Activities This Week

An effective approach to the Diophantine problem of enumerating all points on curves with non-abelian fundamental groups, such as those of genus greater than 1, is provided (conjecturally always) by the Chabauty-Kim method. The central object in this method is a Selmer scheme associated to the initial curve of interest and generalizing the association of Selmer groups to elliptic curves. In this talk, we’ll show that arithmetic dualities produce (derived) symplectic and Lagrangian structures on associated spaces which reflect certain expectations coming from “arithmetic topology”. In addition to some Diophantine utility, this should be viewed as foundational towards a “TQFT” approach to L-functions and related invariants analogous to a parallel story producing knot invariants from structures on character varieties which will be elaborated upon.

Self-testing is the strongest form of quantum functionality verification, which allows one to deduce the quantum state and measurements of an entangled system from its classically observed statistics. From a mathematical perspective (which will be the perspective of this talk), self-testing is an intriguing uniqueness phenomenon, pertaining to functional analysis, moment problems, convexity and representation theory. This talk addresses basic motivation and ideas behind self-testing, and discusses which states and measurements can be self-tested. In particular, the talk focuses on how tuples of projections adding to a scalar multiple of identity, and Jordan algebras find its way into this corner of quantum information theory. Based on joint work with Ranyiliu Chen and Laura Mančinska.

To shake things up a little we’ll talk about the Ising model. I will explain a phenomenon in thermodynamics called the “infrared bound”, and what it is usually good for. The only known way to prove this bound on a graph is using a property called “reflection positivity”. But this basically limits the graph in question to Z^d, the Euclidean lattice.

Recently with Tom Meyerovitch we have been thinking of a new method of proving the infrared bound on other (transitive) graphs. I will present a necessary and sufficient condition for something called “Gaussian domination” which in turn implies the infrared bound. The main idea of the talk is to present the different ideas that arise in these kinds of thermodynamic models.

No background is assumed.

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.