Let X be a complex algebraic variety containing a point x. One of the central ideas of deformation theory is that the local structure of X near the point x can be encoded by a differential graded Lie algebra. In this talk, Jacob Lurie will explain this idea and discuss some generalizations to more exotic contexts.
One of the big ideas in linear algebra is {\em eigenvalues}. Most matrices become in some basis {\em diagonal} matrices; so a lot of information about the matrix (which is specified by $n^2$ matrix entries) is encoded by by just $n$ eigenvalues. The fact that lots of different matrices can have the same eigenvalues reflects the fact that matrix multiplication is not commutative.
I'll look at how to make these vague statements (``lots of different matrices...") more precise; how to extend them from matrices to abstract symmetry groups; and how to relate abstract symmetry groups to matrices.
The vast majority of the World's documented meteorite specimens have been collected from Antarctica. This is due to Antarctica’s ice dynamics, which allows for the significant concentration of meteorites onto ice surfaces known as Meteorite Stranding Zones. However, meteorite collection data shows a significant anomaly exists: the proportion of iron-based meteorites are under-represented compared to those found in the rest of the World. Here I explain that englacial solar warming provides a plausible explanation for this shortfall: as meteorites are transported up towards the surface of the ice they become exposed to increasing amounts of solar radiation, meaning it is possible for meteorites with a high-enough thermal conductivity (such as iron) to reach a depth at which they melt their underlying ice and sink back downwards, offsetting the upwards transportation. An enticing consequence of this mechanism is that a sparse layer of meteorites lies just beneath the surface of these Meteorite Stranding Zones...
Mining massive amounts of sequentially ordered data and inferring structural properties is nowadays a standard task (in finance, etc). I will present some results that combine and extend ideas from rough paths and machine learning that allow to give a general non-parametric approach with strong theoretical guarantees. Joint works with F. Kiraly and T. Lyons.
T cells are important white blood cells that continually circulate in the body in search of the molecular signatures ('antigens') of infection and cancer. We (and many other labs) are trying to construct models of the T cell signalling network that can be used to predict how ligand binding (at the surface of the cell) controls gene express (in the nucleus). To do this, we stimulate T cells with various ligands (input) and measure products of gene expression (output) and then try to determine which model must be invoked to explain the data. The challenge that we face is finding 1) unique models and 2) scaling the method to many different input and outputs.
The computation of multidimensional persistent homology is one of the major open problems in topological data analysis.
One can define r-dimensional persistent homology to be a functor from the poset category N^r, where N is the poset of natural numbers, to the category of modules over a commutative ring with identity. While 1-dimensional persistent homology is theoretically well-understood and has been successfully applied to many real-world problems, the theory of r-dimensional persistent homology is much harder, as it amounts to understanding representations of quivers of wild type.
In this talk I will introduce persistent homology, give some motivation for how it is related to the study of data, and present recent results related to the classification of multidimensional persistent homology.
The concept of pseudofinite dimension for ultraproducts of finite structures was introduced by Hrushovski and Wagner. In this talk, I will present joint work with D. Macpherson and C. Steinhorn in which we explored conditions on the (fine) pseudofinite dimension that guarantee simplicity or supersimplicity of the underlying theory of an ultraproduct of finite structures, as well as a characterization of forking in terms of droping of the pseudofinite dimension. Also, under a suitable assumption, it can be shown that a measure-theoretic condition is equivalent to loc
Let $a_1, \cdots, a_N$ be the sequence of y-smooth numbers up to x (i.e. composed only of primes up to y). When y is a small power of x, what can one say about the size of the gaps $a_{j+1}-a_j$? In particular, what about
$$\sum_1^N (a_{j+1}-a_j)^2?$$
We focus on the mathematical structure of systemic risk measures as proposed by Chen, Iyengar, and Moallemi (2013). In order to clarify the interplay between local and global risk assessment, we study the local specification of a systemic risk measure by a consistent family of conditional risk measures for smaller subsystems, and we discuss the appearance of phase transitions at the global level. This extends the analysis of spatial risk measures in Föllmer and Klϋppelberg (2015).
Our everyday usage of the Internet generates huge amounts of data on how humans collect and exchange information worldwide. In this talk, I will outline recent work in which we investigate whether data from sources such as Google, Wikipedia and Flickr can be used to gain new insight into real world human behaviour. I will provide case studies from a range of domains, including disease detection, crowd size estimation, and evaluating whether the beauty of the environment we live in might affect our health.
For linear dynamical systems, model reduction has achieved great success. In the case of linear dynamics, we know how to construct, at a modest cost, (locally) optimal, input-independent reduced models; that is, reduced models that are uniformly good over all inputs having bounded energy. In addition, in some cases we can achieve this goal using only input/output data without a priori knowledge of internal dynamics. Even though model reduction has been successfully and effectively applied to nonlinear dynamical systems as well, in this setting, bot the reduction process and the reduced models are input dependent and the high fidelity of the resulting approximation is generically restricted to the training input/data. In this talk, we will offer remedies to this situation.
We will explore the many guises under which groups of 2x2 matrices appear, such as isometries of the hyperbolic plane, mapping class groups and the modular group. Along the way we will learn some interesting and perhaps surprising facts.
A Steiner Triple System on a set X is a collection T of 3-element subsets of X such that every pair of elements of X is contained in exactly one of the triples in T. An example considered by Plücker in 1835 is the affine plane of order three, which consists of 12 triples on a set of 9 points. Plücker observed that a necessary condition for the existence of a Steiner Triple System on a set with n elements is that n be congruent to 1 or 3 mod 6. In 1846, Kirkman showed that this necessary condition is also sufficient. In 1974, Wilson conjectured an approximate formula for the number of such systems. We will outline a proof of this
conjecture, and a more general estimate for the number of Steiner systems. Our main tool is the technique of Randomised Algebraic Construction, which
we introduced to resolve a question of Steiner from 1853 on the existence of designs.
I will present a definition of symplectic homology groups for pairs of Liouville cobordisms with fillings, and explain how these fit into a formalism of homology theory similar to that of Eilenberg and Steenrod. This construction allows to understand form a unified point of view many structural results involving Floer homology groups, and yields new applications. Joint work with Kai Cieliebak.
We purify and generalize some techniques which were successful in the limit theory of graphs and other discrete structures. We demonstrate how this technique can be used for solving different combinatorial problems, by defining the limit problems of the Manickam--Miklós--Singhi Conjecture, the Kikuta–Ruckle Conjecture and Alpern's Caching Game.