Past

Bayesian optimization (BO) is a powerful tool for sequentially optimizing black-box functions that are expensive to evaluate, and has extensive applications including automatic hyperparameter tuning, environmental monitoring, and robotics. The problem of level-set estimation (LSE) with Gaussian processes is closely related; instead of performing optimization, one seeks to classify the whole domain according to whether the function lies above or below a given threshold, which is also of direct interest in applications.

In this talk, we present a new algorithm, truncated variance reduction (TruVaR) that addresses Bayesian optimization and level-set estimation in a unified fashion. The algorithm greedily shrinks a sum of truncated variances within a set of potential maximizers (BO) or unclassified points (LSE), which is updated based on confidence bounds. TruVaR is effective in several important settings that are typically non-trivial to incorporate into myopic algorithms, including pointwise costs, non-uniform noise, and multi-task settings. We provide a general theoretical guarantee for TruVaR covering these phenomena, and use it to obtain regret bounds for several specific settings. We demonstrate the effectiveness of the algorithm on both synthetic and real-world data sets.

I review work done in collaboration with Siegel and Zwiebach,  in which a doubled geometry is developed that provides a spacetime  action containing the standard gravity theory for graviton, Kalb-Ramond field and dilaton plus higher-derivative corrections. In this framework the T-duality O(d,d) invariance is manifest and exact to all orders in $\alpha'$.  This theory by itself does not correspond to a standard string theory, but it does encode the Green-Schwarz deformation characteristic of heterotic string theory  to first order in $\alpha'$ and a Riemann-cube correction to second order in  $\alpha'$. I outline how this theory may be extended to include arbitrary string theories. 

This is will be a progress report on our long-ongoing joint work with Bezrukavnikov on lifting the monodromy of the quantum differential equation for symplectic resolutions to automorphisms of their derived categories of coherent sheaves. I will attempt to define the ingredient that go both into the problem and into its solution.
 

Following an idea of Gaiotto, a symplectic representation of a complex Lie group G defines a complex Lagrangian subvariety inside the moduli space of G-Higgs bundles. The talk will discuss the case of G=SL(2) and its link with determinant  divisors, or equivalently Brill-Noether loci, in the moduli space of semistable SL(2)-bundles. 

Given a (-1)-shifted symplectic derived scheme or stack (X,w) over C equipped with an orientation, we explain how to construct a perverse sheaf P on the classical truncation of X so that its hypercohomology H*(P) can be regarded as a categorification of (or linearisation of) X. Given also a Lagrangian morphism L -> X equipped with a relative orientation, we outline a programme in progress to construct a natural morphism of constructible complexes on the truncation of L from the (shifted) constant complex on L to a suitable pullback of P to L. The morphisms and resulting hypercohomology classes are expected to satisfy various identities under products, composition of Lagrangian correspondences, etc. This programme will have interesting applications, such as proving associativity of a Kontsevich-Soibelman type COHA multiplication on H*(P) when X is the derived moduli stack of coherent sheaves on a Calabi-Yau 3-fold Y, and defining Lagrangian Floer cohomology and the Fukaya cat!
 egory of an algebraic or complex symplectic manifold S.

How can we explain the patterns of genetic variation in the world around us? The genetic composition of a population can be changed by natural selection, mutation, mating, and other genetic, ecological and evolutionary mechanisms. How do they interact with one another, and what was their relative importance in shaping the patterns we see today? 

Whereas the pioneers of the field could only observe genetic variation indirectly, by looking at traits of individuals in a population, researchers today have direct access to DNA sequences. But making sense of this wealth of data presents a major scientific challenge and mathematical models play a decisive role. This lecture will distil our understanding into workable models and explore the remarkable power of simple mathematical caricatures in interrogating modern genetic data.

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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.

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.

Continuation of the last talk.

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).