Math Colloquia

In this talk I will present mathematical advances achieved on a reflector design problem. Due to its applications in electro-magnetics and optics, the reflector design is a very practical problem, and has been extensively studied in engineering. Mathematically it can be reduced to the solvability of a fully nonlinear partial differential equation of Monge-Ampere type, subject to a second boundary condition. We give sharp conditions that guarantee the smoothness of the weak solutions as well as applications to the far-field problem. This is a joint work with Xu-Jia Wang.

Ding-Geiges proved that any contact 3-manifold can be obtained by surgery along a knot in the standard contact 3-sphere. Giroux showed that any contact manifold can be identified with an open book decomposition. In this talk, I will explain how braid theory in an open book decomposition can be an effective tool for classifying transverse knots in a contact manifold. In particular, I will explain how, using this tool, we can better understand the self-linking number which is a fundamental transverse knot invariant.

The cubic hypersurfaces are special algebraic varieties with rich geometry, whose study led to significant developments in algebraic geometry. In this talk I will do a survey of what is known about the moduli of cubics, with a special emphasis on the case of cubic fourfolds.

An old theorem of Chern, Hirzebruch and Serre asserts that the signature of closed oriented manifolds is multiplicative in fiber bundles with trivial monodromy action (i.e., bundles for which the fundamental group of the base acts trivially on the cohomology of the fiber). The contribution of monodromy to the signature of a fiber bundle was later described by Atiyah. In this talk I will survey various extensions of these results to the singular setting, and discuss parametrized versions of them in the complex algebraic context. The talk will be suitable for a general audience.

(joint work with Richard Oberlin and Terence Tao) Last year, Zeev Dvir proved an old conjecture in combinatorial geometry/ discrete harmonic analysis / additive number theory, the "Kakeya conjecture over finite fields:" a subset of F_q^n containing a line in every direction has size at most c_n q^n, where c_n is a constant independent of q. This problem was posed by Wolff as an analogy to a still-unsolved conjecture in harmonic analysis, which holds that a subset of R^n containing a unit line segment in every direction has Minkowski and Lebesgue measure n. Dvir's proof is surprisingly easy once you have the idea -- so easy I can present it in full in this talk. I'll explain Dvir's "polynomial method," and describe how we combine the method with other analytic tools in order to prove a finite field analogue of the more general Kakeya maximal function conjecture on an arbitrary subvariety of affine space. If time permits, I'll talk about some further variants of the conjecture (restriction to linear subvarieties of dimension greater than 1, rings more general than F_p, etc.) I intend the talk to be intelligible to non-analysts, especially myself. Terry's blog post is an excellent introduction to the circle of ideas.

A covering space provides an "unwrapping" of a manifold M. Lifting the dynamics generated by a function f to a covering space often clarifies and simplifies its study. For example, motion around loops in the manifold become translations in a covering space. The character of the lifted dynamics is intimately connected with the action of f on the fundamental group of M and its various quotients. The talk will start with a brief history, going back to Poincaré, of the uses of covering spaces in dynamics and then will focus mainly on the interplay between lifted dynamics, base dynamics and the action on first homology in the study of Thurston's pseudoAnosov homeomorphisms on surfaces.

Let (X,L) be a polarized algebraic manifold. I have recently proved that the Mabuchi energy of (X,L) is bounded from below along any degeneration if and only if the Hyperdiscriminant polytope contains the Chow polytope (with respect to the various Kodaira embeddings). This completes the analysis initiated by Ding and Tian in their 1992 Inventiones paper "Kahler Einstein metrics and the Generalized Futaki Invariant", and therefore gives a new perspective on Tian's fundamental notion of K-semistability.

The integrated information theory (Tononi 2004) starts from phenomenology and makes use of thought experiments to claim that consciousness is integrated information. First: the quantity of consciousness corresponds to the amount of integrated information generated by a system of elements. Information is quantified by taking the current state as a measurement the system performs on itself, which specifies a repertoire of prior states that cause (lead to) the current state. Integrated information (phi) is quantified by computing the repertoire specified by the system as a whole relative to the repertoires specified independently by its parts. Second: the quality of an experience is completely specified by the set of informational relationships generated within that system. The set of all repertoires generated by subsystems of a system is represented in a geometric object, the quale. Informational relationships between points in the quale characterize how the measurements resulting from interactions in the system give structure to a particular experience. After describing the theory in some detail, I will discuss how several neurobiological observations fall naturally into place in the framework: the association of consciousness with certain neural systems rather than with others; the fact that neural processes underlying consciousness can influence or be influenced by neural processes that remain unconscious; and the reduction of consciousness during dreamless sleep and generalized seizures. Furthermore, features of the quale can be related to features of conscious experience, such as modalities and submodalities, and can explain the distinct roles of different cortical subsystems in affecting the quality of experience.

Consider a classical 2D gravity wave (studied by Euler, Poisson, Cauchy, Airy, Stokes, Levi-Civita,...) with an arbitrary vorticity function. Consider such a wave traveling at a constant speed over a flat bed. Using local and global bifurcation theory and topological degree following the ideas of Paul Rabinowitz, one can prove that there exist many such waves of large amplitude. I will outline the existence proof (joint with Adrian Constantin) and also exhibit some recent computations (with Joy Ko) of the waves using numerical continuation. The computations illustrate certain relationships between the amplitude, energy and mass flux of the waves. If the vorticity is sufficiently large, the first stagnation point of the wave occurs not at the crest (as with the much-studied irrotational flows) but on the bed directly below the crest or else in the interior of the fluid.

Mark Kac's question "Can you hear the shape of a drum?" asks the extent to which the geometry of a plane domain, viewed as a vibrating membrane, is encoded in the Dirichlet eigenvalue spectrum of the associated Laplacian, equivalently, in the characteristic frequencies of vibration. More generally, one asks the extent to which spectral data on a Riemannian manifold encodes the geometry. We will survey techniques for constructing manifolds with the same spectral data and look at examples in order to identify geometric invariants that are not spectrally determined.

In many applications, one often has fewer equations than unknowns. While this seems hopeless, the premise that the object we wish to recover is sparse or nearly sparse radically changes the problem, making the search for solutions feasible. This lecture will introduce sparsity as a key modeling tool, and introduce a series of little miracles touching on many areas of data processing. These examples show that finding that solution to an underdetermined system of linear equations with minimum L1 norm, often returns the "right" answer.

One of the central tenets of signal processing and data acquisition is the Shannon/Nyquist sampling theory: the number of samples needed to capture a signal is dictated by its bandwidth. This talk introduces a novel sampling or sensing theory which goes against this conventional wisdom. This theory now known as Compressed Sensing or Compressive Sampling’’ allows the faithful recovery of signals and images from what appear to be highly incomplete sets of data, i.e. from far fewer measurements or data bits than traditional methods use. We will present the key ideas underlying this new sampling or sensing theory, and will survey some of the most important results. We will emphasize the practicality and the broad applicability of this technique, and discuss what we believe are far reaching implications; e.g. procedures for sensing and compressing data simultaneously and much faster. Finally, there are already many ongoing efforts to build a new generation of sensing devices based on compressed sensing and we will discuss remarkable recent progress in this area as well.

This talk considers a problem of considerable practical interest: the recovery of a data matrix from a sampling of its entries. In partially filled out surveys, for instance, we would like to infer the many missing entries. In the area of recommender systems, users submit ratings on a subset of entries in a database, and the vendor provides recommendations based on the user's preferences. Because users only rate a few items, we would like to infer their preference for unrated items (this is the famous Netflix problem). Formally, suppose that we observe m entries selected uniformly at random from a matrix. Can we complete the matrix and recover the entries that we have not seen? We show that perhaps surprisingly, one can recover low-rank matrices exactly from what appear to be highly incomplete sets of sampled entries; that is, from a comparably small number of entries. Further, perfect recovery is possible by solving a simple convex optimization program, namely, a convenient semidefinite program (SDP). This result hinges on powerful techniques in probability theory. We will also present a very efficient algorithm based on iterative singular value thresholding, which can complete matrices with about a billion entries in a matter of minutes on a personal computer.

Our perceptual systems (visual and auditory) are confronted with data in very high dimensional spaces. Yet we are able to learn how to recognize faces, objects, phonemes, words, and so on without running into the "curse of dimensionality". How might we get machines to replicate this ability? What might be plausible principles of learning in high dimensional spaces and what is its relevance to biological learning? I will explore these questions with a geometric point of view. My central thesis is this: in high dimensional spaces, natural data occupies a tiny sliver of the space --- most of the ambient space is empty. The geometric structure of the data allows us to build intrinsic and invariant representations, to define suitable classes of functions with which to operate, and ultimately to learn effectively. This point of view motivates new geometrically oriented learning algorithms, new theoretical questions that surround their analysis, and new models and metaphors to reason about perceptual systems.

When can we say two things are the "same?" What, if anything, does this imply about their being "different?" The idea of a category -- a set of objects sharing common properties -- is a fundamental concept in many fields, including mathematics, artificial intelligence, and cognitive and neuroscience. Numerous frameworks, for example, in machine learning and linguistics, rest upon the simple presumption that categories are well-defined. This is slightly worrisome, as the many attempts formalizing categories have met with equally many attempts shooting them down. Instead of approaching this issue head on, I derive a robust theory of "similarity," from a biologically-inspired approach to perception in animals. The very idea of creating categories assumes some implicit notion of similarity, but it is rarely examined in isolation. However, doing so is worthwhile, and I demonstrate the theory's applicability to a variety of natural and artificial learning problems. Even when faced with Wantanbe's "Ugly Duckling" theorem or Wolpert's stingy cafeteria (serving the famous "No Free Lunch" theorems), one can make significant progress toward formalizing a theory of categories by examining their often unstated properties. I demonstrate practical applications of this work in several domains, including unsupervised machine learning, ensemble clustering, image segmentation, human acquisition of language, and cognitive neuroscience. (Joint work with M.H.Ansari)

In polyhedral combinatorices, there is (1) the famous max flow - min cut theorem. There is also the almost as famous (2) shortest path - max cut packing (it's the theory behind Dijkstra's shortest path algorithm). I proved over the years a generalization of (1), a generalization of (2), and that the "blocking" relation between (1) and (2) also holds for the generalizations. There is a (not well known) greedy algorithm for certain linear programming problems which suggests that further generalizatons are possible, but I have no idea how to formulate the potential generalizations.

The irreducible representations of the symmetric group are indexed by partitions. Those representations which remain both irreducible and projective upon reduction modulo p are indexed by p-cores: those partition from which no rim hooks (or ribbons) of size p can be removed. They also correspond to cosets of the finite symmetric group in the affine symmetric group, equivalently to the root lattice, or to the extremal vectors in the basic representation of the affine Lie algebra sl_p. One can also ask which representations remain irreducible (but not necessarily projective) when reduced mod p, or more generally, which modules for the related Hecke algebra stay irreducbile on specialization to a p-th root of unity. This was answered by Fayers in general, and James-Mathas for those modules not equal to their own radical (the p-regular partitions). We give here two new combinatorial descriptions of those partitions, which naturally generalize the description of p-cores. The first is related to the spin of corresponding ribbon tableaux. The second is given in the context of crystal graphs. We also enumerate such partitions, a key ingredient for which is a bijection between p-cores and (p-1)-cores that has several lovely combinatorial descriptions. This is joint work with Chris Berg and Brant Jones.

Machine learning studies the principles governing all learning systems. Human beings and animals are learning systems too, and can be explored using the same mathematical tools. This approach has been fruitful in the last few decades with standard tools such as reinforcement learning, artificial neural networks, and non-parametric Bayesian statistics. We bring the approach one step further with some latest tools in machine learning, and uncover new quantitative findings. In this talk, I will present three examples: (1) Human semi-supervised learning. Consider a child learning animal names. Dad occasionally points to an animal and says "Dog!" (labeled data). But mostly the child observes the world by herself without explicit feedback (unlabeled data). We show that humans learn from both labeled and unlabeled data, and that a simple Gaussian Mixture Model trained using the EM algorithm provides a nice fit to human behaviors. (2) Human active learning. The child may ask "What's that?", i.e. actively selecting items to query the target labels. We show that humans are able to perform good active learning, achieving fast exponential error convergence as predicted by machine learning theory. In contrast, when passively given i.i.d. training data humans learn much slower (polynomial convergence), also predicted by learning theory. (3) Monkey online learning. Rhesus monkeys can learn a "target concept", in the form of a certain shape or color. What if the target concept keeps changing? Adversarial online learning model provides a polynomial mistake bound. Although monkeys perform worse than theory, anecdotal evidence suggests that they follow the concepts better than some graduate students. Finally, I will speculate on a few lessons learned in order to create better machine learning algorithms.

Biological systems make use of feedback in an extraordinary number of ways,on scales ranging from molecules to cells to organisms to ecosystems. In this talk I will discuss the use of concepts from control and dynamical systems in the analysis and design of biological feedback circuits at the molecular level. After a brief survey of advances in synthetic biology, I will present two recent results that combine modeling, identification, design and experimental implementation of biological feedback circuits. In the first example, I will describe the use of intrinsic noise for system identification in transcriptional regulatory networks and its role in identifying active interconnections in genetic networks. The second set of results is in the area of molecular programming, and the development of an in vitro circuit for regulating the rates of transcription of two independent genetic sequences. Finally, using these results as examples, I will discuss some of the open problems and research challenges in the area feedback control using biological circuits.