What is the smallest formula computing a given multivariate polynomial f(x)= In this talk I will present a paradigm for translating the known lower bound proofs for various subclasses of formulas into efficient proper learn= ing algorithms for the same subclass. Many lower bounds proofs for various subclasses of arithmetic formulas redu= ce the problem to showing that any expression for f(x) as a sum of =93simpl= e=94 polynomials T_i(x): f(x) =3D T_1(x) + T_2(x) + =85 + T_s(x), the number s of simple summands is large. For example, each simple summand = T_i could be a product of linear forms or a power of a low degree polynomia= l and so on. The lower bound consists of constructing a vector space of linear maps M, e= ach L in M being a linear map from the set of polynomials F[x] to some vect= or space W (typically W is F[X] itself) with the following two properties: (i) For every simple polynomial T, dim(M*T) is small, say = that dim(M*T) <=3D r. (ii) For the candidate hard polynomial f, dim(M*f) is large,= say that dim(M*f) >=3D R. These two properties immediately imply a lower bound: s >=3D R/r. The corresponding reconstruction/proper learning problem is the following: = given f(x) we want to find the simple summands T_1(x), T_2(x), =85, T_s(x) = which add up to f(x). We will see how such a lower bound proof can often be used to solve the rec= onstruction problem. Our main tool will be an efficient algorithmic solutio= n to the problem of decomposing a pair of vector spaces (U, V) under the simu= ltaneous action of a vector space of linear maps from U to V. Along the way we will also obtain very precise bounds on the size of formul= as computing certain explicit polynomials. For example, we will obtain for = every s, an explicit polynomial f(x) that can be computed by a depth three formula of size s but= not by any depth three formula of size (s-1). Based on joint works with Chandan Saha and Ankit Garg.
Random band matrices (RBM) are natural intermediate models to study eigenvalue statistics and quantum propagation in disordered systems, since they interpolate between mean-field type Wigner matrices and random Schrodinger operators. In particular, RBM can be used to model the Anderson metal-insulator phase transition (crossover) even in 1d. In this talk we will discuss some recent progress in application of the supersymmetric method (SUSY) and transfer matrix approach to the analysis of local spectral characteristics of some specific types of 1d RBM.
Understanding stability of ecosystems and communities has always been major challenge for ecologists. Definitions and measures of stability abound and at times are confusing. Nowadays it is in general accepted that stability is multidimensional and it needs to be measured in different ways. Some of the metrics are used to highlight resistance of ecological systems to a specific type of perturbations (like an invasion of an alien). Others have been developed to highlight the approach to tipping points (that is catastrophic transitions between different dynamical states). As long-term data become increasingly available and experimental approaches are improving, the challenge is how to apply our theoretical metrics on these ecological dynamics to understand stability. In the talk, I will present a possible way for identifying best suitable metrics for measuring stability in ecological communities. More in depth, I will also focus on how changes in dynamical properties of ecological dynamics can be used as early warnings to abrupt ecological changes using examples from ecology and the climate.
One of the features that distinguishes biological systems is the wide range of scales in time and space on which processes and interactions occur. This is a form of complexity, but it is one that can sometimes be turned into an advantage. I will describe models for a couple of systems where my collaborators and I have found that to be the case. The first (from long ago) is a system with ladybugs preying on aphids. The ladybugs (which are highly mobile but reproduce slowly) experience the environment as a system of patches, while the aphids (which are much less mobile but reproduce quickly) experience each patch as spatial continuum. The second (more recent) is a system aimed at describing the evolution of dispersal. Dispersal starts with the movement of individuals, which can be observed by tracks or tracking and described in terms of random walks. That then produces spatial patterns, which then influence ecological interactions within and among populations. Those in turn exert selective pressure on traits that determine the spatial patterns, and finally the selective pressure together with the occasional the appearance of mutants results in the evolution of dispersal traits. All of these processes can, in some cases, operate on different scales in time and space. It turns out that this when this occurs it can be exploited to produce relatively simple models in some situations. The older research I will discuss was conducted in collaboration with Steve Cantrell; the newer was with Steve Cantrell, Mark Lewis, and Yuan Lou