Activities This Week

Remarkably, understanding the behaviour of approximate groups allows us to convert information $A^k$ for some single fixed k into information about the sequence $A^n$ as n tends to infinity, and in particular about the growth of A.

The class of NSOP_1 theories was isolated by Džamonja and Shelah in the mid-90s and later investigated by Shelah and Usvyatsov, but the theorems about this class were mainly restricted to its syntactic properties and the model-theoretic general consensus was that the property SOP_1 was more of an unimportant curiosity than a meaningful dividing line. I’ll describe recent work with Itay Kaplan which upends this view, characterizing NSOP_1 theories in terms of an independence relation called Kim-independence, which generalizes non-forking independence in simple theories. I’ll describe the basic theory and describe several examples of non-simple NSOP_1 theories, such as Frobenius fields and vector spaces with a generic bilinear form.

In the finite-dimensional representation theory of the symmetric groups $S_n$ over the base field $\mathbb{C}$, there is an an interesting phenomena of “stabilization” as $n \to \infty$: some representations of $S_n$ appear in sequences $(V_n)_{n \geq 0}$, where each $V_n$ is a finite-dimensional representation of $S_n$, where $V_n$ become “the same” in a certain sense for $n >> 0$.

One manifestation of this phenomena are sequences $(V_n)_{n \geq 0}$ such that the characters of $S_n$ on $V_n$ are “polynomial in $n$”. More precisely, these sequences satisfy the condition: for $n>>0$, the trace (character) of the automorphism $\sigma \in S_n$ of $V_n$ is given by a polynomial in the variables $x_i$, where $x_i(\sigma)$ is the number of cycles of length $i$ in the permutation $\sigma$.

In particular, such sequences $(V_n)_{n \geq 0}$ satisfy the agreeable property that $\dim(V_n)$ is polynomial in $n$.

Such “polynomial sequences” are encountered in many contexts: cohomologies of configuration spaces of $n$ distinct ordered points on a connected oriented manifold, spaces of polynomials on rank varieties of $n \times n$ matrices, and more. These sequences are called $FI$-modules, and have been studied extensively by Church, Ellenberg, Farb and others, yielding many interesting results on polynomiality in $n$ of dimensions of these spaces.

A stronger version of the stability phenomena is described by the following two settings:

• The algebraic representations of the infinite symmetric group $$S_{\infty} = \bigcup_{n} S_n,$$ where each representation of $$S_{\infty}$$ corresponds to a ``polynomial sequence'' $$(V_n)_{n \geq 0}$$.
• The "polynomial" family of Deligne categories $$Rep(S_t), ~t \in \mathbb{C}$$, where the objects of the category $$Rep(S_t)$$ can be thought of as "continuations of sequences $$(V_n)_{n \geq 0}$$" to complex values of $$t=n$$.

I will describe both settings, show that they are connected, and explain some applications in the representation theory of the symmetric groups.

In mathematics you don’t understand things. You just get used to them.
(ג’ון פון נוימן)

מספר אינסופי (נאמר, בן מניה) של אנשים נכנסים לחדר (גדול מספיק להכיל את כולם) וכל אחד נושא על גבו מספר ממשי. המשתתפים יכולים לראות את כל המספרים שנושאים האחרים – אך, כמובן, לא את המספר על גבם. אחד אחד השחקנים מוצאים מן החדר, וכל אחד בתורו מנחש מה המספר הכתוב על גבו. בתכנון מוקדם – ומבלי לתקשר ביניהם מרגע תחילת המשחק – מצליחים כל השחקנים, פרט למספר סופי, לנחש נכון את המספר.

מהי האסטרטגיה, כיצד יתכן שאסטרטגיה כזו בכלל קיימת? האין כאן סתירה לחוקי ההסתברות?

אקסיומת הבחירה היא חרב הפיפיות המפורסמת ביותר במתמטיקה: מעטים העקרונות המתמטים שלהם מגוון כה רחב של שימושים והשלכות עמוקים ומרחקי לכת כמו לאקסיומת הבחירה; אך יש לה גם השלכות כה מפתיעות שעוררו בעבר (ובמידה פחותה גם היום) ויכוחים סוערים בדבר תקפותה וסבירותה.

בהרצאה נציג את אקסיומת הבחירה ובודדים מבין מאות הניסוחים הידועים כיום כשקולים לה. נסקור כמה מן השימושים החשובים של האקסיומה, ונדון ב”פרדוקסים” שהיא מעוררת. אם יתיר הזמן, נראה שגם ללא אקסיומת הבחירה המתמטיקה בהחלט עדיין יכולה להפתיע.

There are two interesting norms on free groups and surface groups which are invariant under the group of all automorphisms:

A) For free groups we have the primitive norm, i.e., |g|_p = the minimal number of primitive elements one has to multiply to get g.

B) For fundamental group of genus g surface we have the simple curves norm, i.e., |g|_s = the minimal number of simple closed curves one need to concatenate to get g.

In our recent paper with M. Brandenbursky we prove the following dichotomy: either |g^n| is bounded or growths linearly with n. For free groups and surface groups we give an explicit characterisation of (un)bounded elements.

In two talks I will explain the idea of the proof and draw a number of consequences. The proof uses the theory of mapping class groups (i.e. Nielsen-Thurston normal form, Birman embedding) and quasimorphisms.