פעילויות השבוע

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.

If $X$ is a curve of genus at least $2$ defined over the rational numbers, we know by Faltings‘s Theorem that the set $X(Q)$ of rational points is finite but we don‘t know how to systematically compute this set. In 2005, Minhyong Kim proposed a new framework for studying rational (or S-integral) points on curves, called the Chabauty–Kim method. It aims to produce $p$-adic analytic functions on $X(Q_p)$ containing the rational points $X(Q)$ in their zero locus. We apply this method to solve the S-unit equation for S={2,3} and computationally verify Kim‘s Conjecture for many choices of the auxiliary prime $p$.

In the past, the discovery of ultra-rare compute bugs such as incorrect divisions by Pentium chips or cryptographic hash collisions have set headlines and rocked stock markets. All while the Table Maker‘s Dilemma bug, which causes many (but not all) mathematical functions to unexpectedly return slightly incorrect results, remains largely unknown. In my talk I hope to shed some light on this widespread yet poorly understood bug. I will outline a new theoretical framework for modeling its behavior in ”the real world“, and raise some open questions.