Activities This Week

The study of Fourier decay for stationary measures is a classical problem, with roots in Erdős’ 1939 work on Bernoulli convolutions. Over the years, tools and ideas from number theory, smooth dynamics, and arithmetic combinatorics have led to substantial progress, yet several fundamental questions remain open. After introducing the problem and its history, I will present recent joint work with Federico Rodriguez Hertz and Zhiren Wang, establishing essentially optimal results for a large class of stationary measures with respect to smooth nonlinear maps.

The mod p local Langlands correspondence is expected to associate, in a functorial manner, a representation \pi(r) of the group GL_n(F) over a field of characteristic p to every n-dimensional mod p representation r of the absolute Galois group of F; here F is a p-adic field. The correspondence is only known for GL_2(Q_p). It is natural to expect it to be realized in the mod p cohomology of Shimura varieties, but any such construction of \pi(r) depends on many global choices and, apart from the case of GL_2(Q_p), is never known to depend only on r. When F is unramified over Q_p and n = 2 and r is generic, it is known that certain invariants of any \pi(r) arising in cohomology depend only on r. Moreover, Breuil has defined a functor from mod p representations of GL_n(F) to representations of the absolute Galois group of Q_p. In the above case, the known invariants of \pi(r) are enough to compute its image under the functor, which turns out to be the tensor induction of r from F to Q_p, at least up to restriction to inertia.

The talk will discuss these ideas and present some new results partially extending them to the case where F is a ramified quadratic extension of Q_p. The arguments are essentially orthogonal to those of the unramified case. A key element of the proof is the determination of (enough of) the submodule structure of mod p principal series representations of GL_2 over some finite quotients of the valuation ring of F. This structure turns out to admit a combinatorial description in terms of the columns where carries are performed when adding certain integers in base p.

The talk discusses joint works with R. Waxman and with S. Morra. Familiarity with addition with carrying will be assumed, but not familiarity with the other notions mentioned above.

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.