The seminar meets on Tuesdays, 14:30-15:30, in Math -101

2022–23–A meetings

Oct 25 TBA Departamental meeting
Nov 8 Word maps and word measures: probability and geometry Itay Glazer (Northwestern University)

Given a word w in a free group F_r on a set of r elements (e.g. the commutator word w=xyx^(-1)y^(-1)), and a group G, one can associate a word map w:G^r–>G. For g in G, it is natural to ask whether the equation w(x1,…,xr)=g has a solution in G^r, and to estimate the “size” of this solution set, in a suitable sense. When G is finite, or more generally a compact group, this becomes a probabilistic problem of analyzing the distribution of w(x_1,…,x_r), for Haar-random elements x_1,…,x_r in G. When G is an algebraic group, such as SLn(C), it is natural to study the geometry of the fibers of w. Such problems have been extensively studied in the last few decades, in various settings such as finite simple groups, compact p-adic groups, compact Lie groups, simple algebraic groups, and arithmetic groups. Analogous problems have been studied for Lie algebra word maps as well. In this talk, I will mention some of these results, and explain the tight connections between the probabilistic and algebraic approaches.

Based on joint works with Yotam Hendel and Nir Avni.

Nov 15 Definably semisimple groups interpretable in p-adically closed fields (Joint work with Assaf Hasson and Ya’acov Peterzil) Yatir Halevi (Haifa University)

Identifying and characterizing the groups and fields one can define in various first order structures has had multiple applications within model theory and in other branches of mathematics. We focus here on p-adically closed fields. Let K be a p-adically closed field (for example, Q_p). We will discuss some recent results regarding interpretable groups and interpretable fields in K:

1) Let G be an interpretable group. If G is definably semisimple (i.e. G has no definable infinite normal abelian subgroups) group, then there exists a finite normal subgroup H such that G/H is definably isomorphic to a K-linear group.

2) Let F be an interpretable field. Then F is definably isomorphic to a finite extension of K.

No knowledge in model theory will be assumed, but some basic knowledge in logic will help.

Nov 22 On classification of semigroups by algebraic, logical and topological tools Grigory Mashevitzky (BGU)

ההרצאה תתקיים לכבוד פרישתו לגמלאות של פרופ’ גרגורי משביצקי.

Nov 29 Non-Rigidity of Horocycle Orbit Closures in Geometrically Infinite Surfaces Or Landesberg (Yale University)

Horospherical group actions on homogeneous spaces are famously known to be extremely rigid. In finite volume homogeneous spaces, it is a special case of Ratner’s theorems that all horospherical orbit closures are homogeneous. Rigidity further extends in rank-one to infinite volume but geometrically finite spaces. The geometrically infinite setting is far less understood.

We consider $\mathbb{Z}$-covers of compact hyperbolic surfaces and show that they support quite exotic horocycle orbit closures. Surprisingly, the topology of such orbit closures delicately depends on the choice of a hyperbolic metric on the covered compact surface. In particular, our constructions provide the first examples of geometrically infinite spaces where a complete description of horocycle orbit closures is known. Based on an ongoing joint work with James Farre and Yair Minsky.

Dec 6 Stable mappings of manifolds (stable mappings of henselian germs of schemes) Dmitry Kerner (BGU)

Whitney studied the embeddings of (C^\infty) manifolds into R^N. A simple initial idea is: start from a map M-> R^N, and deform it generically. Hopefully one gets an embedding, at least an immersion. This fails totally because of the “stable maps”. They are non-immersions, but are preserved in small deformations. The theory of stable maps was constructed in 50’s-60’s by Thom, Mather and others. The participating groups are infinite-dimensional, and the engine of the theory was vector fields integration. This chained all the results to the real/complex-analytic case. I will discuss the classical case, then report on the new results, extending the theory to the arbitrary field (of any characteristic).

Dec 13 Random Manifolds and Knots Chaim Even Zohar (Technion)

We introduce a combinatorial method of generating random submanifolds of a given manifold in all dimensions and codimensions. The method is based on associating random colors to vertices, as in recent work by Sheffield and Yadin on curves in 3-space. We determine conditions on which submanifolds can arise in which ambient manifolds, and study the properties of random submanifolds that typically arise. In particular, we investigate the knotting of random curves in 3-manifolds, and discuss some other applications.

Joint work with Joel Hass

Dec 20 New insights on the Nevo–Zimmer Theorem Guy Salomon (Weizmann Institute)

Let G be a higher-rank Lie group (for example, SL_n(R) for n>2). Nevo and Zimmer’s structure theorem describes certain nonsingular actions that naturally arise when studying lattices. This theorem is very powerful and manifests rigidity phenomena. For example, it implies the celebrated Margulis Normal Subgroup Theorem, which classifies all normal subgroups of irreducible lattices of G. The original proof of Nevo–Zimmer Theorem heavily uses the structure of Lie groups.

In this talk, I will present a new theorem on general groups that immediately implies the Nevo–Zimmer Theorem (when restricting to the higher-rank Lie case). I will also explain how the generality of our theorem allows us to adapt it to the setup of normal unital completely positive maps on von Neumann algebras.

The talk is based on joint work with Uri Bader.

Dec 27 Some recent developments in mathematical quasicrystals Tobias Hartnick (KIT)

40 years after the discovery of quasicrystals, the mathematical theory around these objects is currently entering into a new phase. While the original goal of mathematical modelling of quasicrystalline materials has largely been achieved, many open questions remain.

One fundamental insight from the early days of quasicrystals is that questions about discrete structures can be attacked by methods from dynamical systems, ergodic theory and harmonic analysis. Over the last decade, attempts were made to use this dynamical approach in a broader context, for example to study quasicrystal-like discrete structures in non-Euclidean geometries. This has created new connections to different areas of mathematics, including rigidity theory of lattices, quasimorphisms, model theory, and non-abelian harmonic analysis, which as a byproduct also provide us with new tools to study classical problems. At the same time, recent progress in the theory of point processes has also lead to the discovery of new phenomena concerning classical quasicrystals, for example some surprising connections to diophantine approximation.

We thus believe that it is a good time to take another look at the classical theory of quasicrystals and see what modern methods have to say about some of the classical problems in the area. We will start from the early beginnings of the theory and then point out a few of the many recent discoveries.

Jan 3 THE AMPLITUHEDRON BCFW TRIANGULATION Tsviqa Lakrec (University of Zurich)

The (tree) amplituhedron was introduced in 2013 by Arkani-Hamed and Trnka in their study of N=4 SYM scattering amplitudes. A central conjecture in the field was to prove that the m=4 amplituhedron is triangulated by the images of certain positroid cells, called the BCFW cells. In this talk I will describe a resolution of this conjecture. The talk is based on joint work with Chaim Even-Zohar and Ran Tessler.

Jan 10 Multi-Bubble Isoperimetric Problems - Old and New Emanuel Milman (Technion)

The classical isoperimetric inequality in Euclidean space $\mathbb{R}^n$ states that among all sets (“bubbles”) of prescribed volume, the Euclidean ball minimizes surface area. One may similarly consider isoperimetric problems for more general metric-measure spaces, such as on the $n$-sphere $\mathbb{S}^n$ and on $n$-dimensional Gaussian space $\mathbb{G}^n$ (i.e. $\mathbb{R}^n$ endowed with the standard Gaussian measure). Furthermore, one may consider the ``multi-bubble” isoperimetric problem, in which one prescribes the volume of $p \geq 2$ bubbles (possibly disconnected) and minimizes their total surface area – as any mutual interface will only be counted once, the bubbles are now incentivized to clump together. The classical case, referred to as the single-bubble isoperimetric problem, corresponds to $p=1$; the case $p=2$ is called the double-bubble problem, and so on.

In 2000, Hutchings, Morgan, Ritor'e and Ros resolved the double-bubble conjecture in Euclidean space $\mathbb{R}^3$ (and this was subsequently resolved in $\mathbb{R}^n$ as well) – the boundary of a minimizing double-bubble is given by three spherical caps meeting at $120^\circ$-degree angles. A more general conjecture of J.~Sullivan from the 1990’s asserts that when $p \leq n+1$, the optimal multi-bubble in $\mathbb{R}^n$ (as well as in $\mathbb{S}^n$) is obtained by taking the Voronoi cells of $p+1$ equidistant points in $\mathbb{S}^{n}$ and applying appropriate stereographic projections to $\mathbb{R}^n$ (and backwards).

In 2018, together with Joe Neeman, we resolved the analogous multi-bubble conjecture for $p \leq n$ bubbles in Gaussian space $\mathbb{G}^n$ – the unique partition which minimizes the total Gaussian surface area is given by the Voronoi cells of (appropriately translated) $p+1$ equidistant points. In the talk, we describe our approach in that work, as well as recent progress on the multi-bubble problem on $\mathbb{R}^n$ and $\mathbb{S}^n$. In particular, we show that minimizing bubbles in $\mathbb{R}^n$ and $\mathbb{S}^n$ are always spherical when $p \leq n$, and we resolve the latter conjectures when in addition $p \leq 5$ (e.g. the triple-bubble conjectures when $n\geq 3$ and the quadruple-bubble conjectures when $n\geq 4$).