# Faculty Activities

*Abstract:*

A number of methods of the algebraic graph theory were influenced by the spectral theory of Riemann surfaces. We pay it back, and take some classical results for graphs to the continuous setting. In particular, I will talk about colorings, average distance and discrete random walks on surfaces. Based on joint works with E. DeCorte and A. Kamber.

*Abstract:*

Abstract: The aim of the talk is to explain the concept of a minimal representative of a dynamical system: A system possessing only periodic orbits that exist in any system in its isotopy class.This concept allows one to use topological methods to study dynamical systems in low dimensions. We'll review the use of minimal representatives in dimensions one and two, and discuss some new ideas that may allow one to apply this concept in dimension three.

*Abstract:*

We introduce an intersection theory problem for maps into a smooth manifold equipped with a stratification. We investigate the problem in the special case when the target is the unitary group and the domain is a circle. The first main result is an index theorem that equates a global intersection index with a finite sum of locally defined intersection indices. The local indices are integers arising from the geometry of the stratification.

The result is used to study a well-known problem in chemical physics, namely, the problem of enumerating the electronic excitations (excitons) of a molecule equipped with scattering data. We provide a lower bound for this number. The bound is shown to be sharp in a limiting case.

*Abstract:*

Let $(M,d)$ be a metric space and let $Y$ be a Banach space. Suppose that for each point $x$ of $M$ we are given a compact convex subset $F(x)$ in $Y$ of dimension at most $m$. A ``Lipschitz selection'' for the family $\{F(x): x\in M\}$ is a Lipschitz map $f$ from $M$ into $Y$ such that $f(x)$ belongs to $F(x)$ for each $x\in M$. The talk explains how one can decide whether a Lipschitz selection exists. We discuss the following ``Finiteness Principle'' for the existence of a Lipschitz selection: Suppose that on every subset $M'$ of $M$ consisting of at most $2^{m+1}$ points, $F$ has a Lipschitz selection with Lipschitz constant at most $1$. Then $F$ has a Lipschitz selection on all of $M$. Furthermore, the Lipschitz constant of this selection is bounded by a certain constant depending only on $m$. The result is joint work with Charles Fefferman.

*Abstract:*

We revisit the old construction of Gromov and Lawson that yields a Riemannian metric of positivescalar curvature on a connected sum of manifolds admitting such metrics. This is joint workwith C. Sormani and J. Basilio. Our refinement is to show that the "tunnel" constructed betweenthe two summands can be made to have arbitrarily small length and volume. We use this tocreate examples of sequences of compact manifolds with positive scalar curvature whose Gromov-Hausdorff limits do not have positive scalar curvature in a certain generalized sense.

*Abstract:*

We propose a high-order compact method for the approximation of the biharmonic and Navier-Stokes equations in planar irregular geometry. This is based on a fourth order Cartesian Embedded scheme for the biharmonic problem, where a bidimensional Lagrange-Hermite polynomial was introduced. A variety of numerical results assure fourth-order convergence rates. In addition, a purely one dimensional procedure was designed for the Navier-Stokes equations. Numerical results demonstrate fourth-order convergence rates. Joint work with M. Ben-Artzi and Jean-Pierre Croisille

*Abstract:*

TBA

*Abstract:*

When the first Betti number $b_{1}(M)$ of a 3-manifold $M$ is greater than one, it follows from Thurston norm theory that if $M$ fibers over the circle, it fibers in infinitely many ways. This talk studies fiberings that are extremal in the sense that the Betti number of the fiber realises the lower bound $b_{1}(M)-1$. It is shown that in hyperbolic manifolds, such fiberings are unique up to isotopy, and can be characterised as having monodromy in a specific normal subgroup of the mapping class group.

***Double feature: please note the special time***

*Abstract:*

The talk is a special Geometry and Topology seminar.

Abstract :

When the first Betti number $b_{1}(M)$ of a 3-manifold $M$ is greater than one, it follows from Thurston norm theory that if $M$ fibers over the circle, it fibers in infinitely many ways. This talk studies fiberings that are extremal in the sense that the Betti number of the fiber realises the lower bound $b_{1}(M)-1$. It is shown that in hyperbolic manifolds, such fiberings are unique up to isotopy, and can be characterised as having monodromy in a specific normal subgroup ofthe mapping class group.

*Abstract:*

We decompose any object in the wrapped Fukaya category of a 2n-dimensional Weinstein manifold as a twisted complex built from the cocores of the n-dimensional handles in a Weinstein handle decomposition. If time permits, we will also discuss how to generalize this result to Weinstein sectors.

This is joint work with Baptiste Chantraine, Georgios Dimitroglou Rizell and Paolo Ghiggini.

*Abstract:*

The deep connection between the Monge optimal transport problem and the foundations of geomtrical optics will be presented. This connection will be applied to classify all the solutions to the phase-from-intensity problem, and even to the construction of an actual phase detector.

Joint work with Gershon Wolansky.

*Announcement:*

**דר' טלי פינסקי**

הפקולטה למתמטיקה

טכניון

**Dr. Tali Pinsky**

The Faculty of Mathematics

Technion

**Math Club 5.12.17**

**אנא שימו לב לשעת ההרצאה הלא שגרתית**

אחרי ההרצאה יתקיים טקס הענקת פרסים של התחרות ע"ש גרוסמן

**P****OSTE****R**

**מחזור שלוש גורר כאוס**

בהרצאה נסתכל על מודל מתמטי להתרבות של אוכלוסיות.

נניח שאם ברגע מסויים מספר החיידקים בצלחת מעבדה הוא x אז מספר החיידקים כעבור דקה הוא f(x). האם אפשר לצפות את מספר החיידקים בצלחת כעבור זמן?

נראה, לפי מאמר של לי ויורק בשם "מחזור שלוש גורר כאוס", שאם מספר החיידקים קופץ בין שלושה ערכים שונים אז התרבות החיידקים היא כאוטית ולא ניתן לצפות אותה.

**Period three implies chaos**

We will consider a mathematical model for population growth.

Suppose that if we have x germs in a sample, the number of germs after one minute is given by f(x).

Is it possible to predict the number of germs as time passes?

Following a paper by Li and York called "period three implies chaos", we will show that if the number of germs oscillates between three different values then the growth is chaotic and the number of germs cannot be predicted

ההרצאה תהיה בעברית

The lecture will be in Hebrew

*Abstract:*

TBA

*Abstract:*

A classical problem in geometry goes as follows. Suppose we are given two sets of $D$ dimensional data, that is, sets of points in $R^D$. The data sets are indexed by the same set, and we know that pairwise distances between corresponding points are equal in the two data sets. In other words, the sets are isometric. Can this correspondence be extended to an isometry of the ambient Euclidean space? In this form the question is not terribly interesting; the answer has long been known to be yes (see [Wells and Williams 1975], for example). But a related question is actually fundamental in data analysis: here the known points are samples from larger, unknown sets -- say, manifolds in $R^D$-- and we seek to know what can be said about the manifolds themselves. A typical example might be a face recognition problem, where all we have is multiple finite images of people's faces from various views. An added complication is that in general we are not given exact distances. We have noise and so we need to demand that instead of the pairwise distances being equal, they should be close in some reasonable metric. Some results on almost isometries in Euclidean spaces can be found in [John 1961; Alestalo et al. 2003]. This talk will consist of two parts. I will discuss various works in progress re this problem with Michael Werman (Hebrew U), Kai Diethelm (Braunschweig) and Charles Fefferman (Princeton). As it turns out the problem relates to the problem of Whitney extensions, interpolation in $R^D$ and bounds for Hilbert transforms. Moreover, for practical algorithms there is a natural deep learning framework as well for both labeled and unlabeled data.

*Abstract:*

Abstract: We show that averages on geometrically finite Fuchsian groups, when embedded via a representation into a space of matrices, have a homogeneous asymptotic limit when properly rescaled. This generalizes some of the results of F. Maucourant to subgroups of infinite co-volume.