פעילויות השבוע

The central limit theorem (CLT) and related results for stationary weakly dependent sequences of random variables have been extensively studied in the past century, starting from a pioneering work of Berenstien (1927). However, in many physical phenomena there are external forces, measurement errors and unknown variables (e.g. storms, the observer effect, the uncertainty principle etc.). This means that the local laws of physics depend on time, and it leads us to studying non-stationary sequences.

The asymptotic behaviour of non-stationary sequences have been studied extensively in the past decades, but it is still developing compared with the theory of stationary processes. In this talk we will focus on inhomogeneous Markov chains. For sufficiently well contracting Markov chains the CLT was first proven by Dobrushin (1956). Since then many results were proven for stationary chains. In 2021 Dolgopyat and Sarig proved local central limit theorems (LCLT) for inhomogeneous Markov chains. In 2022 Dolgopyat and H proved optimal CLT rates in Dobrusin‘s CLT. These results closed a big gap in literature concerning the non-stationary case.

An open problem raised by Dolgopyat and Sarig in their 2021 book concerns limit theorems for Markov shifts, that is when the underlying sequence of functions that forms the partial sums depend on the entire path of the chain. Two circumstances where such dependence arises are products of random matrices and random iterated functions, and there are many other instances when the functionals depend on the entire path.

In this talk we will present our solution to the above problem. More precisely, we prove CLT, optimal CLT rates and LCLT for a wide class of sufficiently well mixing Markov chains and functionals with infinite memory. Even though the inhomogeneous case is more complicated, our results seem to be new already for stationary chains.

ב-1911, Brouwer הוכיח (כמסקנה ממשפט נקודת השבת שלו) שמרחבים אוקלידיים מממדים שונים אינם הומאומורפיים. תוצאה זו הובילה למושג של ממד טופולוגי שהוא מספר טבעי שמוגדר לכל מרחב מטרי. נציג גישות שונות להגדרת ממד טופולוגי ותוצאות קלאסיות בתורת הממד.

The conceptual completeness theorem answers a fundamental question: given the category of models of a theory, how much of the theory’s syntax can be recovered? After a gentle introduction to categorical logic and classifying categories, I will explain Makkai’s remarkable result showing that the semantic category, together with ultraproduct-preserving functors, determines the theory up to definitional equivalence.

Given a natural number $n \geq 1$, the odometer semigroup $O_n$, also known as the adding machine or the Baumslag–Solitar monoid with two generators, is a well-known object in group theory. This talk will examine the odometer semigroup in relation to representations of bounded linear operators. We will focus on noncommutative operators and show that contractive representations of $O_n$ always admit nicer representations. A complete description of representations of $O_n$ on the Fock space will be presented, along with connections to odometer lifting and subrepresentations. Along the way, we will also classify Nica–covariant representations of $O_n$.

The Kazhdan–Lusztig isomorphism, relating the affine Hecke algebra of a p-adic group to the equivariant K-theory of the Steinberg variety of its Langlands dual, played a key role in the proof of the Deligne–Langlands conjectures concerning the classification of tamely ramified irreducible representations. In this talk, I will recall the statement of the Kazhdan–Lusztig isomorphism. I will also introduce the relative Langlands duality and propose a conjectural relative version of the Kazhdan–Lusztig isomorphism. I will focus on specific examples which we can prove.

The theory of characters of infinite groups, initiated by Thoma, is a generalization of the representation theory of finite groups. More explicitly, a character of a group is an (extremal) conjugation invariant positive definite function. A group said to be character rigid if every character of the group is either supported on the center or comes from a finite dimensional representation. Connes conjecture that any irreducible lattice in a higher rank Lie group is character rigid. Surprisingly, this conjecture is a generalization of the celebrated Margulis normal subgroup theorem and of the Stuck-Zimmer conjecture on IRS rigidity. I will discuss a recent joint work with Alon Dogon, Yuval Gorfine, Liam Hanany, and Arie Levit showing that any nonuniform higher rank lattice is character rigid, proving the Stuck-Zimmer conjecture for such lattices.