Activities This Week

A family of lines through the origin in Euclidean space is called equiangular if any pair of lines defines the same angle. The problem of estimating the maximum cardinality of such a family in $R^n$ was extensively studied for the last 70 years. Answering a question of Lemmens and Seidel from 1973, in this talk we show that for every fixed angle$\theta$ and sufficiently large $n$ there are at most $2n-2$ lines in$R^n$ with common angle $\theta$. Moreover, this is achievable only when $\theta =\arccos \frac{1}{3}$. Various extensions of this result to the more general settings of lines with $k$ fixed angles and of spherical codes will be discussed as well. Joint work with I. Balla, F. Drexler and P. Keevash.

A Gaussian stationary process is a random function f:R–>R or f:C–>C, whose distribution is invariant under real shifts, and whose evaluation at any finite number of points is a centered Gaussian random vector. The mathematical study of these random functions goes back at least 75 years, with pioneering works by Kac, Rice and Wiener. Nonethelss, many basic questions about them, such as the fluctuations of their number of zeroes, or the probability of having no zeroes in a large region, remained unanswered for many years.

In this talk, we will give an introduction to Gaussian stationary processes, and describe how a spectral perspective combined with tools from harmonic, real and complex analysis, yields new results about such long-lasting questions.

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.