Activities This Week

C*-algebras have been intensely studied in recent years, especially through the lens of classification via K-theoretic invariants. Prominent advances include results for Cuntz-Krieger algebras of directed graphs. One such result of Cuntz and Krieger shows that the K-theory groups of such algebras essentially coincide with Bowen-Franks groups of the subshift of finite type associated to the graph.

On the other hand, classifying non-self-adjoint operator algebras is an effort initiated by Arveson in his late 60s paper on algebras arising from one-sided measure preserving dynamics. This was later taken up by Davidson and Katsoulis in the topological scenario, where they classified non-self-adjoint operator algebras arising from multidimensional one-sided dynamical systems on compact Hausdroff spaces.

In this talk we will connect, through examples, these traditionally unrelated classification schemes. We survey some pertinent results from the literature and uncover a striking hierarchy of classification for irreversible and reversible operator algebras.

One of the classical enumerative problems in algebraic geometry is that of counting of complex or real rational curves through a collection of points in a toric variety.

We explain this counting procedure as a construction of certain cycles on moduli of rigid tropical curves. Cycles on these moduli turn out to be closely related to Lie algebras.

In particular, counting of both complex and real curves is related to the quantum torus Lie algebra. More complicated counting invariants (the so-called Gromov-Witten descendants) are similarly related to the super-Lie structure on the quantum torus.

[No preliminary knowledge of tropical geometry or the quantum torus algebra is expected.]

משוואות דיפרנציאליות רגילות ומשוואות דיפרנציאליות חלקיות לרוב נלמדות בגישות שונות זו מזו. בהרצאה אציג קשר מיוחד ביניהן, כך שמשוואה חלקית תיראה כמו מערכת של אינסוף משוואות דיפרנציאליות רגילות. אנו נראה זאת על ידי פתרון אלגנטי של מערכת משוואות מהצורה $Mx''+Kx=0$ עבור $K$ ו-$M$ מטריצות מוגדרות חיובית וסימטריות, ונסביר אילו עקרונות בדרך פתרון זו יכולים לסייע בפתרון משוואת הגלים. בדרך זו, “יצוץ” הצורך להשתמש בפירוק פורייה של פונקציה כדי לפתור משוואות חלקיות.

In my talk, I will describe how simultaneous stable reduction and tropical geometry can be used to construct degenerations of plane curves. This is the main ingredient in a new proof for irreducibility of Severi varieties of the projective plane. The crucial feature of this construction is that it works in positive characteristic, where the other known methods fail. The talk will be a follow up on last week’s talk and is based on joint work with Xiang He and Ilya Tyomkin.

The mixing time is a fundamental quantity measuring the rate of convergence of a Markov chain towards its stationary distribution. We will discuss the problem of estimating the mixing time from one single long trajectory of observations. The reversible setting was addressed using spectral methods by Hsu et al. (2015), who left the general case as an open problem. In the reversible setting, the analysis is greatly facilitated by the fact that the Markov operator is self-adjoint, and Weyl’s inequality allows for dimension-free perturbation analysis of the empirical eigenvalues. In the absence of reversibility, the existing perturbation analysis has a worst-case exponential dependence on the number of states. Furthermore, even if an eigenvalue perturbation analysis with better dependence on the number of states were available, in the non-reversible case the connection between the spectral gap and the mixing time is not nearly as straightforward as in the reversible case. We design a procedure, using spectral methods, that allows us to overcome the loss of self-adjointness and to recover a sample size with a polynomial dependence in some natural complexity parameters of the chain. Additionally, we will present an alternative estimation procedure that moves away from spectral methods entirely and is instead based on a generalized version of Dobrushin’s contraction. Joint work with Aryeh Kontorovich.

Estimating the Mixing Time of Ergodic Markov Chains Geoffrey Wolfer, Aryeh Kontorovich - COLT2019 http://proceedings.mlr.press/v99/wolfer19a.html
https://arxiv.org/abs/1902.01224

Mixing Time Estimation in Ergodic Markov Chains from a Single Trajectory with Contraction Methods Geoffrey Wolfer - ALT2020 https://arxiv.org/abs/1912.06845