# Faculty Activities

*Abstract:*

An inhomogeneous Markov chain is a Markov chain {X_i} whose sets of states and transition kernels depend on time. An additive functional is a sum of the form

S_n=f_1(X_1,X_2)+...+f_n(X_n,X_{n+1})

A local limit theorem is an asymptotic statement on the behavior of probabilities of the form

Prob(S_n-z_n\in (a,b))

(for given sequences of constants z_n). Under the assumptions that {X_i} is uniformly elliptic and that f_i are uniformly boundedf, we identify a nearly optimal set of sufficient conditions for a validity of the local central limit theorem, and describe what happens when these conditions fail. (Joint with D. Dolgopyat)

*Abstract:*

An aperiodic tiling of R^d has a hull, which typically is a compact foliated space X. The hull comes equipped with an invariant transverse measure, corresponding to a Z^d-invariant measure on a totally disconnected transversal N. Mathematical physicists attach two types of invariants to this situation. On the one hand, the invariant transverse measure gives rise to a real-valued map on H^d(X; R) (Cech cohomology) which corresponds to the diffraction pattern aspect of tilings. On the other hand. it also gives rise to a real-valued map on K_0(C^*G(X)), the topological K-theory of the C^*-algebra of the holonomy groupoid of the tiling, which corresponds to the spectral (of Hamiltonians defined on such tilings) aspect of tilings. We use the Index Theorem for foliated spaces to prove that under some general assumptions that these points of view are equivalent.

This is joint work with Eric Akkermans (physics, the Technion) and Jonathan Rosenberg (math, U. Maryland).

*Abstract:*

A Banach limit is a classical way in functional analysis to assign "limits" to bounded non-convergent sequences. The method can be extended to assign limits to bounded sequences of convex bodies. The geometric mean of two convex bodies is a new construct studied by myself and V. Milman following a construction of Firey. In this talk we will see the interplay between the two notions. On the one hand, we will see how the geometric mean can be used to create a Banach limit that commutes with duality. On the other hand, we will see how Banach limits can be used to create a new version of the geometric mean with more desirable properties. No previous knowledge of Banach limits or of convex geometry will be assumed.

*Abstract:*

For any element w in the free group on k generators and a group G, there is a map G^k --> G is defined by substitution. These so-called word maps are the analogs of polynomial maps in the category of groups. I will talk about the images of word maps in arithmetic groups, and then segue into the model theory of these groups ending with new rigidity phenomenon for them. The non-survey parts of this work are joint with Alex Lubotzky and Chen Meiri.

*Abstract:*

Dispersion of particles in chaotic, turbulent or random flows has beenstudied for a long time. It is known that the action of advection on largespatial and temporal scales typically can be described as an (anisotropic)normal diffusion process. In random but strongly correlated velocityfields, an anomalous diffusion is possible. Anomalous diffusion ispossible also in spatially regular velocity fields in the presence ofLagrangian chaos.It is less known that an anomalous transport can take place in steadytwo-dimensional flows, in the absence of any kind of chaos. In the presenttalk, we discuss two examples of such behavior.The first example is the deterministic advection in spatially periodic,steady two-dimensional velocity fields, which include stagnation pointsor solid obstacles, so that the passage time is infinite along somestreamlines. The large-time asymptotics of the dispersion law is analyzedusing the special flow construction (a flow built over the mapping).Depending on the type of the passage time singularity, the asymptoticdispersion law can correspond to either subdiffusion or superdiffusion.The analytical predictions match the results of numerical simulations.The second example is the diffusion-advection problem in spatiallyperiodic, steady two-dimensional flows that contain closed cells, possiblyseparated by jets. The anomalous dispersion is predicted and foundnumerically on an intermediate time interval. On the large time scale, anormal diffusion (enhanced by the flow) takes place. The dispersiondisplays peculiar aging properties.

Joint work with M.A. Zaks, P. Poeschke and I.M. Sokolov, Humboldt University of Berlin, Germany

*Announcement:*

**מועדון מתמטי 18.12.19**

**מרצה: ***ד"ר מיכאל חנבסקי, טכניון*

**נושא: ***בעיה איזופרימטרית*** **

**תקציר**: לפי המיתוס, העיר קרתגו נוסדה כשהנסיכה דידו ביקשה לקנות אדמה בגודל "שניתן להקיף בעורו של שור מת". דידו חתכה את העור לרצועות דקות והקיפה עימן גבעה שעליה הקימה את העיר.

הסיפור מעלה בעיה מתמטית - להקיף שטח מרבי בעזרת עקומה באורך נתון - שמוכרת בשם בעיה איזופרימטרית או בעית דידו. הפתרון לבעיה - מעגל - היה ידוע כבר ביוון העתיקה (ולכאורה גם לדידו), אך הוכחה מלאה ראשונה התקבלה רק במאה ה-19. אנחנו נדון בגרסאות שונות של שאלה זו ורעיונות להוכחה.

**אחרי ההרצאה יתקיים טקס הענקת הפרסים של התחרות ע"ש גרוסמן **

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TBA

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In this talk I will discuss obstructions to having a Riemannian metric with non-positive sectional curvature on a locally CAT(0) manifold. I will focus on the obstruction in dimension = 4 given by Davis-Januszkiewicz-Lafont and show how their method can be extended to construct new examples of locally CAT(0) 4-manifolds M that do not have a Riemannian smoothing. The universal covers of these manifolds satisfy the isolated flats condition and contain a collection of 2-dimensional flats with the property that their boundaries at infinity form non-trivial links in the boundary 3-sphere.

*Abstract:*

The nodal set of a nice function defined on a smooth manifold or the Euclidean space is its zero set. The study of nodal sets of Gaussian random fields has positioned itself in the heart of several disciplines, including probability theory and spectral geometry, and, more recently, it has exhibited connections to number theory. We are interested in the asymptotic topology and geometry of the nodal lines in the high energy limit.

In the first part of the talk I will give an overview of the classical results in this field, and the related methods. In the second part of the talk I will describe the more recent progress,related to percolation properties of the nodal lines, borrowing from percolation theory, inspired by the beautiful percolation model due to Bogomolny-Schmit. Finally, I will describethe recent results obtained in a joint work with D. Beliaev and S. Muirhead on the relation between the percolation properties of the nodal sets and their connectivity measures, that were defined and whose existence was established in a joint work with P. Sarnak.

*Abstract:*

**Advisor: **Ron Rosenthal

**Abstract**: A random matrix is said to be sampled from the Ginibre ensemble if all of its entries are i.i.d., complex normal random variables with mean zero. In this work, we study the asymptotic behavior of the angles between pairs of eigenvectors of such matrices. In particular, we compute the limiting distribution of the angle for fixed pairs of eigenvalues, obtain precise bounds on the typical behavior of the maximal angle with high probability and find the limiting distribution for the location of the eigenvalues which attain the maximal angle with high probability. The talk will present the main results and demonstrate some of the techniques used to reach them.

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There are many formulas that express interesting properties of a finite group G in terms of sums over its characters. For estimating these sums, one of the most salient quantities to understand is the **character ratio**:

Trace(ρ(g)) / dim(ρ),

for an irreducible representation ρ of G and an element g of G. For example, Diaconis and Shahshahani stated a formula of the mentioned type for analyzing certain random walks on G.

Recently (https://www.youtube.com/watch?v=EfVCWWWNxvg&feature=youtu.be), we discovered that for classical groups G over finite fields there is a natural invariant of representations that provides strong information on the character ratio. We call this invariant **rank**.

Rank suggests a new organization of representations based on the very few “Small” ones. This stands in contrast to Harish-Chandra’s “philosophy of cusp forms” (P-of-CF), which is (since the 60s) the main organization principle, and is based on the (huge collection) of “Large” representations.

This talk will discuss the notion of rank for the group GL(n) over finite fields, demonstrate how it controls the character ratio, and explain how one can apply the results to verify mixing time and rate for random walks.

This is joint work with Roger Howe (Yale and Texas A&M). The numerics for this work was carried by Steve Goldstein (Madison).

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Let M be an orientable hyperbolic surface without boundary and let c be a closed geodesic in M. We prove that any side of any triangle formed by distinct lifts of c in the hyperbolic plane is shorter than c.

The talk will be presented for advanced undergraduate and beginning graduate students.

*Abstract:*

After summarizing 1D periodic Jacobi matrices, I will define periodic Jacobi matrices on infinite trees. I'll discuss the few known results and some interesting examples and then discuss lots and lots of interesting conjectures. This is joint work mainly with Nir Avni and Jonathan Breuer but also with Jacob Christensen, Gil Kilai and Maxim Zinchenko.

It is on the spectral theory of a class of operators on trees, for which there has been literature on the random case even in the theoretical physics literature but I am not aware of any application to anything close to real physics so this is probably better as a math talk but I leave it to you to sort it out if you are interested. I don’t care at all which it is called or even if it is jointly sponsored.