(Click here for the schedule)


Conic integral geometry: from simple principles to useful applications

Integral geometry is a classical subject with roots that date back at to the mid-19th century. Despite this long history, the spherical/conic case has received little attention on its own, although this setting is more natural and useful (but often more difficult to analyze). We will build up this theory from simple principles and explain the resulting kinematic calculus, which in this conic form achieves a remarkable simplicity. We will also explain some interesting connections to the theory of hyperplane arrangements and to random matrix theory, and if time permits discuss some intriguing open problems.


Topological complexity of real and complex varieties.

Upper bounds on topological invariants  (such as the Betti numbers) of real varieties in terms of degrees and the dimension of the ambient space are usually smaller than the corresponding bounds in the complex case. The same holds true in random algebraic geometry as well where it is usual that the expected value of some topological invariant (such as the sum of the Betti numbers) is at most (and often the square root) of the corresponding value in the complex case. In this talk I will explain certain situations in which this picture is inverted. These are all deterministic examples, but potentially could be studied from a random viewpoint as well.


Determinantal point processes and energy in the n-sphere.

A method to generate points in a manifold M is to choose a subspace of L^2(M) and use the determinantal point process associated to its projection kernel. This process usually has nice properties, mainly related to the mutual repulsion of the random points. In this talk I will present some determinantal point processes in the n-sphere that allow us to improve the bounds on the very famous energy minimization problem (find N points in the sphere whose energy - electrostatic potential or generalizations - is minimal).

All the elementary concepts needed to understand the methods and the results will be presented during this talk.




Nodal domains and percolation, comments, remarks, and questions

The talk is a mini review of connections between nodal domain problems for Gaussian random functions and critical non-correlated percolation.

Differences between universal and non-universal relations are highlighted.


Critical points and Euler-Poincaré characteristic of random spherical harmonics.

We discuss the limiting distribution and the asymptotic fluctuations of the number of critical points of random Gaussian spherical harmonics in the high energy limit. Our results require a careful investigation of the validity of the Kac-Rice formula in nonstandard circumstances. We also establish a Quantitative Central Limit Theorem (in Wasserstein distance) for the Euler-Poincaré Characteristic of excursion sets.  

Based on joint works with Domenico Marinucci and Igor Wigman.  


Random Functions and Fundamental Cosmology

I will survey three critical questions in modern cosmology i) Is the effective initial state of the universe set by a phase of accelerated expansion (“inflation”) immediately after the Big Bang and, if so, what are the likely values of associated cosmological observables, ii) what mechanism sets the value of the cosmological constant (or “dark energy”) observed in the present epoch and iii) is the vacuum state that defines the dark energy stable against quantum tunnelling via the nucleation of expanding bubbles of lower energy vacua? in many cosmological scenarios, the answers to all three questions depend on the properties of a complicated, multidimensional potential which can be treated as an (effectively) random function. I will discuss how these questions can, in principle, be related to statements about the extrema and saddles of random functions and summarise recent progress on this topic.




Percolation of random nodal lines

In this talk, I will present a percolation phenomenon for random nodal lines: for any connected bounded open set U in the real plane and two arcs g and g' in its boundary, there exists c>0 such that for any L>0, with probability at least c, there exists a connected component of the zero set of a random function f, joining Lg to Lg' in LU. Here f belongs to a natural infinite dimensional space of real analytic functions related to the Bargmann-Fock space. This is a joint work with Vincent Beffara.


How many equilibria will a large complex system have?

In 1972 Robert May argued that generic complex systems equipped with stability feedback mechanisms become unstable to small displacements from equilibria as the system complexity (as measured by the interaction strength and the number of degrees of freedom) increases. In search of a global signature of this local instability, we consider a class of nonlinear dynamical systems whereby N degrees of freedom are coupled via a smooth homogeneous Gaussian vector field with both gradient and divergence-free components. Our analysis shows such systems undergo an abrupt change from a simple set of equilibria (typically, a single equilibrium for N large) to a complex set of equilibria, with their total number growing exponentially with N. This suggests that the local instability transition manifests itself on the global scale in an exponential explosion in the number of equilibria. Our analysis utilises the Kac-Rice formula for counting zeros of random fields and invokes statistical properties of the elliptic ensemble of real asymmetric matrices. [This is joint work with Yan Fyodorov]


Random lemniscates

A (rational) lemniscate is the level set of the modulus of a rational function. While sampling from an ensemble of random lemniscates that is invariant under rotations of the Riemann sphere, we study basic geometric and topological properties. This quickly leads to simple-to-state questions that take a fresh viewpoint on classical themes. For instance, what is the average (spherical) length of a random lemniscate? How many connected components are there on average and how are they arranged in the plane?  We discuss answers to these questions.  This is joint work with Antonio Lerario.


Top eigenvalue of a random matrix: Tracy-Widom distribution and third order phase

Tracy-Widom distribution describes the probability distribution of the typical fluctuations of the top eigenvalue of a Gaussian (NxN) random matrix. Over the  last decade, the same distribution has surfaced in a wide variety of problems from KPZ surface growth, directed polymer, random permutations, all the way to large $N$ gauge theory and wireless communications, with some of these problems having no apriori connection to random matrices. Why is the Tracy-Widom distribution so ubiquitous? In statistical physics, universality is usually accompanied by a phase transition--near a critical point often the details become completely irrelevant. So, is there an underlying phase transition associated with the Tracy-Widom distribution? In this talk, I will demonstrate that for large but finite N, indeed there is an underlying third order phase transition from a `strong' coupling to a `weak' coupling phase--the Tracy-Widom distribution turns out to be the universal crossover function between these two phases for finite but large N. Several examples of this third order phase transition will be discussed.


On sparse polynomials, toric varieties, condition length and randomness.

A space of sparse complex polynomials can be specified through a monomial basis and a Hermitian inner product. It will be argued that if one fixes a n-tuple of such spaces then the common roots belong to a certain toric variety. This variety admits a canonical Finsler structure to be used in the definition of condition numbers. I shall then report on a recent theorem relating the algorithmic complexity of path-following to the condition length. Open questions about the condition number of a random systems will be discussed at the end.


Statistics of Axion Diameters in Calabi-Yau Compactifications

An important problem in early universe cosmology is to determine whether quantum gravity permits a primordial gravitational wave signal that is intense enough to be seen in the cosmic microwave background.  This is equivalent to determining whether field space diameters larger than the Planck mass are possible in quantum gravity.  I will describe an approach to this problem in compactifications of string theory on Calabi-Yau manifolds.  The field space in question is an N-dimensional polytope, and the metric is a constant matrix determined by the intersection numbers and rigid divisors of the Calabi-Yau.  I will examine the statistics of axion diameters through a random matrix model and through enumeration of explicit threefolds.  I will give a partial answer to the question ``what is the expected number of rigid divisors in a Calabi-Yau threefold?''  This result serves as a key input to the program of studying metastability in the landscape of string theory.  


Critical points of random smooth functions: central limit theorems

I will  discuss  the distributions of   critical points of    Gaussian random smooth functions on Euclidean spaces or flat tori, and  show  that when a certain small parameter goes to zero, the number  of critical points  in a given region  satisfies a central limit theorem.




A Broader Class of Measures for Faster Solving

We first point out some restricted but fundamentally important settings where the recent positive solution to Smale's 17th Problem can be considerably improved: For certain highly sparse systems, one can find an approximate root (with constrained  phase) in average-case time polynomial in n + log d, where n is the number  of variables and d is the maximum degree of any input polynomial. Moreover,  one can even allow fat tails for the probability distributions of the  coefficients.  Sub-Gaussian distributions and certain discrete distributions  are thus allowed.

We then consider the case where monomial terms are no longer forbidden, and give new, more flexible estimates for the condition       numbers of random polynomial systems. In this ``non-sparse'' setting, we use techniques that avoid the use of unitary invariance, thus again allowing sub-Gaussian measures. This is joint work with Alperen Ergur,  Grigoris Paouris, and Kaitlyn Phillipson.


Expected Number and Height Distribution of Critical Points of Smooth Isotropic Gaussian Random Fields

We obtain formulae for the expected number and height distribution of critical points of general smooth isotropic Gaussian random fields parameterized on Euclidean space or spheres of arbitrary dimension. The results extend those of Fyodorov (2004) and hold in general in the sense that there are no restrictions on the covariance function of the field except for smoothness and isotropy. The results are based on a characterization of the distribution of the Hessian of the Gaussian field by means of the family of Gaussian orthogonally invariant (GOI) matrices, of which the Gaussian orthogonal ensemble (GOE) is a special case. This is joint work with Dan Cheng.


Real eigenvalues of real random matrices and statistics of spherical spin glasses

I will discuss results on the fluctuations of critical point counts, and connections with the real eigenvalues of non-Hermitian random matrices. Specifically, I will use this connection to present numerical work on the fluctuations of the number of critical points in the spherical p=2 spin glass model with external field. I will describe how this count is identical to the number of real eigenvalues of a certain non-Hermitian random matrix. This is joint work with D. Mehta, J. D. Hauenstein, M. Niemerg and D. A. Stariolo. On the analytical side, I will present my recent work on the distribution of the number of real eigenvalues in a simpler version of this model, namely the real Ginibre ensemble. The main results of the latter are central limit theorems for any even polynomial linear statistic of the real eigenvalues.



Real Zeroes of Random Polynomials (after Ken Soze)

In the talk, I will present two recent discoveries of Ken Soze posted in arxiv.

His first theorem states that with high probability the number of real zeroes of a univariate random polynomial is bounded by the number of vertices on its Newton-Hadamard polygon times the cube of the logarithm of the polynomial degree. The proof of this

theorem is based on the tools from complex and harmonic analysis.

His second theorem says that the expected number of real zeroes of univariate random polynomials with real independent identically distributed coefficients (or more generally, exchangeable coefficients) does not grow faster than the logarithm of the degree. This solves a long-standing open problem. The main ingredients in his proof of this result are Descartes' rule of signs and a new anti concentration inequality for the symmetric group.


Landscape and dynamical complexity in two simple statistical physics model

We compare some energy landscape characteristics, like the number and index  of critical points, for two well known models from statistical physics: a model with a "complex" landscape, but without quenched disorder and a model with a "simple" energy landscape, but with quenched disorder. In the first case, the system has an exponentially large number of stationary points, while the second, "simpler" system, has a number of stationary points that is proportional to the size of the system. We observe that landscape complexity does not lead, by itself, to complex relaxational dynamics. The role of averaging over quenched disorder and the large systems size limit in the ultimate behavior of statistical models is addressed in an attempt to understand the  basic links between complex energy landscapes and relaxational dynamics.


Topologies of nodal sets of random band limited functions

It is shown that the topologies and nestings of the zero and nodal sets of random (Gaussian) band limited functions have universal laws of distribution. Qualitative features of the supports of these distributions are determined. In particular the results apply to random monochromatic waves and to random real algebraic hyper-surfaces in projective space.