Activities This Week

This is about joint work with Shuji Saito. We will begin the talk with a quick introduction to Voevodsky’s theory of motives, and how Kahn-Saito-Yamazaki generalise this theory to allow one to treat cohomology theories that are not necessarily A^1-invariant, and have a notion of ramification. We finish by discussing a blowup formula in Kahn-Saito-Yamazaki’s category of motives with modulus, and how this produces a new proof of this blowup formula for cyclic homology.

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.