism, leading Davidson and Katsoulis to introduce the weaker notion of piecewise conjugacy, conjecturing it to be the correct criterion for classification.

In this talk, we disprove their conjecture in general. By identifying a previously unnoticed topological obstruction to the existence of certain admissible maps into spaces of unitary matrices, we produce an explicit counterexample consisting of two piecewise conjugate 4-variable systems on a two-dimensional compact space whose tensor algebras are not isomorphic. This result leaves the classification problem wide open and highlights the necessity of more refined invariants for tensor-algebra isomorphism.

We examine the arithmetic properties of eigenvalues of random matrices with integer entries, focusing on the irreducibility of their characteristic polynomials and their Galois groups. Rivin, Jouve-Kowalski-Zywina, and Lubotzky-Rosenzweig previously studied characteristic polynomials arising from random walks on the Cayley graphs of Zariski-dense finitely generated subgroups of linear groups, such as SL_n(Z) . Eberhard, resolving conjectures of Babai and Vu-Wood assuming ERH, analyzed discrete random matrices and established that the characteristic polynomial of a matrix with independent entries (say taking the values 0,1 with equal probabilities) is irreducible and has a large Galois group with high probability as the matrix dimension grows. Ferber, Jain, Sah, and Sawhney proved a counterpart of these results to symmetric matrices.

In this talk, I will present recent results joint with Daniele Garzoni and Sasha Sodin on random tridiagonal matrices where the main diagonal consists of independent Bernoulli entries, the superdiagonal and subdiagonal entries are identically one, and all other entries are zero. We show that the characteristic polynomial of such matrices is irreducible and analyze the structure of its Galois group. If time permits, we will discuss applications to the localization of the eigenstates in Anderson’s 1-dim localization model.

A key feature of our approach lies in combining techniques from both the above random walk framework and the above discrete matrix setting. The latter leverages the Extended Riemann Hypothesis (ERH) to reduce the problem to analyzing the distribution of eigenvalues modulo primes p. To achieve strong error bounds in these computations, we exploit the powerful mixing properties of simple groups such as PSL_2(p) , a central tool in the above-mentioned random walk results.

Chernoff’s inequality is one of the most basic inequalities in probability theory. It states that the sum of independent random variables is close to its mean with very high probability. The `Expander-Graph Chernoff inequality’, proven by Ajtai, Komlos, and Szemeredi in 1987, is a powerful extension of Chernoff’s inequality, asserting that Chernoff’s inequality holds even when sampling the random variables using paths in an expander graph instead of sampling them independently.

In this talk, we will discuss further extensions and generalizations of Chernoff’s inequality coming from high dimensional expanders. High dimensional expanders are sparse hypergraphs exhibiting pseudo-random properties. These are more sophisticated structures than the expander graphs of [AKS], but in return the generalization of Chernoff’s bound obtained from them applies in a far more general setup.

We will see why Chernoff-type bounds hold for high dimensional expanders, and how this inequality arises naturally in problems in high dimensional geometry and computational complexity.

This is based on joint work with Max Hopkins. The talk will not assume prior background in complexity theory and is aimed at a broad audience.

בתיאוריה של קשיחות של גרפים, מסתכלים על מבנים שמורכבים מנקודות וממוטות שמחברים ביניהן, ושואלים מתי מבנה כזה הוא קשיח, כלומר לא יכול להתעוות או ״להתקמט״ בלי לשבור את המוטות. בהרצאה אציג את המושג של גרף קשיח, נדבר על תכונות של גרפים כאלה ונדון בשאלה כיצד ניתן לקבוע אם גרף נתון הוא קשיח או לא.