RESEARCH STATEMENT-ERIC SAMANSKY

Transcription

1 RESEARCH STATEMENT-ERIC SAMANSKY Introduction The main topic of my work is geometric measure theory. Specifically, I look at the convergence of certain probability measures, called Gibbs measures, on fractals and algebraic sets and look at the behavior of tubular neighborhoods around these sets. My research has strong connections to the current work of professors Dennis Cox of the Rice University Statistics Department, Robert Hardt of the Rice University Mathematics Department, and Petr Klouček of the University of Neuchâtel Mathematics Department in Switzerland. Their main result, which describes how Gibbs measures converge on somewhat smooth submanifolds, has important applications for areas in statistics and applied mathematics, such as designing algorithms for complicated optimization problems, Markov Chain Monte Carlo algorithms, and equilibrium configurations for functional materials (see [CHK] and its references). In contrast, my results are more centered in the areas of theoretical mathematics, namely the fields of geometric measure theory, probability theory, fractal geometry, real analysis, and algebraic geometry. Key motivation to my research is annealing, a process that certain materials undergo where the material is heated up and then quickly cooled. The annealing process allows material to minimize energy instead of settling at a local minimum. On a molecular level, when the material is heated, the molecules are more spread out than when the material is cooled. If the material could be cooled to absolute zero, the molecules would concentrate only on the material. The idea of achieving a better minimum for energy is behind the numerical analysis tool called simulated annealing (SA), which, to quote [CHK], is a stochastic optimization algorithm that mimics the physical process of a thermodynamic system settling into the state of minimal energy while lowering the temperature. For more on simulated annealing, see [GKV]. In [S] I study a measure associated with SA, called a Gibbs measure, that reveals, for strangely shaped material, where the molecules are located at a certain temperature. For the purposes of my research, we define a Gibbs measure thusly: Definition. Let P λ (B) = B e λj(x) dx e R λj(x) dx, n where J(x) 0 is a continuous function such that the denominator of P λ (B) is finite. B is a Borel set. P λ (B) is a probability measure, called a Gibbs measure. 1

2 2 RESEARCH STATEMENT-ERIC SAMANSKY The thermodynamic interpretation of λ is that it represents (kt ) 1, where T is temperature, and k is Boltzmann s constant, which is approximately joules/kelvin. Boltzmann s constant gives the energy of a particle in the system given a temperature. Cooling a system to absolute zero corresponds to taking λ to infinity, where this measure concentrates more and more on the zero set of J(x), which is called the constraint for SA. We would like to know what happens to P λ (B) when λ (which is when T 0) in terms of weak convergence, which we define now: Definition. A sequence of probability measures {P n } n converges weakly to a probability measure P, denoted P n P, if φdp n φdp for all bounded continuous real-valued functions φ. With this definition in hand, we state a modified version of the theorem from the paper Convergence of Gibbs Measures Associated with Simulated Annealing, Theorem [CHK]. Assume J is a nonnegative C 3 function on R n with J(x) x p for x sufficiently large, p > 0. If the set M = {x R n : J(x) = 0} is nonempty and bounded, and if the Hessian D 2 J has constant rank k near M (on open sets covering M), then M is a C 2,α n k dimensional manifold, and as λ, P λ P where P (B) = M B Λ(y) 1/2 dh n k y M Λ(y) 1/2 dh n k y, where H n k is n k dimensional Hausdorff measure, and Λ(y) is the product of the non-zero eigenvalues of D 2 J. This theorem describes how the resulting measure P concentrates on the zero set M of our function J(x). Notice that the measure does not distribute equally along this zero set, but rather takes into account the curvature in the integration. One can prove that the measure is weighted towards the parts of M with lower curvature. Current and Past Research My research in [S] involves instances where a set M is given, and we must choose a function J such that M is the zero set of J. To that end, I study the distance squared function; that is, J(x) = dist 2 (x, M). For well-behaved sets M, this function is nice enough that the theorem above applies. However, for more complicated M the function J will not be nice, and consequently the resulting measure P is not clear. Looking at our choice of function J, we see that we only have constant eigenvalues of D 2 J, since the function acts locally like J(x) = x 2. Therefore, when we drop the restriction on M having to be a C 2,α manifold, we obtain some nice, and sometimes surprising results.

3 RESEARCH STATEMENT-ERIC SAMANSKY 3 To simplify exposition, I will switch to using J δ (x) = χ Mδ instead of e λdist2 (x,m), where M δ = {x : dist(x, M) < δ}. So J δ (x) is the indicator function of the delta neighborhood around M. We can prove that J δ (x) converges weakly to e λdist2 (x,m) as δ 0 and λ. My most important theorem in [S] proves that if M is a semialgebraic set (defined below), the measure described will distribute evenly over the entire set. Definition. A subset X of R n is called semialgebraic if X = I i=1 j=1 J {x R n : f i,j (x) = 0, g i,j (x) > 0} where f i,j and g i,j are real-valued polynomials on R n for i = 1,..., I and j = 1,...J. These include all algebraic subvarieties as well as ployhedra in R n (For more on semialgebraic sets, see [H]). The main theorem in my thesis proves that measure will not concentrate in any cusps or corners, and will distribute evenly over the entire top dimensional strata (see below) of X. That is: Theorem S1. Semialgebraic Sets: Let A be a compact semialgebraic subset of R n. Let k N be the dimension of A. Then, for all Borel sets B R n, χ B A δ (x)dx χ R n Aδ (x)dx Hk M(B) as δ 0 H k (M) where H k M is k-dimensional Hausdorff measure restricted to the set M. The proof involves tubular neighborhoods of the stratifications of these sets, where for any semialgebraic set A, we can split A into parts that are smooth submanifolds, called strata. See [H] for a further description of stratification. Theorem S1 reveals how the measure distributes over any compact semialgebraic set (of arbitrary dimension) laying in any dimension. The result is similar to others that apply measure theory to objects in algebraic geometry, such as a result included in Federer s book that states that any compact algebraic variety has finite Hausdorff measure (see [Fe], section 3.4.8). In order to gain intuition about the above theorem, I first looked at sets M that had some number of corners, where the curvature was undefined. I saw that for the measure P λ (B) = R R B e λj(x) dx, with J(x) = R n e λj(x) dx dist2 (x, M), the converged measure P = lim λ P λ distributed evenly along M (with respect to the corresponding Hausdorff dimension). These sets include: (1) M = the unit cube in R 2.

4 4 RESEARCH STATEMENT-ERIC SAMANSKY (2) M = C 4 C 4, the 1 4 Cantor set crossed with itself. Here, we are removing the central crosses at each stage (approximation shown above, see [Mo], pages 32-33). (3) M = the Koch Curve (approximation shown above, see [M], pages 65-67). By generalizing these results, I obtain the following: Theorem S2. Self-Similar Fractals: For all self-similar fractals F with Hutchinson s open set condition (see below), we obtain, as λ, P λ P where P (B) = Hdim(F ) F (B). H dim(f ) (F ) Here, Hutchinson s open set condition requires that self-similar parts of the fractal can almost be separated by open sets (see [Hu]). Using the consequence of Theorem S2 to my advantage, I came up with examples where the measure will not just concentrate on the highest dimensional parts of the zero set M. To do this, I used different targets (that is, concentric circles of decreasing radii), and figured out the resulting probability measure P as δ 0, for J δ (x) = χ Mδ instead of e λdist2 (x,m). In the theorem below, we call the centers of the targets on the left a, and the centers of the targets on the right b. Theorem S3. Targets: (1) With M = (circles of radii 1 n, n N), we get P (a) = 1 (and so P (R2 a) = 0).

5 RESEARCH STATEMENT-ERIC SAMANSKY 5 (2) With M = (circles of radii 1 n and circles of radii 1 2n, n N), we get P (a) = 2 3, P (b) = 1 3. (3) With M = (circles of radii 2 1 n and circles of radii 1 + e, n N), we get P (a) = 1 n 2, P (circle) = 1 2, where the 1 2 measure for the circle is distributed equally along all parts of the circle. We see that the measure concentrates in places where the neighborhoods all contain infinite length, which allow us to construct these surprising measures, where we get weighted point masses and even a measure splitting its time between a zero dimensional object and a one dimensional object. These theorems show just some of the surprising general results we can obtain with these interesting probability measures. Future Plans My future plans include extensions of my previous results, suggestions about other directions to take my research, and possible ideas for undergraduate research projects. After I presented my research at the 30th Annual Summer Symposium in Real Analysis at the University of North Carolina at Asheville, Associate Professor Manav Das of the University of Louisville suggested that I try to extend Theorem S2 to include fractals that don t have Hutchinson s open set condition. Specifically, Professor Das was interested in Julia Sets, which are overlapping fractals obtained by looking at attractors of points (see [F], chapter 14). Currently, I have been to able to make some progress with overlapping fractals over R, using a theorem of Falconer s cited in Matilla s book (see [M], p ), and I would like to extend the result to more general overlapping fractals, including Julia Sets. Also I would like to investigate results of these measures on random fractals (see [F], chapter 15), which are fractals obtained by looking at a probability distribution in terms of direction. The idea of tubular neighborhoods in Theorem S1 are included in the definition of Minkowski Content, which we now define, (taken from [Fe], section ).

6 6 RESEARCH STATEMENT-ERIC SAMANSKY Definition. For S R n and each integer m with 0 m n, the m-dimensional Minkowski Content is M m L n {x : dist(x, S) < δ} (S) = lim, δ 0+ α(n m)δ n m where L n is n-dimensional Lebesgue measure and α(n m) is the n m-dimensional volume of the n m-dimensional unit ball. Possibly, the places where the Gibbs measure concentrates depends on the Minkowski Content, and I would like to find a connection between the two. Finally, I plan to check numerical results for different types of sets and different types of measures. Some of this work could include undergraduates. In the VIGRE program at Rice University, I have had experience teaching undergraduates advanced concepts, such as Hausdorff dimension. For possible projects tied to my research, the undergraduates could check measures on interesting sets, and try to extend Theorem S3 on targets (specifically, where the resulting measure was split between a point mass and a circle). We could investigate sets where the measure could be split between sets in zero-, one-, and two-dimensions, try to generalize the result so that the measure would be split between any number of dimensions, and investigate the consequences this would have on the measure. References [CHK] D. Cox, R. Hardt, and P. Klouček, Convergence of Gibbs measures associated with simulated annealing, submitted to SIAM, J. Mathematical Analysis, [F] K.J. Falconer, Fractal geometry: mathematical foundations and applications, John Wiley & Sons, [Fe] H. Federer, Geometric measure theory, Springer, Berlin Heidelberg New York, [GKV] C.D. Gelatt, S. Kirkpatrick, and M.P. Vecchi, Optimization by simulated annealing, Science 220 (1983), [H] R. Hardt, Stratification via corank 1 projections, Singularities, Proc. Symp. in Pure Math., 40 (1983), [Hu] J. E. Hutchinson, Fractals and self similarity, Indiana Univ. Math. J. 30 (1981), [M] P. Matilla, Geometry of sets and measures in Euclidean spaces: Fractals and rectifiability, Cambridge Univ. Press, Cambridge, [Mo] F. Morgan, Geometric measure theory: a beginner s guide, Academic Press, [S] E. Samansky, Convergence of Gibbs measures and the behavior of shrinking tubular neighborhoods of fractals and algebraic sets, Ph.D Thesis, Rice University, in preparation.