Activities This Week

In the study of the connection between automorphism groups of models and the models themselves (or their theories, or their bi-interpretability class), the Small Index Property (SIP) has played a central role. The work of Hodges, Lascar, Shelah and Rubin among others has established in many cases when a model of a first order theory T has the Small Index Property.

With Ghadernezhad, we have studied this property for more general homogeneous classes. We have isolated properties of closure notions that allow to prove the SIP for some non-elementary cases, including Zilber’s pseudo-exponentiation and other examples.

I will present a panorama of these results, including our more recent generalizations of the Lascar-Shelah proof of SIP for uncountable structures. This last part is joint work with Zaniar Ghadernezhad.

This is joint work with Michael Magee. Since the 1970’s, Physicists and Mathematicians who study random matrices in the standard models of GUE or GOE, are aware of intriguing connections between integrals of such random matrices and the enumeration of graphs on surfaces. We establish a new aspect of this theory: for random matrices sampled from the group U(n) of Unitary matrices. The group structure of these matrices allows us to go further and find surprising algebraic quantities hidden in the values of these integrals. The talk will be aimed at graduate students, and all notions will be explained.

In this talk I will describe joint work with Chris Phillips and Qingyun Wang. The weak tracial Rokhlin property for actions of discrete amenable groups on simple unital C-algebras is defined by Qingyun Wang [https://arxiv.org/abs/1410.8170]. We show that the class of simple separable unital exact C-algebras with strict comparison and almost divisible Cuntz semigroup is closed under taking crossed products by such actions. We use this to show that the class of simple separable unital nuclear $\mathcal{Z}$-stable C*-algebras is also preserved.

Examples include the non-commutative Bernoulli shift of any discrete amenable group $\Gamma$ on $\bigotimes_{\Gamma} \mathcal{Z} \cong \mathcal{Z}$ and others.

כולנו מכירים (ואם לא ,אז רוב הסיכויים שאנחנו יכולים להגדיר לבד) את המושג חציון של קבוצה סופית של מספרים ממשיים. האם יש הכללה מעניינת של מושג החציון במרחבים אויקלידיים ממימד גבוה יותר מאחד?

כולנו יכולים וודאי להראות שאם באוסף סופי כלשהו של אינטרוולים כל שניים בעלי חיתוך לא ריק אז לכל האוסף חיתוך לא ריק. הנה הכללה של העובדה הזו (המשפט הקלאסי של HELLY מתחילת המאה ה-20):

אם אוסף סופי $F$ של קבוצות קמורות במימד $d$ מקיים שכל ${d+1}$ קבוצות ממנו בעלות חיתוך לא ריק אז לכל האוסף חיתוך לא ריק.

מה הקשר בין המשפט הנ”ל למושג החציון וקרובי משפחתו? על כל זאת ועוד בהרצאה.

One way of viewing coarse geometry is that it is the perfect tool for treating metric spaces like groups and doing representation theoretic/harmonic analytical techniques in a much larger setting. R. Willett and G. Yu define a property (T) for coarse spaces using the uniform Roe algebra. In this talk, I plan to define coarse spaces and show that the uniform Roe algebra is nice tool that acts like the group C* algebra. I will define geometric property (T) and discuss some of its properties.