you kind of some numerical simulations of this thresholding scheme so that you get a better intuition of it. So here again is the scheme in its simplest version. So it's a time discretization of mean curvature flow and it proceeds by kind of looking at the characteristic function of the set, the boundary of which evolves by mean curvature and the time step size is h like what I did on the blackboard. So you're given the characteristic function of the set at time step n minus one, you convolve it with the heat kernel at time h which means I could also say you solve the heat equation with these initial data for time h so now you get a smooth function and then you look where this resulting function is less or larger than one half and that defines a new characteristic function or a new set so that's the that's the scheme and here is here is a numerical simulation so oops so here is a dumbbell shaped set at time step n minus one you blur the picture by convolving you find the level set of where the level set of the function of one half and that defines the new set and you can detect that it's slightly behaving as it should be kind of this this thin part has thickened a little bit as you would expect for mean curvature flow and as I mentioned last time that's a low complexity scheme because convolution can be done very efficiently after spatial discretization with almost optimal complexity and with not difficult programming by the fast Fourier transform that's what I mentioned last time that it satisfies the comparison principle and that's the basis for convergent proofs and here is the multi-phase version and and in a certain sense that's what makes this interesting that exactly essentially the same idea the same idea kind of works for multi-phase versions so you're describing a partition of your space by the corresponding characteristic functions that sum up to one you do exactly the same thing as before you do a convolution step where you convolve all the characteristic functions and then you ask where which of the characteristic functions is largest in a given point and and here you see kind of a purely academic numerical simulation where you you have this fairly irregular partition you modify it you ask for which which of the these three functions dominates you get a new arrangement and so on and you can already see that after a single time step you get the right angle condition for free which here in this case is just that all three angles have to be equal and and that makes it so in fact mean curvature was introduced in material science and and this is a kind of the fact that you can use this for many many phases makes it interesting for what's called grain growth in material science so in three-dimensional version of networks and and these three authors have demonstrated that you can really kind of simulate several thousands of grains in in three space dimensions and can do statistics on and but the multi-phase case in a certain sense I mean there's only this recent result by last week's speakers and this week's participant that that at least in the sense of bracket a solution exists okay so this this is was let me briefly recap what we did yesterday so this was I started with telling you about this minimizing movement work of the Georgie which by the right type of interpolation recovers the the exact or the energy the dissipation inequality without loss and and then I formulated two results the first one definitely I want to give you the argument for that this minimizing movement scheme is indeed sorry that the thresholding scheme is indeed a minimizing movement scheme and that this minimizing movement scheme has to do with mean curvature flow in the sense in the one sense I mean before you before proving a convergence result in the sense that the energy functional converges to the perimeter so the area of the interface written in this bv type fashion so those were the two results which I already stated last time so let me give you the argument for at least the first one so so let's look at this functional one has to minimize and so that's the energy e h plus one over two h square d h square u chi n minus one and let's plug in the definitions so that's one over square root of h the integral of u one minus u times g h convolved with u plus one over square root of u u minus chi g h u minus chi so that's I just copied what's what's on the blackboard and now you have to now there is a simple algebraic transformation which you see best by realizing that this expression here defines a symmetric bilinear form so let me just use this notation so this is one minus u comma u plus u minus chi comma u minus chi and if you use kind of elementary linear algebra you see that this can be rewritten as minus u two chi minus one plus chi chi I hope that's correct so this term is sitting here the quadratic term in u cancels with the quadratic term in u here the quadratic term in chi is sitting here and the mixed term between u and chi where the minus sign is sitting here so that's just linear algebra so what does that mean that means we have minus the integral of u g h convolved with two chi minus one plus chi g h convolved with chi this second term is completely irrelevant for this variation problem since it does not depend on you only the first term is relevant for the variation problem in fact it's a you know linear term and using the fact that the Gaussian has integral one I can rewrite this as the convolution of two times g h minus one if if we do this we see two things we see that so we want to minimize this expression so we want to minimize this expression which means we want to maximize this expression which means we can do that if we choose u to be equal to one where this is positive and equal to zero where this is negative so in this case we get that this is equal to minus two times g h chi minus one if u is equal to the characteristic function of the set where the convolution is larger than one-half but but that's really the smallest value this expression can have for any u remember that u is a function which assumes the values between zero and one that's really kind of the the the smallest value it can ever assume so it's this is equal to this value but it's always at least as large as this value and therefore we see that this function here indeed is the minimizer and that was just thresholding so so by very simple calculation you see that that thresholding indeed is is has the interpretation of solving this this almost trivial variational problem so so that's that's certainly very very easy argument and now kind of the work consists in in using this structure and the first the first part of the work kind of consists in understanding the energy functional and that's proposition one which tells you why the energy functional indeed has to do something with with what you would expect for mean curvature flow which is the gradient flow of the interfacial energy because this energy functional as this parameter h converges to zero converges to the parameter function and and this this lemma essentially sorry this proposition essentially relies on let's say for the most interesting part relies on the following lemma so that should be lemma three which i want to state now so for any characteristic function we have the following three statements we have that this functional this this energy functional which is defined up there indeed is always bounded by a buff by the parameter functional and we have monotonicity in the sense that as this parameter so this holds in a point wise point wise in the configuration for any natural number n so so if this of course is not really full monotonicity in this parameter n here because it's some kind of integer monotonicity but in a certain sense what these two what these two inequalities tell you is that for fixed configuration these energy functional monotonically converge to the parameter functional and and that's as people who are a little bit familiar with gamma convergence will know is already kind of a main step for the for the gamma convergence essentially that's that's essentially all all it relies I mean one needs one needs the consistency one also needs to know that for fixed chi e of h chi actually converges to this value as h goes to zero that requires a kind of a little bit of the approximation is a classical argument but in fact these two inequalities are the most interesting part and then there is a third property which I'm going to use later which I want to state here that again for an arbitrary configuration if you look at the l1 norm of the gradient of the convolution it's estimated up to some constant by so the energy even for finite h is at least strong enough to control the bv norm of the convolved configuration so those are all three statements for fixed configuration let me let me let me show you the proof at least of of the first two ones and then we'll see how much you want to see so so let's start with with the first the first inequality so I'm going to write down the definition of the energy functional and I'm using something which I already used over there that this is kind of symmetric in these two arguments so I can also write it like this and now I'm using for beanie in the sense that I this is an integral in x and if I'm not saying anything it's always in x when it's integrated in x it's always over the unit cell but now there is a second integral in here which is which comes from the convolution and I'm exchanging the orders of integration and pull the convolution integral outside so therefore I get the integral over the entire rd so integrals over z are always to be supposed implicitly supposed to be over the entire rd and now here we have the unit cell and we get 1 minus chi times x chi times x minus z plus chi times x 1 minus chi x minus z dx dz and I'm just I've just rewritten the the functional and now you realize that this expression because chi is a characteristic function this expression is nothing else than the the difference the modulus of the difference of chi at x and chi at x minus z and now because of periodicity so here you're looking you're looking in at an l1 modulus of continuity of your characteristic function chi you're looking at the l1 difference between chi and shifted chi and therefore by if you want the mean value property you can estimate that by the l1 norm of the gradient and if you think a little bit about it what you get is as an intake as an integrand you get the direction in which you shift times the normal vector so nu is like always in the bv context the measure theoretic normal and so that's the estimate which you get and now you I mean here you here we used for beanie and kind of exchanging and pulling the x integration outside now you pull the z integration outside like it was originally sorry the so we pulled the z integration outside now we pulled in and you see that you get so I should write it like this we get this expression and so now we have to look at we have to look at this at this integral here so first of all by rotational symmetry of the Gaussian you realize that it does not depend on in which on which direction this unit vector points so could just take the without loss of generality I could take the first the first direction so that's using the rotational symmetry now I'm changing variables recalling that the Gaussian of variance h was nothing else than the Gaussian of variance 1 rescaled by square root of h so if I change variables and I bring this in here you realize that this is just the integral of the unit Gaussian times the modulus of the first component and now you use the fact that the Gaussian in fact is something which is tensorial so it's the one-dimensional Gaussian in the z1 variable times the d minus one-dimensional Gaussian in the other variables and you use for beanie here you write that as an integral in z1 integral and d z prime and the integral in in z prime that is equal to one because of the normalization of the Gaussian so this in fact is let me continue here is just a one-dimensional integral of the 1d Gaussian of unit variance z1 d z1 and I forgot somewhere the factor of one-half which now is which came in here which now is important because I can write this by symmetry of the Gaussian as the integral from 0 to infinity of this expression so the 1d Gaussian is equal to 1 over square root of 2 pi exponential minus z1 square over 2 and if you multiplied with z1 I'm sure you have often seen or you have seen this calculation this is nothing else than the derivative of the 1d Gaussian so therefore up to a minus sign so therefore this integral is let me continue here this integral is is easy it's just the 1d Gaussian evaluated at the origin and now you look once more at the formula and you see that this is equal to 1 over 2 pi square root which we called c0 so so that shows that in a certain sense explains why in this gamma convergence result this constant c0 which is 1 over square root 1 over the square root of 2 pi comes up it's already present in the simple in equal sharp inequality which holds for any characteristic configuration okay let me also because that's that's perhaps more interesting let me also show you this monotonicity property here which uses essentially a similar trick so now I'm sorry so so let's let's think of chi as being chi as being smooth the so we're looking at this integral here now it's over the period cell but it's easier to think about the whole space and so you write this here as the integral from 0 to 1 gradient chi x minus tz dot z dt and now you use jensen in the dt integral and you get that this is equal to the integral gradient chi x minus tz dot z dx dt now that by translation variance of the lebesgue integral I can get rid of this translation and and this here I can write as the integral of z chi over modulus of chi times grad chi and that's the that's exactly the formula and now now you do an approximation argument so chi in fact is a characteristic function of a cacciopoly of b so you use the fact that that you can approximate bv functions by smooth functions and that in this case this this classical expression turns into the measure theoretic normal okay so let's look at this monotonicity so let me start with the left hand side and let me use this representation which we already had here and so this would be one over n square h square root the integral g n h square of z the integral of chi of x minus z minus chi of x dx dz let's do another change of variables which you already did over there to get the unit Gaussian here so this gives one over n one over square root of h g1 z so I should call it z hat but x minus n times square root of h z minus chi of x dx dz so so far this is an exact identity and now it's clear what you're doing you're just using the triangle inequality in l1 to write this here as the sum going from n middle n from 1 to infinity chi x minus n square root of h z minus chi x minus n minus 1 square root of h times z dx and you use translation in variants of the integral on the torus to see that all these terms are actually the same term and so with the sum it's just n times that term and now you see that these two n's cancel and indeed you get that this is by by the same formula that this is less than the energy of chi so so so this this monotonicity is is a very robust property and in fact it also holds nicely for the multi-phase case you just need a triangle inequality for your different interfacial surfaces interface as you would expect so so as I said everything here generate I mean I'm always presenting the results and the proofs in the case of just two phases or a single boundary but things generate to generalize to the general general case okay so now I guess I'm going to ask you what what other proofs you want to see before starting on the proof so there's a second lemma which I need in the order to prove a compactness result which is what it finishes the first chapter of preliminaries so the so I'm giving you lemma 4 in second so the compactness statement reads so let chi zero the initial data be such that the parameter energy is finite and let chi n n and n the solution well in fact I could even write the solution of the thresholding scheme with initial data chi zero and time step size h consider the piecewise linear interpolation so we're not yet doing the george's interpolation so piecewise linear interpolation means that chi superscript h at some time t this characteristic function is equal to chi n the nth step for t between n h and n plus one h so the if this is the time axis zero h two h three h you put chi zero here chi one there and so on chi two there then chi h then as the time step size goes to zero chi h is compact the l one but then also any lp topology in time space so that's the torus that's the time interval and since I can't really go with to infinite times I have to put a lock here or a finite time horizon so so this fifth lemma tells you that we can always there's always a limit we can select a subsequence I mean for any for any sequence of time step sizes that go to zero we can select a subsequence of the outcome of the scheme and it will converge to an evolving set I mean the strong convergence here is important for the limit to be a characteristic function again so so that of that's always in a certain sense an easy part in this type of analysis in probability theory you would call it tightness and and then of course all the work consists in identifying the limit and in order to prove this we need a fourth lemma which is just just three statements on on the relation between the energy functional and the metric so the first relation is that if you're interested in the difference between the convolved and the unconvolved function you can control this by by the energy this is little of square root of h if the energy stays bounded there's a second statement that if you look at two configurations u and u prime then the l2 difference the squared l2 difference is estimated by something which involves the the distance I'm looking for the right pre-factor 1 over 2 squared of h dh square u u prime plus this term for the both energies and then it's also convenient to have that the if you look at the differences of energies for two configurations and you look at the square then this is estimated by 1 over 2 square root of h to the power 3 or h to the power 3 halves times dh square u u prime and this last inequality is we'll need it at some point for the variational interpolation and but at this stage it tells you that for fixed h e is indeed a continuous functional even ellipse it's continuous functional with respect to the metric dh okay and the and the the idea is now that we're going to prove lemma 5 if you want lemma 5 based on on this here and on these two things so that that would be kind of now getting more into details alternatively I could start telling you about the convergence proof which is kind of the next would be the next section so so now you have the choice of either seeing kind of a bit more details I mean getting some getting some intuition on these on these functionals and some of easy manipulations and how you get a semi interesting result this compactness result or moving onwards who's in favor of seeing the details who is in favor of moving onwards that's pretty equal sir so um uh so for a certain class and that's something selim is a W has worked out for a certain class of anisotropies you keep kind of good both I mean you can you can get that from a thresholding scheme with a kernel which has kind of both good properties namely of being non-negative in real space and non-negative in four-year space so in a certain sense for this for this minimizing movement interpretation the non-negativity in four-year space is perhaps even more important because only that guarantees you that you can take kind of a square root which is important for for the metric so uh so if one's willing to give up more I think I think essentially you can do things for all for all anisotropies but they're nicer anisotropies where definitely everything should be should go through we didn't and in the work with tim laux we didn't really look at the anisotropic case we treated the multi-phase case and the case where every kind of grain boundary may have a different isotropic interfacial tension and that's that in this case there are no practically no changes and in the other case there would be some changes but my feeling is that for this class of zonoedal and isotropies as selium calls them it should be fine more questions yes yes so so even if it would slightly deviate from one half uh on the natural on the physical timescale your set would shrink or invade the entire space I mean if if the so in fact that's something which tim laux worked out that if um uh if you perturb it by around one half with uh I mean if you if you twist one half by something that's of the order of I guess square root of h uh or h square now I forgot but if you I mean if you if you modify it in a very controlled way uh then you get motion by mean curvature plus kind of a constant speed so so and in fact and but this this was already known before you can also get the motion by mean motion by area motion by volume preserving mean curvature flow by modifying the this uh thresholding scheme by insisting that you choose the threshold so that your new answer has exactly the same volume as the old answer and and also in this case you get uh things can be uh can be can be adapted plus a constant velocity oh I think there is no uh there is no problem of of uh kind of deciding a number whether a number is larger less than one half to machine precision and then of course if if square if h now is smaller than the machine precision then things might go wrong but I I don't think that whenever operates uh operates in that and that regime so I mean threat I mean uh so uh I mean stability with respect to spatial discretization errors and so that of course is important but uh but that I mean thresholding I think yes yes yes I mean so so the deviation would have to be larger than some power of h some moderate power of h now forgot h square or square root of h and typically that I mean that's uh that's not the machine precision more questions okay so now I got a split vote on uh on on what to uh what to present you so uh I don't know is it so I mean I wrote down I wrote down the proofs by hand is it possible to scan that and put it on the web uh and then uh and then you uh you can look at it and uh and then you can still ask me uh ask me tomorrow by tomorrow I mean if is it possible to get them on onto the web today okay then uh then you can tell me uh can tell me tomorrow whether you want to see whether you want to see some of the proofs and the other situation might be that I run out with what I've what I've prepared and then and then I will use the remaining time to give you give you the proofs okay so so so I'm going to erase these statements now anyway they're also I have them also on the screen here is around these two things and uh and now let me start with the uh with the second chapter which contains the main results and uh and the connection to the work by Andrin Taylor and Wong and uh and the uh the bracket and notion so and and and by the way I mean so you I can also give proofs after I mean I have a second session um this um this afternoon so if you come to me and tell me I really want I really want to see uh some of these proofs I can be can be easily convinced so uh main result okay so uh so let me start with the uh with an informal observation so it is kind of well appreciated by uh by by the mean curvature flow community that mean curvature flow in fact is a gradient flow formally what you would expect uh if I use the bv notation I write it like this but of course classically it's just the uh hostile measure of the uh of the boundary of the set and uh why why why is not surprising that something gives like this gives rise to curvature because the first variation and so let me use this notation for the first variation so I had one notation for metric slope and one notation for a first variation so if you look at the first variation of uh of the interfacial area in direction of a vector field xi so xi will always in my case denote a time or time independent vector field uh so here which of course lives on the same domain uh is uh as uh many of you will know is given by integrating uh the weak formulation is given by integrating the uh tangential divergence which I can write as the ambient divergence minus the uh normal normal part uh which uh classically is just the uh integrating the mean curvature um times uh nu xi so in my in my notation the mean curvature is a scalar and uh that's the mean curvature vector so uh that's the first variation so the first variation brings up mean curvature so uh mean curvature flow is a gradient flow of the area functional uh uh if uh you uh if you consider the metric so with respect to uh the infinitesimal inner product of the uh our two space on the on the boundary so as it stands this is just almost tautological and uh um and the first problem uh with this type of uh if now so if now you wanted to use this insight to write down a minimizing movement scheme uh you run into the following problem uh so namely the problem that the uh distance function which comes from this infinitesimal metric from this infinitesimal from this Romanian structure is degenerate induced distance function this formal Romanian structure is degenerate in this sense and that's uh I think something which uh was which many people were aware of but uh which Michaud and uh Mumford pointed out uh not that long ago and therefore if you want to write down a minimizing movement scheme uh without knowing the thresholding algorithm you have to come up with a proxy so hence for uh minimizing movement scheme a la de georgie proxy or substitute something which acts a little bit like what you would expect the uh induced distance to be um for the induced distance or induced metric and that's exactly what uh uh what Almgren, Taylor and Wang proposed in 93 by saying that Chi N minimizes i in fact there is uh for for um for technical reasons it's convenient to look at mean curvature flow up to a factor of two so the normal velocity two times the normal velocity is given by uh by mean curvature then things come out nicely with thresholding so v always denotes the normal velocity and h the mean curvature in the sense that it's the trace of the exterior curvature I mean the vine garden map and uh so Chi N minimizes uh the parameter functional plus uh one over two h and then comes the following expression uh you take uh the uh um non-negative so the unsigned distance function in uh rd to uh your previous um uh to the boundary of your previous configuration and you integrate this uh distance function over the symmetric difference between the new set and uh the previous set and I think here you will need a factor of four which uh comes from uh from this tool here so uh so that's the uh that's the famous Almgren, Taylor, Wang scheme so you minimize if I would write this geometrically you minimize the surface area of the boundary and keeping keeping track of this this type of distance to the previous time step and uh so um uh in the original paper they had a had a kind of uh a very conditional convergence proof of this and that was uh to my understanding improved but still uh uh being conditional by uh lookhouse and Sturzenhecker 95 and the statement is uh is as follows so uh let uh consider a sequence of time steps that goes to zero and uh a limiting configuration uh which satisfies that uh the uh piecewise linear uh sorry piecewise constant interpolation converges in l1 to chi let's say over a finite time horizon up to capital T and uh so this is something which is a benign assumption because uh well I mean here we have to work more but uh kind of compactness is uh is is also not a problem in this setting but now comes the uh which only makes it a conditional convergence result namely that um uh if you look at the parameter of the uh approximating sequence so the sequence which comes out of the scheme then this should converge to uh the parameter of the limit and uh and that is indeed uh uh kind of uh uh a non-trivial assumption which cannot typically not be checked uh at least presently not be checked for the scheme so let me just comment on it there is one direction which you get for free here so by lower semi-continuity you always get for free that the uh time integrated um bv norm of the limit is uh less that of the approximating sequence so uh in assuming this uh the main assumption you're making is that the parameter does not drop in in the limit and what this does for you is that it morally rules out ghost interfaces so it shouldn't be the case that on the positive xh level uh you have two uh two interfaces which get closer and closer they don't have to be straight and in the uh limit so here is chi is equal to one chi is equal to one and chi is equal to zero and in the limit it's gone because you're looking at it not from a very full point of view uh where you count multiplicity but you look at it from uh from a bv point of view where you disregard multiplicities and uh and that's the situation I think that is this that is essentially what you're ruling out by this assumption but if you make this assumption then um then you're fine uh so then uh there exists a normal velocity v uh which is square integrable with respect to the interfacial measure in the sense of uh that it's really uh uh it it's it's it's the normal velocity of the characteristic function in the distributional sense so that means that uh v times zeta v times grad chi uh plus the integral of dT zeta times chi is equal to minus the integral at t is equal to zero of chi zero and here I need uh spacetime integrals for any test function zeta so there exists there exists such a normal velocity and uh and uh we have uh the normal velocity is equal to the mean curvature in this sense which is uh suggested by uh by this weak formulation of mean curvature that means uh we have a gather spacetime integral divergence chi minus nu d chi nu minus two v chi nu chi dT is equal to zero for any test vector field psi so that's uh that's uh that's their uh their result uh so uh this um greentailer-wong scheme uh which is based on the idea of minimizing movements but historically it might even have been the other way around that um greentailer-wong wrote down the scheme and then the georgey uh kind of embedded this uh uh in in the more general theory of minimizing movements so anyway so uh that this type of minimizing movement scheme for mean curvature flow uh converges in this conditional sense in the sense that if you make this uh uh this extra assumption of no ghost interfaces then for any uh for all times it converges to weak solution of of mean curvature flow in uh in in the specified sense and uh and that is a pretty involved work I mean it uses to some extent uh uh regularity theory for minimal surfaces to get uh uh to get that these sets are nice behave not uniform in age but for a fixed age and uh um and in a certain sense the most delicate term is to recover this uh uh this metric term in the limit and now what we did but that's not the result which I'm which I wanted to focus on is essentially the same result uh for the uh for the thresholding scheme so uh so this here is the piecewise constant interpolation of greentailer-wong and now our theorem which was published uh two years ago uh is exactly the same thing just with the only uh uh with the with the following difference so um so uh now this is the uh piecewise constant interpolation thresholding here we have to take the moral equivalent so uh we're integrating these energy functionals e h over time and we want them to converge to the limit so that's uh that's uh the uh kind of the moral uh substitute in our framework uh of the uh assumption of Lukhausen-Sturzenhecker and it's essentially the same type of assumption then we get the same result and in fact I would say our proof is uh because uh the thresholding scheme involves less subtle geometry like looking at distance functions to boundaries of set our proof is uh is perhaps uh less technical more robust so we definitely don't use I mean we just use well I mean a little bit ideas of kind of excess and but then essentially structure theorem for for bv functions and again the term one has to work most in both proofs is is to pass to the limit here okay so that uh that is uh that is uh what's uh what's around what has been around so far but then we ask the question um so since both uh since both kind of convergence results start from minimizing movements so start from natural discretizations of gradient flows either in this academic way of the arm grain Taylor-Wong scheme or in kind of this numerically relevant way of thresholding at least the limit should satisfy the dissipation inequality so the question is is it true that for the limit if you look at the uh rate by which the interfacial energy decreases and I have to put a factor of two here because of this factor of two uh do I get that this is at least as an inequality given by integrating the square over the mean curvature and uh and that's not I mean this uh this in a certain sense is not strong enough to derive that uh dissipation inequality so the notion this notion of uh weak solution is not strong enough and and this is a little bit uh dissatisfactory right because uh you start from gradient flow the dissipation inequality is hardwired into gradient flow you're using kind of a natural time discretization for a gradient flow but uh you're deriving a kind of a notion of a limit that doesn't even allow for for this basic property of a gradient flow and uh uh and therefore so so that was uh that was dissatisfactory and therefore uh we um uh uh uh well we wanted to get this and we were kind of uh uh we're aware that um this issue of getting the right dissipation is in fact at the basis of uh Brachy Brachy's notion of uh um mean curvature flow so uh this is so the dissipation inequality in a localized version is uh is just uh is is is what uh what Brachy's notion is built on but let me remind you what uh uh what this means so uh uh and but let me still use this bv notation so given a localizing function zeta which uh is uh let's say smooth and of course periodic and non-negative you want to monitor the rate by which um uh by which this localized surface measure changes and uh and so by purely kinematic uh calculation or classical calculation you see that uh there are um now more than one configuration um contribution there is what you would expect namely the localized uh the localized term uh which comes from the localized term which which looks looks like this one here but then there must be another it's clear that there must be another term because even uh um even if uh uh uh even you know I mean even if nothing dramatic would happen your uh interface could move out of the region where this function is looking at so there is also transport term and sorry I so I wanted to write down the purely kinematics formulation so this would be normal velocity times uh mean curvature but then also the um you're moving out of uh the um what you see by the cutoff function so you get a transport term like this so that's a curvature effect and that's a transport effect and uh so this one you clearly see that uh so if zeta is constant then you uh then you kind of end up with the classical inside that the um uh that the um first variation of the area functional is given by mean curvature so that the rate of change is given by normal velocity times mean curvature and uh but now since due to the cutoff function due to the non constant and the non constancy of zeta you get such a transport term and therefore uh the the notion of mean curvature flow is encoded uh by requiring now let me put in the factor of two again this here is lesser equal to minus zeta h square plus gradient zeta dot new times h red chi for all non negative test function and it's uh in a certain sense at the basis of uh of bracker's uh notion of solution that for smooth evolutions this here really characterizes despite the fact that you just have an inequality sign characterizes mean curvature that's uh that's the uh that's the inside then of course he used very folds to formulate that in a kind of more robust way that's something I will not be using I wish we could but that's something which I will not be using here uh so I will stick to this bv type notation and the bv setting but uh but even before that the main idea is that you can encode uh an equality pde by inequality and that's something which uh you know I mean clearly is also present in the general gradient flow theory of uh of um uh um the georgi but uh is is present in in many other situations so for instance this notion of uh weak solutions to compressible Euler equation for instance edward fire riser is kind of one of the main uh people working in that area also works with inequality so it's not unusual that uh that pde's can be characterized by inequalities and that's um that's one instance okay so so therefore we were right away a little bit a little bit more ambitious and asked the question well can we instead of just getting this global dissipation inequality can we get this entire family of dissipation inequalities and that's indeed the result and that's kind of the result which I would like to explain a bit better in this course so that's theorem three that we posted it last year but now there is kind of a new version and again that works in the multi-phase case so uh so the assumption is very much like there so let me put it again for completeness so uh for uh chi zero uh for initial condition with finite uh parameter uh let uh chi n uh denote the solution of the thresholding scheme at uh for time uh with initially with these initial conditions and time step size h which i'm suppressing here in the notation let uh chi h denote the piecewise linear interpolation the sort of piecewise constant so what what i defined in lemma five uh let uh chi be such that for a subsequence of h going to zero we have exactly these two statements that the uh interpolation of the thresholding scheme converges uh in uh l one on time space and that uh we have not just an inequality here but uh inequality so that's the unfortunate assumption we we need uh but then we're fine so then there exists uh mean curvature the mean curvature which is square integrable on the boundary in time space and it's characterized in the usual weak way which is almost sitting here so uh it holds that uh zero t uh time integral of the tangential divergence of a test vector field uh minus h times the normal component of this test vector field integrated over the boundary integrated over time is equal to zero for any test vector field well let's take compact support in time but that's not necessary uh such that uh mean curvature flow in brackets and in form of brackets inequality is satisfied so uh by this uh i mean uh um now do i what how do i go i'm going to write it down let me write it down right away uh like uh like this so so that the gradient uh that zeta times the gradient chi at some point t plus the integral from zero to t the integral over space of h square then i guess i need a minus sign here and probably i forgot a two no i didn't uh that's okay uh minus but then i need the two here minus uh gradient zeta uh gradient sorry yeah gradient zeta dot nu times h uh gradient chi uh integrated over time is lesser equal to two times gradient chi zero and this is uh zeta and this holds for all zeta uh which uh which is smooth and periodic of course which are non-negative and it holds for almost every t okay so that's uh that's the uh that's if i didn't mess up with signs uh that's exactly the time into the time integrated version of this formal inequality okay so uh so that's the uh that's the that's the result i would like to uh i would like to explain and let me uh let me comment that this the proof of this is much easier than the proof of this so this is a much softer result and that's uh thanks to the tools by the georgie so uh so here uh uh here is here is where kind of the set of ideas comes together uh on the one hand uh this numerical scheme this thresholding scheme which uh which is uh you know real practical and successful numerical scheme for mean curvature flow uh this uh this very intelligent notion of encoding mean curvature flow as an inequality which is due to bracket and which is a localized which kind of uses the localized energy dissipation inequality and therefore is clearly related to the gradient flow structure and the tools by the georgie on minimizing movements so this is where we're kind of these three areas which i said in the beginning come together okay so uh how much what's when i do have to sir more or less over okay probably that that's a friendly way of saying it's over time already okay okay so right so that's the uh that's in a certain sense the those are that's really now the main statement of what i want to do in the rest and the rest of in the rest of the time and now you can you can either tell me before this afternoon you want to see some of the uh ideas from the easier statements i gave before in order to get a little bit of better feeling of what these kind of objects really do uh if you don't give me this uh strong impression we can post this uh tonight and you can have a look at that and uh and then if so if i don't get this this feedback i would start telling you uh how to uh go from the very first lemma by the georgie uh with the variational interpolation uh to uh to this inequality and uh and the main step is is again a very simple observation on the thresholding scheme namely that the thresholding scheme doesn't just satisfy a single um minimizing movement hasn't hasn't does not just have a single minimizing movement property but it has kind of a whole family of localized minimizing movement properties and it's this nice feature of of the thresholding scheme uh which uh which we use to localize to get to this kind of localized statement and here i forgot the zeta so uh so that's that would be uh that would be for uh for this afternoon if you uh if you don't tell me otherwise okay sorry for being over time and uh we'll continue over over over


I want to thank everyone for joining us online as well as the participants in this conversation. As was described, this is an event to explore how to hack DC more directly. What is the future of cybersecurity and cyber threats? And to discuss it, we've put together just a fantastic panel of leading experts on this topic. To begin, we're joined by Nicole Pearlroth, who covers cybersecurity for the New York Times, and also looks set to win Room Raider with her background there. It's phenomenal. She's the recipient of a number of journalism awards for her reporting on topics like Chinese government, intellectual property theft, etc. She's also the author of a forthcoming book. This is how they tell me the world ends. That is the untold story of the cyber weapons market. Then we have Senator Angus King, who is Senator from Maine and before that Governor of Maine. And along with him is Representative Mike Gallagher, who is represent from Wisconsin and before that United States Marine. And they are the co-chairs of the Cyber Space Solarium Commission, which is a bipartisan commission of leaders and experts that issued some 80 recommendations on how to improve US cybersecurity strategy. So it's a phenomenal group and the flow of this is rather than having it be each person giving a presentation. We're gonna take advantage of the fact that we can all be online in exchange. And so we're just gonna kick questions back and forth to each other. And so I'm gonna start it off. And the very first question is essentially tackling right on that topic. I'd love to hear from each of you what is one way that you think cybersecurity is going to be different in terms of the threats of the future. So let's just go in the order that I introduced. Nicole, why don't you start us off? Well, I think in the future, the big obvious one that comes to mind is Internet of Things. And we are just plugging everything we can online. We have been for the last decade. And so all of these threats we're seeing right now in terms of ransomware. I think in some of the more destructive attacks, I think we're going to unfortunately see an IoT component to those. In the future, I think right now we haven't seen a number of attacks on IoT systems beyond some of the targeted attacks. There was the recent attack on the water facility that Israel has blamed on Iran. There was the attempt on the wrong Bowman Dam in New York. But we haven't seen these things at scale. And so I think that is really where the threat's going to move over the next decade. Senator King. Well, I think everybody's thinking about elections right now. And the more I think about it, the more vulnerable I think our elections are. We keep having warnings, but things don't necessarily change. I mean, last week in Georgia, apparently it was a semi-disaster. Wisconsin wasn't good. 2016, we know what happened. Our election system, the conventional wisdom, is our election system is such a hairball that it's nothing to worry about because it's so decentralized. I'm not so sure about that. I mean, a sophisticated actor can tell you, anybody can tell you which counties in Florida you should go for, which counties in Wisconsin or Michigan. And it really doesn't, wouldn't take a hell of a lot. And I think we are way more vulnerable than we think. I've had hearings with secretaries of state and chief election officers and they're almost cocky about how invulnerable they are. And I just don't believe it. And I'm very worried about the election in 2020. I think it could just be, and I guess the final point would be, if you think about it, elections are exercises in trust. The day after the election, somebody tells you, here's who won. And you say, oh, OK, and you move on. What if that whole basis of trust is undermined one way or another by manipulation of how results were reported or what actually were voted or who got to vote or lines at the polling place, all of those things. And then, of course, you add the coronavirus on top. I'm very worried about the vulnerability. Can I ask you a follow-up on that? My nightmare scenario related to that is not just attacks on it, but you phrased it as, in the normal times, you find out the night of the election, the results. But if what we've seen out of, for example, right now in Georgia, it seems more likely we're going to see an extended period of time of just the basic counting of them. And that's in a scenario in Georgia where you didn't have. It looks like a major cyber attack. How do you think we're going to be able to weather the storm if we have a delay in terms of the reporting, particularly coming from battleground states or districts? What would that look like in your space? Well, every day that goes by will be a new layer of uncertainty and suspicion. Conspiracy theories will be rampant. People will be saying, what's going on in Broward County or what's going on in Green Bay? And it's a, I just, again, I go back, we take so much for granted in our country. And one of the things we take for granted is you go and vote. Your vote's counted that night. You find out who wins. Suddenly, if the trust and confidence in that system is undermined, that invites, I hate to say it, but it invites violence. It invites people going into the streets who feel that their votes weren't counted or something crooked happened. This is, we had an election in Maine in 1880 that was so close that we ended up with Joshua Chamberlain locking himself in the state house in order to talk down an angry mob. I mean, we almost had a civil war in the state over an election. And it just, I just don't think we're anywhere near ready. I hate to be doom and gloom about it, but that's what worries me right now. And it may be a foreign actor, it may be a domestic actor, or maybe several foreign actors. All they got to do is undermine that confidence and trust and they're 90% home. Representative Gallagher, do you want to get on on this point or a different scenario? Well, I just want to note a fun fact, which is that 1880 election was the first vote that a young Angus King actually cast. So, personal experience. Okay, I'm joking Angus. Hey for that, Gallagher. No, I very much agree with everything Angus said. We had a lot of recommendations in the initial report related to election security. And then our pandemic acts, Annex not only talks about that, but also has a recommendation related to internet of things which Nicole talked about. And I agree very much with what she said. So, you know, if I am trying to think about the future of cybersecurity, I mean, I think it's obvious to suggest that it will be AI enabled, it will be faster, adaptive, intuitive. And that means that the threshold for access to tools will be lower and a lot more non-state actors that will be able to do serious critical infrastructure damage. We were talking the other week about, you know, the next horrifying scenario, the next Columbine doesn't need to be carried out with a gun. It can be carried out by an ambitious young person with knowledge of cyber weapons and tools. And so that really concerns me, but I also think it's bound up in this bigger question, which we're starting to ask ourselves and for which there's a ton of proposals emerging on the left and the right about how we responsibly decouple our economy in critical hardware and software from China. And I think what the coronavirus crisis has revealed is how dangerously dependent on China we are. And I would suggest that regardless of who wins in November, and regardless of who's president in 2021 or 2024, I think we are on an inevitable process of selective economic decoupling from China that has important ramifications for cyber that's going to be extraordinarily difficult. And then the final thing I'd say to bring it back to the scenario that Angus was talking about in terms of that uncertainty surrounded the next election, I mean, add into that mix, you know, state-sponsored propaganda, whether it's Chinese Communist Party, apparatchiks on Twitter, spreading misinformation or whether it's Russian actors spreading misinformation. And so unless we get our act together now, I think we're going to be in a very difficult situation over the next couple of years. Can I follow up on that? You said unless we get our act together now, what does that mean? What specifically ought we to do? So at least in the pandemic annex that we talked about, we really emphasize a lot of reforms, the Election Assistance Commission, for example, needs to be able to distribute grant funds to state and local governments so that they can ensure that alternative methods of votings can move forward so that the administrators are prepared for possible rescheduling of voting. We suggest that the pandemic also illustrates the need and importance of paper-based systems to sufficiently audit and certify a vote. We believe we can't afford to have several ballots be voided and that a paper-based system to check that any electronic system is accurate and makes sense. And we recognize the irony of a fancy cyber commission recommending paper balloting. But there's a variety of proposals related to election security in particular, but then we talk more broadly about the need for civics education, cyber education in K through 12 education so that we can educate the next generation about the nature of propaganda and interference, but also the importance of voting and things like that. And I defer to Angus for anything else I missed there. Well, Peter, one of the big problems we're facing is the disorganization of the federal government in terms of dealing with this. One of our, you asked a question, what should we do? And my glib answer is read our report. We have about 80 recommendations. But one of the major ones is the creation of a position of national cyber director in the executive office of the president, Senate confirm, presidential appointed, Senate confirm. There needs to be somebody overseeing the crazy quilt of cyber activities throughout the federal government and involving the federal government and the private sector. 85% of the target space in cyber is in the private sector. So the relationship between the private sector and the federal government is really crucial, but right now there's no one in charge and there's no one who the president can go to and say, work on this or you're in charge of this, you're responsible for this, you're accountable for this. We think that's a very important recommendation. We're working, the White House is resistant to it. The national security advisor, I suspect doesn't like it. No national security advisor would because it's some diminution of their authority. But I think it's one of the most important recommendations we have, the other is to develop a strategy of continuity of the economy, planning. We learn from the pandemic. We should be thinking in advance about the unthinkable. What happens if the entire Northeast energy grid goes down or if there's a simultaneous attack on water supplies across the country? What do we do? What are the steps that we take? Who do you call? Who do you activate? All of those kinds of things. So the two of the really important recommendations are national cyber director and setting up a really strong structure for developing continuity of the economy. So Nicole, I've been watching you. You've been nodding. How do you assess, so we've heard these are the various things that have been proposed. These are the things we ought to do. You're an observer of this space. How do you assess the likelihood of them happening in time? Well, I was actually quite impressed with the report. I had just finished my book and I just finished the epilogue in my book where I'd come up with solutions. And then the report came out and I put them together and theirs was better. They nailed it. It was the we need a coordinator in the White House. We need someone running point on this. I don't know who that person is right this moment. We need someone, especially with the election threats I've been documenting over the last six months. It is not just Russia anymore, although Russia has been extremely active in the disinformation space and amplifying the arguments we're already having with one another and starting with one another. But Iran is targeting the Trump campaign. China was recently caught trying to break into Biden staffers' personal Gmail accounts. We know this is happening. This is very real. It's the only difference this time is that there's going to be many more players. And so I was really nodding along with what the senator was saying about, we're constantly being told that the one thing that saves us from election interference is just how decentralized and tangled up it is. And after what happened in 2016, I totally disagree with that and agree with the senator. All it would take is something like we saw from software errors, not a cyber attack in Durham County when people showed up to what in that case was a blue county in an otherwise red or swing state and couldn't vote because the e-poll books had shut down. And then later we learned through these leaks that the e-poll book company had been breached by Russia. And we didn't know until six months ago, so almost three years after the election, that it was actually technical errors that were to blame. And there was a lot of tension there between the state, the Secretary of State in North Carolina, and Department of Homeland Security just taking a forensic look at their systems. And it just stems me that we only found out less than a year ahead of the next election that actually now it was software errors and IT configuration issues to blame in a situation that had this terrible recipe for an election that we would all continue to doubt. And one of the words I hear tossed around to describe what you were saying earlier, Peter, about the delays is this idea of a perception hack. Right now, we are more divided as a country than any other time I can remember in modern history. And all it would take is a situation like we saw in Iowa with the Iowa Democrats primary, where the caucus results were delayed for a couple of days for people to doubt the outcome of the election. And if you start doubting the outcome of an election in a political climate where we have a president that continues to use the word rigged to describe mail-in ballots, you really do have a recipe for a disaster. I was nodding along. I am equally terrified. Yeah, I think there's an interesting connection between what you raised on IoT and also in discussing the emerging threats to elections is that while the narrative in both is sort of the whole system goes down, the all the power goes down or in 2016, as Representative Gallagher alluded to we saw disinformation campaigns that targeted the entire U.S. My sense of what looms in both IoT targeting but also on the election side is more specific kind of micro-targeting. Whereas, you know, so it's not all smart cars are hacked. It's rather a specific numbers of them. All the way down to, you know, we play this out in the burning book, a single smart home being hacked as a new kind of crime, committing arson against one home, or it's not every single water treatment plan in the entire U.S. because there's this patchwork quality to it, but I go after one. Same thing in elections. I don't try and spread disinformation targeting the whole because that's a lot easier to detect now. I go after this specific district or this specific slice of voters or maybe it's the voting machines in a certain location. And that the intelligence side of it's gonna be a lot harder to detect. I wanna throw the question back. Peter, can I ask on that? Yeah, go ahead, jump on in.  I mean, for those of you less versed in nerd culture, you know, the idea that, you know, the Galacta is the only ship that survives because it's the least advanced, right? So as we ponder IoT and 5G future where everything is connected and the attack surface for bad actors is essentially unlimited, do we, particularly in the military domain, need to build in more fail safes or maybe less networked equipment or something else to minimize the risk of catastrophic cyber attacks? I think there's a different, you know, while I love and we could spend the whole time doing, you know, BSG and frack and talk like that. To me, the parallel is we want escalators that fail in a way that doesn't mean the system collapses. So you want a system that allows you to have the most advanced, but if it doesn't work, you still have a fallback. Peter, there was a perfect example of that in Ukraine in December of 2015 when we're pretty sure the Russians hacked the Ukrainian electric grid and they brought it down and they flooded the call center and they brought down the electric grid, the lights went out and it only, but it only lasted six hours and the analysis of why did it only last six hours was that the Ukrainian electric grid wasn't fully digitized and there were some old fashioned analog switches and, you know, people who had to go and throw breakers and it's exactly what Mike was talking about. It was, you know, back to the future that because, and actually, we passed a bill here about a year ago that is asking the Idaho National Lab to look at, are there places in the grid where we could undigitize, not to screw up the whole architecture and, you know, literally go back to the fifties, but critical places where it can be isolated more easily and a hacker can't control it because somebody physically on the ground has to do something. So that's, I don't think that's an unrealistic proposal at all and in fact, it's what saved the Ukrainians during that situation back three, four years ago, five years ago. And I would just add, I was in Ukraine last year and they were about to have their elections and it's all done on paper. The idea there that you would ever digitize any part of the election is just laughable, laughable over there. And if you remember, they had a hack back in 2014 where they didn't hack the election, but they hacked the reporting system to project a victory for a far-right fringe candidate and they caught it just in time. But ever since then, they've completely rolled back any effort to digitize any part of their election and I think that's a model for us. Nicole, can I ask a question back at you? So we asked you to evaluate sort of what you see of the space. Let's self-evaluate. How is the media equipped to report these kinds of stories, whether it's IoT hacks or election hacks? Do you think the media is well-equipped to tell the story the right way or will it either spread fear, uncertainty or doubt or even in some situations aid the attacker in their very goal to spread distrust as the two members of Congress have laid out? I knew you were gonna ask this. So I think, no, I don't think the media is prepared. And I think, I mean, one of the things I've tweeted publicly and I've said publicly is we need to be very prepared in the media for when the hack and leaks come. And I do expect they will come this election season. We need to have a big red box at the top of those stories to tell our readers where that material came from, if we plan on reporting on it the way that the media reported on some of those hacks and leaks in 2016. That's just one example. I think people really don't understand, let's just look at Russian disinformation right now. I don't think people, and I don't think the media really understand what it looks like. I think when we got that report on February that Russia was once again trying to meddle to reelect the president and also to boost Senator Sanders' candidacy. I don't think people really understood what that looked like on the ground. And what it looks like on the ground right now, I'm just working on the story that I think will be out today if it's not already out, is it's just, Russia is just pushing distrust, they're just pushing populists versus the establishment. The ultimate goal is total is that we'll be tied up in so much of our own political infighting on both sides of the political spectrum that we won't be equipped to check Russia as it maneuvers however it wants. And I think that's actually already playing out. And so I don't think the media fully understands what that dynamic looks like in that partisan infighting and these vitriolic partisan battles we see playing out every day is the goal. And I think if we had some broader informed perspective about what the ultimate end goal of these Russian disinformation campaigns look like, we might be able to at least have a couple of paragraphs in some of these stories that offer some of that context and right now I don't think we're anywhere there. So I wanna toss this back to the members of Congress. You are part of what feels like one of the last or bipartisan efforts, at least observing from NASA. Do you think that is possible in our space? Well, let me respond first because last week, Peter, we had the markup of the National Defense Authorization Act and we got I think 10 or 11 of our recommendations into that markup and there's room for several more as we move forward in that process. And that was all on a bipartisan basis. This commission was a very interesting one. There were 14 members, four members of Congress. And of course we know their party. There were Mike as a Republican and Jim Mangevin in the house, a Democrat, Ben Sasse Republican, I'm an independent. And so that was the make up there. The other 10 members of the commission, which came from the administration or from the executive and from the private sector, I haven't the faintest idea of their politics. I haven't the faintest idea of who belongs to which party, how they voted, who they supported. And that's the way it ought to be. We had over, I think we had 30 meetings and lots of discussion, lots of debate and differences of opinion. Partisanship was never, there wasn't a whiff of it and that's how it should work. And I think we're gonna be able to do a lot of the recommendations on that basis. This doesn't seem to be a partisan issue. I'll never forget one day in the intelligence committee when we were talking about, you know, the Russians and the Trump campaign, Marco Rubio said, look folks, Putin is not a Republican, he's an opportunist. And next time it could be us. And I thought that was an important insight that somehow right now, you know, at least in 16 the Russians were attempting to help Mr. Trump. But next time they may not, they may go in the other direction. This is dangerous, no matter what the party is. And so I don't see this as a partisan issue. It has a partisan flavor now, particularly when you talk about Russia because the president has never acknowledged that Russia played an active role in that election, unfortunately, but by and large, I think people understand this is a danger. And individual candidates are starting, we're starting to see and feel this either from other countries or, you know, the whole thing about deep fake and conspiracy theories and wild accusations altered photographs, all of those kinds of things, that's starting to happen to us as individual candidates. And that's when people understand the nature of the threat. I was just gonna say, I think Angus really set the tone from the top of the outset of a bipartisan tone or a transpartisan or whatever the term is. And it was his leadership that I think got everyone to work together. There were a lot of times when I is sort of, you know, I'm supposed to be the evil hawkish Republican on the commission, I felt like the dove.  But I think ultimately what we're trying to do in the report, and Nicole, you're incredibly kind with that assessment, was to lay out a strategy that we call layered cyber deterrence and a set of reform measures that will endure beyond the November election or at least convince a group of people in the White House and Congress to study these issues with not just a short-term, reactive perspective, but a long-term, how do we restore some semblance of deterrence and cyberspace and cultivate that expertise both within the executive branch and the legislative branch that we think is necessary to defend the country in this space. And so it creates a lot of very strange bedfellows. Some of our strongest allies when it comes to the National Cyber Director recommendations are actually Republicans in the Senate and the whole thing is just, you know, it really, there's no clear ideological or partisan fault lines a lot of the time. I wanted to weave in a question that Anand Shah posted on the chat which is about the coronavirus pandemic. And they framed it in terms of how did it lead us to think about critical infrastructure in a different way. And Senator King, why don't you lead off, not just in terms of, why don't we frame it this way? How did the pandemic affect your thinking? So you had this report come out, 80 major recommendations, and then we've moved into a new phase in American life and politics, and you've recently had a series of follow-on recommendations. So how did the pandemic affect the way that you and the commission were thinking about these issues? Well, that's a really good question. We, in fact, when we found ourselves in the midst of the pandemic, the report was released. The release of our report was one of the last large gatherings in Congress. I think it was something like March 11th. And if it had been two or three days later, we probably wouldn't have had that meeting up in the heart at Senate office building with a couple of hundred people. But we then found ourselves in the middle of this and we said, well, what can we learn from this? So we did additional study and discussion and created an appendix that was released a few weeks ago about lessons learned from the pandemic. One clear lesson is the whole work at home idea, which has been, which has really sustained us through this thing has created a whole new set of targets or at least an expanded set of targets. I mean, imagine for a moment what the effect on our economy would be if we didn't have the technology to work at home. We had, you know, 40 million people out of work. It would have been more like a hundred million if all the people that work for insurance companies and engineering firms and those kinds of things couldn't have worked from home. So that's created a whole new target space. That's one thing. It also pointed up the importance of, gets back to the national cyber director, having somebody in charge, having somebody in charge, not only of the response but of the planning. It goes back to the, it sort of underlined our belief that there has to be continuity of the economy planning. I mean, in some ways the pandemic was a wake up call or even more than had been before that saying, you know, here's how interconnected you are. Here are the risk levels. Here's what you should be thinking about and that you gotta be planning ahead. You gotta think about the, it really is think about the unthinkable. And, you know, that's how you learn. I mean, in the military, you always have an after action review and we ought to be looking at the pandemic and the response, which by the way, is not over by any means and say, okay, what do we learn from this and how do we apply it to a similar kind of catastrophe only on the cyber side? Representative Gallagher. Well, we had our first cute child alert in the background there. We did. We brought in extra cybersecurity advice and expertise. Very good.  So she may have been unaware of my distance, but it's been awesome to see all of her success in her great work. So I agree emphatically with everything Angus just said. I think if anything, the pandemic, not only I think reinforce recommendations on reorganization, National Cyber Director, but made us realize that we undervalued security of internet of things to tie it back to something Nicole mentioned at the outset. So one of our new recommendations is to ensure what to get Congress to pass an IoT security law focusing on known challenges like insecurity of wifi routers and mandate that devices have reasonable security measures such as those outlined under NIST's recommendations for IoT device manufacturers. And I think that recommendation kind of gets to this, this needle we tried to thread in the report of, we very much did not want to take a prescriptive, top-down, heavy government mandate and regulation approach. We want the market to work, but we do believe that the federal government needs to create incentives in this space. And certainly when it comes to cloud security and IoT, that's an area where we try to do, we try to achieve that incentivization without being too heavy handed from the federal government. So one of the questions that has been posed in the chat is our workforce capabilities. This is both the technology and the people problem. And as a number of reports have alluded to, we have insufficient talent in this space. There's a number of scrunch. What do you think are the best ways to go after that? And how are we doing in terms of meeting our current and future needs? Well, it's a crucial requirement. We discussed the workforce pretty extensively in the report. We've just got to think in a whole lot of different ways. We need to have scholarships. I mean, this is talk about back to the future. My mother went to college in the 20s on a scholarship, which basically said for every year, you get to go to William and Mary. You have to teach for a year in Virginia public schools. It was a kind of indentured servitude, but we need to be thinking about those kinds of things. You know, the ROTC, we need a cyber ROTC, where we provide scholarship help and support for students who make a commitment to public service when they get out on the cyber side. We also, and I've talked with the military about this, we have to think about what the real requirements are for a cyber warrior. Do you really need to be able to do 100 push-ups in order to be a cyber warrior? Maybe, you know, I don't mean lower the standards, but just we need to think a little bit differently. There's an enormous amount of cyber talent out there. Think of all the kids who at this very moment are playing pretty sophisticated video games. We need to figure out how to tap that talent. I was told the other day there's something like 35,000 IT related jobs in the federal government that are empty. So to say there's a need is a really understatement. So it's got to be a kind of all hands on deck. There's no one solution, education reform, scholarships, marketing, all of those things, but it is a serious problem. And it's one that we need to address in a, you know, no single solution, but in a multiplicity of ways.  Well, I just think we have a lot of recommendations on, you know, 20% increase in the cyber core scholarship program, trying to get the military to improve its transition assistance program. So when you transition off active duty, at least when I went through it, the program was terrible. It doesn't set you up for success, particularly for a young Lance corporal, you know, seeing if you can spot and identify the people that have talent and immediately as they come off active duty, slot them into a cybersecurity role. But I think, and maybe Angus would disagree on this, but I think as we looked at this, we kind of came to the conclusion that even in the best case scenario, if we really improve pay, improve hiring flexibility, you know, allow that young kid with purple hair who doesn't wanna do, you know, push-ups or get a high and tight to nonetheless be a cyber warrior, there's still, we get to this point where we still are never gonna be able to compete with Google and whoever on pay and lifestyle, right? But we can't compete on mission from the federal government, right? The NSA has a cool mission. I mean, you can do things there that you can't do anywhere else. Similarly, if we elevate SZA and make its mission just as appealing and as sexy as Cyber Command and NSA, we believe that an organization like SZA, which is charged with defending our critical infrastructure and our domestic networks, can occasionally compete with and attract talent and beat Google and Facebook and the other companies that might be able to offer a far bigger paycheck than the federal government ever will be able to. Can I follow up? We've talked about the pipeline into the military or we talked about SZA, why do we not have an auxiliary for that in terms of, you know, not people becoming direct cyber warriors, but a broader, you might think of it as a reserve corps. You think about what Estonia has in the Cyber Defense League. What we have at certain state levels, Michigan has a cyber corps that's, again, not part of their national guard. It just strikes me as such a gap that the Coast Guard has an auxiliary that helps it do boat inspections. We have the Civil Air Patrol that can mobilize to help with aviation related emergencies. And yet we don't have a cyber equivalent. What would fill that gap? Because again, if it's just sending more people into the military or directly joining government, it still leaves a massive gap. Peter, we almost do in the sense, there's a lot of cyber capacity in the national guard. People who work in the private sector as engineers, IT engineers and other things who are in the national guard and there are guard units around the country that have tremendous cyber capability. So that's not a full answer to what you're talking about, but I think there is activity going on in that area. And I agree with what Mike said about making the transition out better. Peter, I don't wanna miss since our time is running out, talking about the international aspects of this and norms and standards. Part of our report talks about the importance of having international norms about cyber crime and use of cyber so that it's part of the deterrent strategy. We want militia cyber actors to know that they're gonna be international pariahs if they do so and they're gonna pay a price internationally. And that's an important part of our recommendation. I just wanted to be sure we got that message across. Can I chime in? We do ask SZA to look at the creation of a civilian cyber reserve, ideally creating something that might, when I got out of the Marine Corps, I did the reserves for a little bit, but I ended up just not wanting to do it. And I think there are a lot of people for whom they wanna serve, but maybe the national guard or the reserves isn't an attractive option. So theoretically SZA could create something that's a bit more attractive. But I'd be curious, Nicole, if you mentioned the epilogue of your book and kind of recommendations you laid out, did you arrive at any hypotheses about the human element of this and how we can improve recruitment? So I do think it's the most important part. It's all about getting- I agree and thanks for asking. I think one thing that we have to always remember is we're at such a disadvantage in a capitalist society and a democracy. We don't have the same advantage that China does where it can tap some of its best engineers in the private sector to do some of its moonlighting and contracting after hours. And we don't have the benefit of being a Russia where we're just gonna let an entire cyber crime industry do whatever it wants, so long as it doesn't hack inside Mother Russia and does favors for the government every once in a while. We just don't have that. So we are already at a huge disadvantage in this case. And I think some of the efforts, I think they are still baby steps, but I think they're moving the right direction and might be a model for scaling up is within the Pentagon right now, they are doing within the, and I'm sorry, I'm blinking on the name of it, but it's the digital defense agents, something. You can look it up. They basically, they are taking people for one two-year tours of duty from Microsoft, from some of the security firms just to take a look at some of the low-hanging fruit on .gov and .mil and locking up those systems. So that's a start. They are also engaging SINAC, for instance, and some of the private bug bounties where they're calling in some of the best engineers from Microsoft, from the security space, from the hacking community, and asking them to take a look at the networks that are attached to our weapons systems, for instance, and patching up those holes. But I actually think we could do something even more ambitious, something like a Google Project Zero where we recruit some of the best security people at these companies. It wouldn't even have to be two years. It could be a year. Spend a year doing the service for your country, get Google to help with some of the pay on the backend or some of these technology companies or something like the Linux Foundation, which is really interested in locking up our open source code and get people who are big names in the space to volunteer to do a year of duty, basically hacking the government and seeing where we are most vulnerable or, alternatively, if we do find ourselves in a situation where we do need to call up for a private reserve, have those people be a part of that natural talent pool that you would tap? So I want to hit a couple more of the questions that we're seeing in the chat here. So this is going to go, let's just go round Robin on it. We've created a new branch of the military, Space Command, Space Force. It should have been command in my own personal opinion. So is there a need to create a new military service on the cyber side? We've got it, we've got it. Cyber Command, the question is whether it's strong enough. There are about 6,000 people. One of our recommendations is that the Pentagon do an analysis of whether that's enough. The commander of Cyber Command is Paul Nakasone, who's also the head of the NSA. Traditionally, those have been two different jobs held by the same person, so-called dual hat arrangement. Whether that is sensible going on into the future is something that's debated practically every year, but we do have Cyber Command. I personally don't think it's strong enough. I mean, this raises a sort of fundamental question of are we fighting the last war? Are we preparing for a war that is unlikely to take place in terms of guns and missiles and ships and airplanes and not sufficiently preparing for what is, I think, more likely the next war, which will be bits and bytes? So, Representative Gallagher, I'm gonna put that in a two-part question. And one, should we have cyber as a separate service, given that we did so for space? Why would we have a space one and not a cyber one? And second, dual hat or not, should we continue with the current double structure of the leadership within NSA? Well, so, first of all, it should have been space core and it should have been subordinate to the Navy. It's fundamentally a naval mission. Look no further than the rank structure in Star Trek. It's Captain John Luke Picard. I mean, these are Navy ranks, so I rest my case on that. I agree with what Angus said about it's less a matter of creating, you know, a new service than it is. I mean, it's hard to create two new services in less than a decade than it is adequately resourcing cyber command, giving new acquisition and budget authorities to cyber command and seeing what the new four structure assessment comes out with. We, for example, are recommending allowing existing agencies to do threat hunting on defense industrial based networks, allowing CISA to do threat hunting on .gov networks in the way cybercom can do it on .mil networks. So I think it's, we sort of took the approach of taking what we have right now and elevating and empowering existing agencies rather than creating a bunch of new ones. I asked for the dual hat question, very controversial, but we've asked DOD to tell us where it has met certain expectations, you know, and where it hasn't. And so I'm punting on that one politically right now. Look, if I have my choice, I'd put Paul Nakasone in charge of several other parts of the government. I mean, he's one of the most talented people that we have. I will not punt on it. I do not believe that Space Force was needed if we were gonna create a new service that should have been on the cyber side given that the space issue was mostly, was an acquisitions problem versus cyber. It was everything from acquisitions to it's highly operational. Also, if you're gonna talk about creating a new service that means a whole new service culture and the like and clearly as we're going back to the human side, cyber is one where having a very different culture makes a lot more sense than space given that despite all the advertising, no one from Space Force is going into space anytime soon versus the cyber side, they're all operational. And so having a very different identity promotion and the like made more sense. And I also think we should split the dual hat that having a simultaneous head of NSA and cyber command is like having the same person be general manager of the patriots and coach of the Celtics. They're two different games. Even so, even if you have a talented person and sorry for represent Gallagher, I didn't use the idea of having the same general manager and coach reference for the Packers because we saw how that went previously. Well, it worked under Lombardi, he had full authority for all of that. Many, many generations back. So let's jump into, I think we've got time for one last question. And actually we'll ping off of on the Packers side. So there is a saying that if you have two good quarterbacks, you actually have none, which is something about to play out in terms of draft strategy there. So in terms of priorities, of all the different priorities that have been set out there, all the different actions that could be taken, what do each of you think is the most important single thing that ought to be done in terms of US policy? So what is the one, so for the members of the commission you had 80, what's the most important thing that you think should be done?  What's the single most thing out of that? So Senator King, why don't you start us out? I'm gonna modify the question a bit by saying the two most important things. I think the first is one that we really haven't discussed at all. And that is the establishment of a clear articulated public doctrine of deterrents. Right now, our adversaries essentially pay no price for attacking us. They, you know, some sanctions here or there, but there's no cost imposed for an attack beneath the level of a physical attack. And I think that's the problem. I mean, that's why we keep, they keep coming after us. Why wouldn't they? It's cheap. Putin can pay 8,000 hackers for the price of one jet fighter. And there's no deterrence. I want the people in the Politburo to say, gee, if we do this, if we go after those American elections, some bad things are liable to happen to us and maybe we ought to not do that. In other words, I want them to know there's a cost imposed. So that's number one. I think there needs to be a clear, establishment, establish of a deterrent posture. It's not the same as nuclear deterrence, but in the realm of if you do this, something bad will happen. Number two is the organization, the National Cyber Director. I think one of their problems right now is the sort of incoherent structure that we have. One of my principles of management is messy organization produces messy policy. And right now there's no central driving force. And so I think a National Cyber Director, not with a huge office, but with an opportunity to coordinate, oversee the other agencies. Those are the two things that I think are the most important of all the things that we talked about. And how optimistic are you about that making it into the NDA? Well, there's a placeholder in the committee. We voted last week and there is a provision that talks about the National Cyber Director, but it's literally a placeholder so that there can be further discussions with Mike Rounds, who's the chair of the Cyber Subcommittee with the administration. So essentially we're still working on it. I'm really hopeful.  And as I think Mike mentioned earlier, we have strong support, for example, from Ron Johnson, the Republican chair of the Homeland Security Committee. He believes this is important too. So I'm hopeful and basically it's gonna be persuading the administration, I think. And this isn't about President Trump. This is about any president. This is a favor to the president, giving them someone that they can hold accountable in this area. So I think there's a reasonable shot at it. As far as the declaratory deterrent policy, again, that's really up to the administration and the president. Congress can talk about those things, but ultimately the executive has to promulgate what our strategy is. Representative Gallagher? Well, I agree with everything Angus said, but maybe just to kind of add to that, if you get the organization piece right, I mean, the goal is for a better organized federal government to do better public-private collaboration, right? Because in cyber, the federal government, I would argue, is not the main effort. It is the supporting effort. And as Angus alluded to before, most of our critical infrastructure is in the private space. And so I really think that some of the most important recommendations are all of those in a report that aim at elevating and empowering CISA. CISA is not a household name like NSA or Cybercom, but CISA is supposed to be that interface with the private sector. The private sector's preferred collaborating partner. And I think if we can get that piece right, if we can make CISA cool, not that it isn't right now, I think better public-private collaboration flows from that small part of reorganization that we're recommending. Nicole? Well, and I'll just add to my point. I mean, I think a lot of people have had a lot of skepticism about Homeland Security being the agency charged with our defense in many ways. But I think over the last few years, CISA, I think people have put their skepticism aside. I think Chris Krebs has done a really great job and a good job being the face of that agency and coming from Microsoft, being able to recruit more people from the technical community where he needs them. So just I'll throw him a bone on this Zoom meeting. But I think a lot of people really roll their eyes when they think about regulation being the answer to all our digital problems right now. And I roll my eyes sometimes too. But I think one thing that really surprised me in my own research that I think would be really helpful to provide for some of this is there was a global vulnerability report that was done about almost 10 years ago now by some academic researchers. And they just looked at country by country. What are the countries that are most secure in terms of their cybersecurity? And I think the metric they used was what percent of attacks on its machines are successful. And the safest country in the world is Norway. So Norway is extremely digitized. But what they have on us is they have made digital literacy and cybersecurity quarter their curriculum, which I know is in your report. They have also they have a master plan. They have a cybersecurity master plan. They update it every year based on current threats. And it does things like the things we've been trying to pass through legislation unsuccessfully for the last decade. It identifies critical infrastructure. It sets up mandatory requirements for those critical infrastructure operators. There are liabilities and fines for critical infrastructure operators who do not meet that bar. It requires them to go through basic common sense things like penetration testing and to use encryption and multi-factor authentication. And it's not rocket science, but Norway is the safest country in the world when it comes to cybersecurity. And then another really useful case study they did was actually Japan, which over the course of one year upped its cybersecurity score by something astronomical, like 50%. And what they when they saw when they looked at what was the difference that year was Japan came up with a master similar master plan for cybersecurity. And they really put a lot of effort into digital literacy, cybersecurity training, both in the public and private sector, fines. They were the only national cybersecurity policy to mention the word airgapping. They put a lot of emphasis on setting up perimeters, non-digitized perimeters around some of their core critical infrastructure. And this is something we could easily do here. We just unfortunately have a lot of strong lobbies and lobbyists who've argued that some of these really basic things that we would require from our critical infrastructure operators or to burn some on the private sector. And that's just nonsense at this point. When you look at the cost of cyber attacks related to terrorism these days, the cost of cyber attacks is now by some estimates in the trillions, whereas we see the cost of terrorism going down. So it's time to really focus on this issue and think about what we're losing in terms of intellectual property and business every day from cyber attacks and ransomware attacks and intellectual property theft. So yes, master plan, look at what some of these other countries have done and implemented here. So I'll close by, I think it brings this full circle. For me it's to, in that master plan, go after the risks that you identified in your answer to the very first question, which is a particular focus on internet of things. Essentially, my feel is that we are recreating almost all the mistakes that we made with internet security when it was the last generation in terms of it being used for communication. We're basically recreating all of those mistakes as the internet alters into an internet of things where it's about operation. And the numbers tend to back that up when it comes to how little security we are baking in because of this absence of requirements. I mean, one study found that 98% of IoT traffic is unencrypted and 57% of the devices out there right now are vulnerable to medium to high level threat. And as we play out in the Burn-In book, but also we can steer the audience to a new info graphic that we have on the New America website. You, it basically lays out 10 different ways that you could hack the city of Washington DC by going after a number of these different IoT elements that we've talked about in the discussion today, whether it's smart homes, whether it's the move into having drones, whether it's trains to water treatment plants. The reference of what Israel suffered in the last week, someone going after a system and targeting the chlorine levels. I can tell you from actually having done the research, if you think that the small towns and small businesses that do water treatment upriver on the Potomac have better cybersecurity than the Israeli government, they've got really, really bad news for you. And so we, in my sense, we need to, we're baking in vulnerability right now that we're gonna regret for the next 15 years, much like played out with the last 15 years related to cybersecurity. So I'd love us to get on top of that. This has been just a fun, rich conversation. We've hit on so many different topics. So I wanna thank again all the attendees who joined us, but in particular, our distinguished panel Hill. Thank you very much for giving up your time to have this conversation with us and to everybody out there, stay well.


Suicide affects all of us. Building resilience and preventing suicide requires us to actively communicate with each other. It starts with everyday actions to build trust and meaningful connections with our shipmates, making sure they know they are never alone. Life in the Navy includes being in unfamiliar places separated from loved ones and friends in their support while executing demanding missions in a stressful environment. Successfully navigating this challenge includes leaning on the support of your peers, your leaders, and your community. Working to stop suicide is an ongoing, everyday, all year effort. This year's Suicide Prevention Month theme is Every Sailor Every Day. Be the first to break the silence. Start the conversation if you notice someone having difficulty navigating stress. Take time to find out how your shipmates are doing. It's about being there for Every Sailor Every Day by Every Sailor Every Day.


Good morning! My name is Angie Johnson. I am the Steel County Extension Agent through the NDSU Extension Program, and this morning I'm going to be talking about how I'm getting through COVID-19 and working for my producers. So with COVID-19, we are really looking at a multitude of germs and virus. It feels like that's all we've been talking about, but we can actually do something about it. The CDC guidelines recommend warm soapy water and also disinfecting wipes to clean your office and supplies within your house. Well, my office is inside this tractor, and I share this tractor with other employees here on the farm, so it's really important that if my office is being shared, I need to be disinfecting it after each time one of us employees uses it. So it's as simple as starting with warm soapy water and a disinfecting cloth. Making sure to grab and touch and clean every high surface area, including hydraulic arms, steering wheels, light switches, and gear shifting levers. Don't forget your throttle increase, your PTO knob or lever, and all window handles and interior and exterior door handles. Together we will get through COVID-19.


Hi everyone, I am Dr. Aarish Poverty DNA consultant gastroenterologist at Manipal Hospital Jainagar. I trust you and your family members are doing fine. These are very tough times for all of us and we should stay safe and be positive to tide over this crisis scenario. For any kind of medical advice for already existing problems or any new problems for you or your family members, we are always there to support you people. You can feel free to contact at the numbers given below at any point of time. Reassuring you and your family for all the healthy life and stay safe. Thank you.