Integrated Calendar

<p>Stellar evolution makes us believe that we have over 10 million stellar-mass black holes (BH) in our own Galaxy, whose total mass should far exceed the mass of the central black hole. For half a century we have known that stellar-mass BHs exist, from the few dozen X-ray binaries, where tidally torn material from a very close stellar companions accretes onto the black hole and makes it shine? But is there actually this vast population of dormant BHs, either in&nbsp;wide binaries with a normal star or just free floating? The hunt for these BHs is now on, using&nbsp;ESA’s Gaia mission and other facilities: we have now detected the first dormant BHs in binary&nbsp;systems, after some spectacular earlier misidentification of BH impostors. And, there is&nbsp;first direct evidence for free-floating BHs by means of microlensing. These first discoveries already pose interesting puzzles about how these BH systems could have formed. The next few years offer spectacular prospects of finding far more dormant BHs, whether they are&nbsp;free-floating or in binaries, which should teach us how and when stellar-mass black holes form.</p>

We study opinion dynamics on finite graphs. One of the classical examples is the voter model, in which each vertex is given some initial opinion, and opinions evolve with time by interactions between neighbouring vertices in which one side convinces the other. It is not hard to see that on a finite graph, the system will eventually reach a consensus, which can be seen as the output of this system.

In this talk we will introduce a new opinion dynamics model, which we call the Campaign model. In this model vertices interact as in the voter model but for a limited amount of time T ("the campaign"), and then a majority vote is taken ("elections"). The behaviour of this model depends on some geometric properties of the underlying graph.

We will discuss the behaviour of the Campaign model, and in particular ask at which times T the system loses correlation to the majority of the initial opinions. Next, we will study the dynamical noise sensitivity of the model, and ask when it exhibits a non-trivial threshold when noising the dynamics.

This is a joint work in progress with Gideon Amir, Omer Angel and Omri Marcus.

<p>To reach real sustainability for today’s preferred ‘sustainable’ types of energy, viz. electrical ones as solar (photovoltaic, PV, &amp; wind; thermal solar &amp; hydroelectric, which also store) and chemical ones as reduced CO2 [food], H2 &amp; batteries,<em> also the enabling&nbsp;materials&nbsp;need to be sustainable</em>. Alas, mostly they are not, and that is a problem.&nbsp; As sustainability implies long life spans, it is thought to be incompatible with modern society’s pillars of&nbsp;<em>continuing growth &amp; consumerism. </em>This is a serious issue that, while outside the scope of this lecture<em>,</em> adds to the science &amp; technology challenge to return to repairable&nbsp;devices, and&nbsp;repair-friendly designs, with as best option <em>self-healing</em>,** the most relevant option for micro- and macro-electronic device materials. The sustainable materials challenge is reminiscent of the sustainable energy one, i.e., we need to go from science fiction to reality. Starting&nbsp;with&nbsp;bio-solar conversion, via ionics and organic material self-healing, we get to inorganic light&nbsp;ßà&nbsp;electricity conversion compounds. Emphasis will be on PV materials, as in hindsight those already provide (confirm) some material self-healing criteria. Once (many) more experimental properties data will become available (&amp; accessible), deep learning may guide further discovery.</p>

Let $\mathcal{R} : L^\infty(\mathbb{S}^{n-1}) \rightarrow L^\infty(\mathbb{S}^{n-1})$ denote the spherical Radon transform, defined as $\mathcal{R}(f)(\theta) = \int_{\mathbb{S}^{n-1} \cap \theta^{\perp}} f(u) d\sigma(u)$. A long-standing question in non-linear harmonic analysis due to Lutwak, Gardner, and Fish--Nazarov--Ryabogin--Zvavitch, is to characterize those non-negative $\rho \in L^\infty(\mathbb{S}^{n-1})$ so that $\mathcal{R}(\rho^{n-1}) = c \rho$ when $n\geq 3$. We show that this holds iff $\rho$ is constant, and moreover, $\mathcal{R}(\mathcal{R}(\rho^{n-1})^{n-1}) = c \rho$ iff $\rho$ is either identically zero or is the reciprocal of some Euclidean norm. Our proof recasts the problem in a geometric language using the intersection body operator $I$,  introduced by Lutwak following the work of Busemann, which plays a central role in the dual Brunn-Minkowski theory. We show that for any star-body $K$ in $\mathbb{R}^n$ when $n \geq 3$, $I^2 K = c K$ iff $K$ is a centered ellipsoid, and hence $I K = c K$ iff $K$ is a centered Euclidean ball. To this end, we interpret the iterated intersection body equation as an Euler-Lagrange equation for a certain volume functional under radial perturbations, derive new formulas for the volume of $I K$, and introduce a continuous version of Steiner symmetrization for Lipschitz star-bodies, which (surprisingly) yields a useful radial perturbation exactly when $n\geq 3$.

Joint work with Shahar Shabelman and Amir Yehudayoff.