Activities This Week

n stable homotopy theory, one attempts to understand the category of spectra. This category is morally akin to the derived category of abelian groups. But it lends itself to a form of localization that is not available in classical algebra - chromatic localization. The study of chromatically localized spectra is central to stable homotopy theory. I will explain a novel approach for such study (pioneered by Barthel-Schlank-Stapleton-Weinstein): the use of the isomorphism of the Lubin-Tate and Drinfeld towers of rigid analytic geometry. I shall explain a structural result, accounting for much of the rational behavior of chromatically localized spectra, obtained in collaboration with Shay Ben-Moshe.

Given a commuting $d$-tuple of matrices or operators, we immediately get a homomorphism from the polynomial ring in $d$ variables. An extension of such a homomorphism is called a “functional calculus.” On the other hand, viewing a commutative algebra as functions on its character space is a fruitful approach that goes back at least to Gelfand. However, matrices and polynomials tend not to commute. Hence, the natural object of study in this case is the free algebra $\mathbb{C}\langle z_1,\ldots, z_d \rangle$. By the first analogy, we will call the elements of the free algebra noncommutative polynomials. The second analogy tells us to treat them as functions. The natural analog of the affine space is the collection of all $d$-tuples of matrices of all sizes. We want to understand algebraic relations between noncommutative polynomials through their values. For example, given $f,g \in \mathbb{C}\langle z_1,\ldots, z_d \rangle$, what can we say about $f$ and $g$, if for every $d$-tuples of matrices $X = (X_1,\ldots,X_d)$, $f(X)$ has the same spectrum as $g(X)$. What if $f(X)$ is always similar to $g(X)$? We will answer these questions and others.  

אחד הכלים השימושיים באלגבראות אופרטורים הוא הדילציה. זהו תהליך בו אנו ״מחליפים״ מטריצה או אופרטור במטריצה או אופרטור הפועלים על מרחב גדול יותר עם תכונות טובות יותר. השאלה היא מה זה עוזר לנו? בהרצאה אראה איך אפשר להוכיח משפט קלאסי בכלים של אלגברה לינארית 2 (ועוד קצת), אציג את משפט הדילציה של Sz. Nagy, ואסיק ממנו תוצאה מפתיעה שנקראת אי-שוויון פון נוימן. את שאר הזמן נקדיש לשימושים נוספים ושאלה שפתוחה כבר הרבה מאוד שנים.