Activities This Week

It is well known that a function $K: \Omega \times \Omega \to \mathcal{L}(\mathcal{Y})$ (where $\mathcal{L}(\mathcal{Y}$) is the set of all bounded linear operators on a Hilbert space $\mathcal Y$) being (1) a positive kernel in the sense of Aronszajn (i.e. $\sum_{i,j=1}^N \langle K(\omega_i , \omega_j) y_j, y_i \rangle \geq 0$ for all $\omega_1, \dots, \omega_N \in \Omega$, $y_1, \dots, y_N \in \mathcal Y$, and $N=1,2,\dots$) is equivalent to (2) $K$ being the reproducing kernel for a reproducing kernel Hilbert space $\mathcal H (K)$, and (3) $K$ having a Kolmogorov decomposition $K(\omega, \zeta)=H(\omega)H(\zeta)^*$ for an operator-valued function $H: \Omega \to \mathcal{L}(\mathcal X, \mathcal Y)$ where $\mathcal X$ is an auxiliary Hilbert space.

Last time, I introduced free noncommutative function theory and wrote down the analogue of the result above for noncommutative kernels. In part two, I will give a sketch of our proof and discuss some well-known results (e.g. Stinespring’s dilation theorem for completely positive maps) which follow as corollaries. With any remaining time, I will talk about applications and more recent related results.

נתחיל את ההרצאה בתזכורת על קבוצות קמורות, פונקציות קמורות, ולמה קמירות זה דבר טוב. נדבר טיפונת על אופטימיזציה, על תכנון מוגדר חיובית למחצה, ועל בעיות אופטימיזציה שאינן תלויות מימד — המביאות אותנו באופן טבעי לחקור את הקמירות בהקשר של משתנים שהם מטריצות ולא סקלרים. נבקר כמה תוצאות, חלקן קלאסיות וחלקן עכשוויות, אשר יראו לנו את ההבדלים המהותיים בין קמירות סקלרית לבין קמירות מטריציאלית ואת ההשלכות מרחיקות הלכת של זו האחרונה. אם יוותר קצת זמן, נאמר כמה מלים על התפתחויות אחרות משני העשורים האחרונים בתחום של “מתמטיקה לא מתחלפת חופשית”.

Random walks on lattices have been studied for decades and are by now very well understood. In this talk we will define a random walk on the primitive points of a lattice and discuss its properties. The random walk is obtained in a similar manner to the classical one with the difference that one divides by the gcd at each step. Subject to suitable conditions on the measure generating the walk, we will see how these random walks correspond to positive recurrent Markov chains. In particular we will see that there is a unique stationary distribution for these random walks.

I will describe a link between tame geometry and diophantine geometry that has been unfolding in the past decade following the fundamental theorem of Pila-Wilkie in the theory of o-minimal structures. In particular I will describe how this theorem has been used in proofs of the Manin-Mumford conjecture (by Pila-Zannier), the Andre-Oort conjecture for modular curves (by Pila) and many other questions of “unlikely intersections” in diophantine geometry. I will then discuss questions related to effectivity of the Pila-Wilkie theorem and its implications for the diophantine applications. In particular I will discuss our recent proof (joint with Novikov) of the restricted form of Wilkie’s conjecture, and more recent results on effectivity for the larger class of semi-Noetherian sets.

The Birch and Swinnerton-Dyer conjecture predicts that the group of rational points on an elliptic curve E over Q has rank equal to the order of vanishing of the L-function of E. A generalization of this conjecture to all geometric Galois representations V was formulated by Bloch and Kato. I will explain a proof of a version of the Bloch-Kato conjecture in p-adic coefficients, when V is attached to a p-ordinary Hilbert modular form of any weight and the order of vanishing is 1. The case of elliptic curves corresponds to classical modular forms of weight two, and was treated by Perrin-Riou in 1987 using the modular points on E(Q) constructed by Heegner. The proof in the general case relies on the universal p-adic deformation of Heegner points and a formula for its height.