__16:00-17:00__

**First Talk:****On uniform Hilbert Schmidt stability of groups**

__Danil Akhtyamoff and Alon Dogon:__**Consider the following question: Given a group G, and a map ϕ from G to some unitary group U(n). Knowing that ϕ is close to being a homomorphism (i.e. unitary representation), can we find a true unitary representation that is close to it? This question depends a lot on how one measures distances of matrices in U(n). D. Kazhdan proved the answer is affirmative when G is amenable and the operator norm is used to measure distances. In this work, we consider the question when using the normalized Hilbert Schmidt norm on U(n). We prove that virtually abelian groups satisfy this, and under the assumption of finite generation and residual finiteness, these are the only examples.**

__Abstract:____17:00-18:00__

**Second Talk:**__The Cauchy-Crofton Formula__

**Yael Hacohen:**__Given a finite-length curve γ in the Euclidean plane, we may count — for every line ℓ in the plane — the number of intersections N__

**Abstract:**_{γ}(ℓ) of the curve and the line; this

number may be viewed as a function on the space of lines. The

Cauch-Crofton Formula states that the integral of N

_{γ}, with

respect to an appropriate measure, is proportional to the curve

length.

Similarly, for a continuous real function f:[a,b]→ℝ, we

may count — for every y∈ ℝ — the number of solutions

N

_{f}(y) of the equation f(x)=y; this number is called the Banach

Indicatrix of f. The Banach Indicatrix Theorem states that the

integral of the Banach Indicatrix over ℝ is equal to

*variation*of f (the length of the 1-dimensional curve

x↦ f(x)).

In the talk I will sketch Banach's original proof of his Indicatrix

Theorem, and show how the Cauchy-Crofton Formula may be deduced from

it. I will also describe some

applications of the Cauchy-Crofton Formula.

No advanced background is assumed: we shall use tools and results seen

in standard undergraduate analysis courses, as well as some measure

theory.

The talk is based on the following sources.

1. Staphan Banach, Sur les lignes rectifiables et les surfaces dont

l'aire est finie, Fundamenta Mathematicae, VII (1925) 225-236.

2. S. Ayari and S. Dubuc, La Formule de Cauchy sur la Longueur d'une

Courbe, Canadian Mathematical Bulletin, 40 (1), 1997, 3-9.

3. Dmitry Fuchs and Sergei Tabachnikov, "The Crofton Formula", in:

Fuchs and Tabachnikov, Mathematical Omnibus: Thirty Lectures on

Classical Mathematics, AMS, 2007

----------------------------------------------------

Amichai Lampert is inviting you to a scheduled Zoom meeting.

Topic: Graduate student seminar

Time: Dec 2, 2020 04:00 PM Jerusalem

Join Zoom Meeting

https://huji.zoom.us/j/87073887657?pwd=NmphaHNEYmhGOEt4WmN2ZDlyYkNjQT09

Meeting ID: 870 7388 7657

Passcode: 326653

## Date:

Wed, 02/12/2020 - 16:00 to 18:00