# Seminars & Events for 2015-2016

##### Polynomials and (finite) free probability

Recent work of the speaker with Dan Spielman and Nikhil Srivastava introduced the ``method of interlacing polynomials'' (MOIP) for solving problems in combinatorial linear algebra.

##### Spectral theory of the Laplacian and number theory

In this talk, I will discuss the interplay between the geometry of Riemannian manifolds and properties of eigenfunctions and eigenvalues of the Laplacian. Of particular interest is the relationship between the chaotic behavior of the manifold, in terms of ergodic geodesic flow, and chaotic behaviour of eigenfunctions.

##### Expansion for simplicial complexes

For graphs, the notion of expansion is a highly useful concept that has found applications in various areas, especially in combinatorics and theoretical computer science. In recent years, the success of this concept has inspired the search for a corresponding notion in higher dimensions.

##### Spectral summation formulae and their applications

Starting from the Poisson summation formula, I discuss spectral summation formulae on GL(2) and GL(3) and present a variety of applications to automorphic forms, analytic number theory, and arithmetic.

##### Fixed-point expressions for the Fukaya endomorphism algebra of RP^{2m} and higher genus open invariants

The Atiyah-Bott localization formula has become a valuable tool for computation of symplectic invariants given in terms of integrals on the moduli spaces of holomorphic stable maps.

##### Minimal surfaces with bounded index

We prove a structural theorem that provides a precise local picture of how a sequence of closed embedded minimal surfaces with bounded index on a given three-manifold might degenerate. We then discuss several applications, including some compactness results. (This is joint work with O. Chodosh and D. Ketover)

##### Time-Periodic Einstein--Klein--Gordon Bifurcations Of Kerr

For an open measure set of Klein--Gordon masses mu^2 > 0, we construct one-parameter families of solutions to the Einstein--Klein--Gordon equations bifurcating off the Kerr solution such that the underlying family of spacetimes are each an asymptotically flat, stationary, axisymmetric, black hole spacetime, and such that the corresponding scalar fields are non-zero and time-periodic.

##### Reduced Order Models for the numerical modelling of complex systems

This talk will address the challenge of complexity in numerical simulations of complex physical problems.When the latter exhibit a multiscale and or a multiphysics nature, appropriate mathematical models and accurate numerical methods are required to catch the essential features of the manifold components of the physical solution.

##### Min-max theory and least area minimal hypersurfaces

Minimal hypersurfaces are one important way to understand the structure of the ambient manifold and min-max theory is a powerful method for constructing them. I will present the basic ideas of min-max theory, a theorem of Calabi-Cao as an application and a possible extension of the latter.

##### Directions in hyperbolic and Euclidean lattices

It is well known that the orbit of a lattice in hyperbolic n-space is uniformly distributed when projected radially onto the unit sphere. I consider the fine-scale statistics of the projected lattice points and express the limit distributions in terms of random hyperbolic lattices.

##### The chromatic number of graphs without long holes

Andras Gyarfas conjectured in 1985 that for all k and t, every graph with sufficiently large chromatic number contains either a complete subgraph on k vertices or an induced cycle of length at least t. We prove this conjecture and discuss some related results. Joint work with Maria Chudnovsky and Paul Seymour.

##### Twisted Alexander polynomials and knot concordance

Fox and Milnor showed that the classical Alexander polynomial of a knot K in S^3 can obstruct K from bounding a smooth disk in B ^4 . Twisted Alexander polynomials provides further information about four-dimensional aspects of classical knots. This information is most easily formulated in terms of the knot concordance group.

##### On the Averaged Colmez Conjecture

**PLEASE NOTE ROOM CHANGE FOR THIS DATE ONLY: FINE 224. **The Colmez conjecture expresses the Faltings height of a CM abelian variety in terms of the logarithmic derivatives of certain Artin L-functions. In this talk, I will present an averaged version of the conjecture proved in my joint work with Shou-Wu Zhang.

##### Special Colloquium: Bayesian Inversion for Functions and Geometry

Please note special day (Thursday). Many problems in the physical sciences require the determination of an unknown function from a finite set of indirect measurements. Examples include oceanography, medical imaging, oil recovery, water resource management and weather forecasting. Furthermore there are numerous inverse problems where geometric characteristics, such as interfaces, are key unknown features of the overall inversion. Applications include the determination of layers and faults within subsurface formations, and the detection of unhealthy tissue in medical imaging. We describe a theoretical and computational Bayesian framework relevant to the solution of inverse problems for functions, and for geometric features. We formulate Bayes' theorem on separable Banach spaces, a conceptual approach which leads to a form of probabilistic well-posedness and also to new and efficient MCMC algorithms which exhibit order of magnitude speed-up over standard methodologies. Furthermore the approach can be developed to apply to geometric inverse problems, where the geometry is parameterized finite-dimensionally and, via the level-set method, to infinite-dimensional parameterizations. In the latter case this leads to a well-posedness that is difficult to achieve in classical level-set inversion, but which follows naturally in the probabilistic setting.

[1] A.M. Stuart. Inverse problems: a Bayesian perspective. Acta Numerica 19(2010) 451--559. http://homepages.warwick.ac.uk/~masdr/BOOKCHAPTERS/stuart15c.pdf

[2] M. Dashti and A.M. Stuart. The Bayesian approach to inverse problems.To appear in Handbook of Uncertainty Quantification, Springer, 2016. http://arxiv.org/abs/1302.6989

[3] S.L.Cotter, G.O.Roberts, A.M. Stuart and D. White, MCMC methods for functions: modifying old algorithms to make them faster. Statistical Science, 28 (2013) 424-446. http://homepages.warwick.ac.uk/~masdr/JOURNALPUBS/stuart103.pdf

[4] M.A. Iglesias, K. Lin, A.M. Stuart, "Well-Posed Bayesian Geometric Inverse Problems Arising in Subsurface Flow", Inverse Problems, 30 (2014) 114001. http://arxiv.org/abs/1401.5571

[5] M.A. Iglesias, Y. Lu, A.M. Stuart, "A level-set approach to Bayesian geometric inverse problems", submitted. http://arxiv.org/abs/1504.00313

##### Bayesian Inversion for Functions and Geometry

**This is a joint seminar with the PACM Colloquium. Please note special location. **Many problems in the physical sciences require the determination of an unknown function from a finite set of indirect measurements. Examples include oceanography, medical imaging, oil recovery, water resource management and weather forecasting.

##### Krylov-Evans type theorem for twisted Monge-Amp\'ere equations

Motivated by the pluriclosed flow of Streets and Tian, we establish Evans-Krylov type estimates for parabolic "twisted" Monge-Ampere equations in both the real and complex setting. In particular, a bound on the second derivatives on solutions to these equations yields bounds on Holder norms of the second derivatives.

##### Compactness questions for triholomorphic maps

A triholomorphic map u between hyperKahler manifolds solves the "quaternion del-bar" equation du=I du i + J du j + K du k. Such a map turns out, under suitable assumptions, to be stationary harmonic. We focus on compactness issues regarding the quantization of the Dirichlet energy and the structure of the blow-up set.

##### Harmonic analysis and intrinsic Diophantine approximation

We will describe a recently developed general approach to Diophantine approximation on homogeneous algebraic varieties, and demonstrate it in some familiar natural examples. The approach utilizes harmonic analysis on Lie groups and some arguments in homogeneous dynamics and ergodic theory.

##### The Resilience of the Perceptron

The most widely used optimization method in machine learning practice is the Perceptron Algorithm, also known as the Stochastic Gradient Method (SGM). This method has been used since the fifties to build statistical estimators, iteratively improving models by correcting errors observed on single data points. SGM is not only scalable, robust, and simple to implement, but achieves the state-of-th

##### Quantum graphs and Neumann networks

Quantum graph models are extremely useful but they also have some drawbacks. One of them concerns the physical meaning of the vertex coupling. The self-adjointness requirement leaves a substantial freedom expressed through parameters appearing in the conditions matching the wave function at the graph vertices.