Activities This Week
Algebraic Geometry and Number Theory
A blowup formula for motives with modulus
Jan 3, 15:10—16:30, 2018, Math -101
Speaker
Shane Kelly (FU Berlin)
Abstract
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.
Operator Algebras and Operator Theory
Regular and positive noncommutative rational functions
Jan 8, 16:00—17:00, 2018, -101
Speaker
Jurij Volcic (BGU)
Abstract
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.
אשנב למתמטיקה
על אי-תלות וחילופיות: משפט דה-פינטי
Jan 8, 18:30—20:00, 2018, אולם 101-
Speaker
תום מאירוביץ'
Abstract
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.
BGU Probability and Ergodic Theory (PET) seminar
Automatic sequences as good weights for ergodic theorems
Jan 9, 11:00—12:00, 2018, 201
Speaker
Jakub Konieczny (Hebrew University )
Abstract
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.
Colloquium
An analogue of Borel’s Fixed Point Theorem for finite p-groups
Jan 9, 14:30—15:30, 2018, Math -101
Speaker
George Glauberman (University of Chicago)
Abstract
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.