# Graduate student seminar - Danil Akhtyamoff, Alon Dogon and Yael Hacohen

First Talk: 16:00-17:00

Danil Akhtyamoff and Alon Dogon: On uniform Hilbert Schmidt stability of groups

Abstract: 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.

Second Talk: 17:00-18:00

Yael Hacohen: The Cauchy-Crofton Formula

Abstract:  Given a finite-length curve γ​​ in the Euclidean plane, we may count — for every line ℓ​​ in the plane — the number of intersections Nγ(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 Nγ​​, 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
Nf(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
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Amichai Lampert is inviting you to a scheduled Zoom meeting.