Activities This Week

Hilbert’s 17th problem asked whether a multivariate polynomial, which is positive on all tuples of real numbers, can be written as a sum of squares of rational functions. The positive answer was given by Artin, and the proof techniques presented a cornerstone for real algebra and real algebraic geometry. At the beginning of the millennium, Helton and McCullough solved a free version of H17: if a noncommutative polynomial is positive semidefinite on all tuples of symmetric matrices, then it can be written as a sum of hermitian squares of noncommutative polynomials.

In this talk we shall address the variation of this problem for noncommutative rational functions. By assuming that a rational function is positive semidefinite on all symmetric tuples, one quietly asserts that the function is defined on all symmetric tuples. Such functions are called regular. We will present a characterization of regular noncommutative rational functions in terms of their realizations (from control theory) that can be algorithmically checked. Then we will discuss the proof of the rational version of Helton-McCullough theorem, and its reliance on a ``truncated’’ GNS construction.

מה ניתן לומר על ההתפלגות המשותפת של סידרת ניסויים, אם ידוע שאין חשיבות לסדר של הניסויים?

הקשר בין אי-תלות (במובן ההסתברותי) וחילופיות מובע על ידי משפט של ברונו דה-פינטי — הסתברותן ואקטואר איטלקי שחי ועבד במהלך המאה ה-20.

לעיקרון העומד מאחורי משפט זה יש משמעות רבה בהקשר של הסקה סטטיסטית ולמידה, וגם מגוון מפתיע של שימושים וקשרים בתחומים רחוקים לכאורה של המתמטיקה.

בהרצאה נתאר ונסביר את המשפט של דה-פינטי. ניתן ניסוח הקשור לקמירות, ונתאר מספר מסקנות מעניינות, ככל שהזמן יאפשר.

We study correlation estimates of automatic sequences (that is, sequences computable by finite automata) with polynomial phases. As a consequence, we provide a new class of good weights for classical and polynomial ergodic theorems, not coming themselves from dynamical systems. We show that automatic sequences are good weights in L^2 for polynomial averages and totally ergodic systems. For totally balanced automatic sequences (i.e., sequences converging to zero in mean along arithmetic progressions) the pointwise weighted ergodic theorem in L^1 holds. Moreover, invertible automatic sequences are good weights for the pointwise polynomial ergodic theorem in L^r, r > 1. This talk is based on joint work with Tanja Eisner.

Borel’s Fixed Point Theorem states that a solvable connected algebraic group G acting on a non-empty complete variety V must have a fixed point. Thus, if V consists of subgroups of G, and G acts on V by conjugation, then some subgroup in V is normal in G.

Although G is infinite or trivial here, we can use the method of proof to obtain applications to finite p-groups. We plan to discuss some applications and some open problems. No previous knowledge of algebraic groups is needed.

In this talk, I present an analogue of the Hardy-Littlewood conjecture on the asymptotic distribution of prime constellations in the setting of short intervals in function fields of smooth projective curves over finite fields. I will discuss the definition of a “short interval” on a curve as an additive translation of the space of global sections of a sufficiently positive divisor E by a suitable rational function f, and show how this definition generalizes the definition of a short interval in the polynomial setting. I will give a sketch of the proof which includes a computation of a certain Galois group, and a counting argument, namely, Chebotarev density type theorem.

This is a joint work with Tyler Foster.

De Rham homology and cohomology of algebraic varieties over a field of characteristic 0 were studied by R. Hartshorne in a 1975 paper. In the same paper Hartshorne gave an analogous definition for complete local rings of equicharacterisitc 0 and proved that in this complete local case the properties of de Rham homology and cohomology were similar to the global case. In particular, both in the local and in the global case there exist Hodge-to-deRham spectral sequences for homology and cohomology. In the local case one gets those spectral sequences from surjecting a regular local ring onto the local ring in question (and in the global case by embedding the algebraic variety in question into a regular algebric variety)..

Recently my student Nick Switala proved the following in the complete local case: beginning with the E_2 page the Hodge-to-deRham spectral sequences both for homology and cohomology are finite-dimensional and the isomorphism classes of those spectral sequences depend only the local ring in question, not on the surjection from a regular local ring. I am going to explain Switala’s results in my talk.