פעילויות השבוע

Given a $d$-dimensional ($d < \infty$) operator space $\mathcal{E}$ with basis $\{Q_1, \cdots, Q_d\}$, consider the corresponding noncommutative (nc) operator ball $\mathbb{D}_Q := \{ X \in \mathbb{M}^d : \| \sum_j Q_j \otimes X_j \| < 1 \}$. In this talk, we discuss the problem of extending certain biholomorphic maps between subvarieties $\mathfrak{V}_1$ and $\mathfrak{V}_2$ of nc operator balls $\mathbb{D}_{Q^{(1)}}$ and $\mathbb{D}_{Q^{(2)}}$.

For trivial reasons, such an extension cannot exist in general, and we discuss several examples to showcase the obstructions. When the operator spaces $\mathcal{E}^{(1)}$ and $\mathcal{E}^{(2)}$ are both injective, and the subvarieties $\mathfrak{V}_1$ and $\mathfrak{V}_2$ are both homogeneous, we show that a biholomorphism between $\mathfrak{V}_1$ and $\mathfrak{V}_2$ can be extended to a biholomorphism between $\mathbb{D}_{Q^{(1)}}$ and $\mathbb{D}_{Q^{(2)}}$. Moreover, we show that if such an extension exists then there exists a linear isomorphism between $\mathbb{D}_{Q^{(1)}}$ and $\mathbb{D}_{Q^{(2)}}$ that sends $\mathfrak{V}_1$ to $\mathfrak{V}_2$.

אחת התופעות שאני מאוד אוהב במתמטיקה היא הוכחת משפט בתחום אחד, באמצעות כלים מתחום אחר. בהרצאה נראה איך ניתן להוכיח משפטים בקומבינטוריקה (העונים לשם ”תורת רמזי“) באמצעות דינמיקה טופולוגית. למי שייפתח התיאבון, בסמסטר הבא אעביר קורס (לתארים מתקדמים) שישכלל מאוד את הרעיון הזה.

Topological recursion is a remarkable universal recursive procedure that has been found in many enumerative geometry problems, from combinatorics of maps, to random matrices, Gromov-Witten invariants, Hurwitz numbers, Mirzakhani’s hyperbolic volumes of moduli spaces, knot polynomials. A recursion needs an initial data: a spectral curve, and the recursion defines the sequence of invariants of that spectral curve. There is a duality in topological recursion which allows one to obtain closed formulas for the invariants of the recursion and which has implications in free probability theory and integrable hierarchies. In the talk I will survey recent progress in the topic with the examples from Hurwitz numbers theory, Hodge integrals and combinatorics of maps.

The talk is based on the joint works with A. Alexandrov, P. Dunin-Barkowski, M. Kazarian and S. Shadrin.