Past

Van den Dries has proved the decidability of the ring of algebraic integers, the integral closure of the ring of integers in
the algebraic closure of the rationals.  A well-established analogy leads to ask the same question for the ring of complex polynomials.
This turns out to go the other way, interpreting the rational field.    An interesting structure on the
limit of Jacobians of all complex curves is encountered along the way. 

What can fashionable ideas, blind faith, or pure fantasy have to do with the scientific quest to understand the universe? Surely, scientists are immune to trends, dogmatic beliefs, or flights of fancy? In fact, Roger Penrose argues that researchers working at the extreme frontiers of mathematics and physics are just as susceptible to these forces as anyone else. In this lecture, based on his new book, Roger will argue that fashion, faith, and fantasy, while sometimes productive and even essential, may be leading today's researchers astray, most notably in three of science's most important areas - string theory, quantum mechanics, and cosmology. Yet Roger will also describe how fashion, faith, and fantasy have, ironically, also been invaluable in shaping his own work.

Roger will be signing copies of his book after the lecture.

This lecture is now SOLD OUT. Any questions, please email: @email

Let $F$ be a binary form of degree $d \geq 3$ with integer coefficients and non-zero discriminant. In this talk we give an asymptotic formula for the quantity $R_F(Z)$, the number of integers in the interval $[-Z,Z]$ representable by the binary form $F$.

This is joint work with C.L. Stewart.

We introduce a novel stochastic volatility model where the squared volatility of the asset return follows a Jacobi process. It contains the Heston model as a limit case. We show that the the joint distribution of any finite sequence of log returns admits a Gram--Charlier A expansion in closed-form. We use this to derive closed-form series representations for option prices whose payoff is a function of the underlying asset price trajectory at finitely many time points. This includes European call, put, and digital options, forward start options, and forward start options on the underlying return. We derive sharp analytical and numerical bounds on the series truncation errors. We illustrate the performance by numerical examples, which show that our approach offers a viable alternative to Fourier transform techniques. This is joint work with Damien Ackerer and Damir Filipovic.

Graham Benham

The Fluid Mechanics of Low-Head Hydropower Illuminated by Particle Image Velocimetry

We study a new type of hydropower which is cost-effective in rivers and tides where there are small pressure drops. The concept goes as follows: The cost of water turbines scales with the flow rate they deal with.  Therefore, in order to render this hydropower desirable, we make use of the Venturi principle, a natural fluid mechanical gear system which involves splitting the flow into two streams. The turbine deals with a small fraction of the flow at slow speed and high pressure, whilst the majority avoids the turbine, going at high speed and low pressure. Now the turbine feels an amplified pressure drop, thus maintaining its power output, whilst becoming much cheaper. But it turns out that the efficiency of the whole system depends strongly on the way in which these streams mix back together again.

Here we discuss some new experimental results and compare them to a simplified mathematical model for the mixing of these streams. The experimental results were achieved using particle image velocimetry (PIV), which is a type of flow visualisation. Using a laser sheet and a high speed camera, we are able to capture flow velocity fields at high resolution. Pressure measurements were also taken. The mathematical model is derived from the Navier Stokes equations using boundary layer theory alongside a flow-averaging method and reduces the problem to solving a set of ODE’s for the bulk components of the flow.

Nabil Fadai

Asymptotic Analysis of a Multiphase Drying Model Motivated by Coffee Bean Roasting

Recent modelling of coffee bean roasting suggests that in the early stages of roasting, within each coffee bean, there are two emergent regions: a dried outer region and a saturated interior region. The two regions are separated by a transition layer (or drying front). In this talk, we consider the asymptotic analysis of a multiphase model of this roasting process which was recently put forth and studied numerically, in order to gain a better understanding of its salient features. The model consists of a PDE system governing the thermal, moisture, and gas pressure profiles throughout the interior of the bean. Obtaining asymptotic expansions for these quantities in relevant limits of the physical parameters, we are able to determine the qualitative behaviour of the outer and interior regions, as well as the dynamics of the drying front. Although a number of simplifications and scaling are used, we take care not to discard aspects of the model which are fundamental to the roasting process. Indeed, we find that for all of the asymptotic limits considered, our approximate solutions faithfully reproduce the qualitative features evident from numerical simulations of the full model. From these asymptotic results we have a better qualitative understanding of the drying front (which is hard to resolve precisely in numerical simulations), and hence of the various mechanisms at play as heating, evaporation, and pressure changes result in a roasted bean. This qualitative understanding of solutions to the multiphase model is essential if one is to create more involved models that incorporate chemical reactions and solid mechanics effects.

We present global rates of convergence for a general class of methods for nonconvex smooth optimization that include linesearch, trust-region and regularisation strategies, but that allow inaccurate problem information. Namely, we assume the local (first- or second-order) models of our function are only sufficiently accurate with a certain probability, and they can be arbitrarily poor otherwise. This framework subsumes certain stochastic gradient analyses and derivative-free techniques based on random sampling of function values. It can also be viewed as a robustness
assessment of deterministic methods and their resilience to inaccurate derivative computation such as due to processor failure in a distribute framework. We show that in terms of the order of the accuracy, the evaluation complexity of such methods is the same as their counterparts that use deterministic accurate models; the use of probabilistic models only increases the complexity by a constant, which depends on the probability of the models being good. Time permitting, we also discuss the case of inaccurate, probabilistic function value information, that arises in stochastic optimization. This work is joint with Katya Scheinberg (Lehigh University, USA).
 

Linear algebra is a widely used tool both in mathematics and computer science, and cryptography is no exception to this rule. Yet, it introduces some particularities, such as dealing with linear systems that are often sparse, or, in other words, linear systems inside which a lot of coefficients are equal to zero. We propose to enlarge this notion to nearly sparse matrices, caracterized by the concatenation of a sparse matrix and some dense columns, and to design an algorithm to solve this kind of problems. Motivated by discrete logarithms computations on medium and high caracteristic finite fields, the Nearly Sparse Linear Algebra briges the gap between classical dense linear algebra problems and sparse linear algebra ones, for which specific methods have already been established. Our algorithm particularly applies on one of the three phases of NFS (Number Field Sieve) which precisely consists in finding a non trivial element of the kernel of a nearly sparse matrix.

'Erdős-Ko-Rado type problems' are well-studied in extremal combinatorics; they concern the sizes of families of objects in which any two (or any $r$) of the objects in the family 'agree', or 'intersect', in some way.

If $X$ is a finite set, the '$p$-biased measure' on the power-set of $X$ is defined as follows: choose a subset $S$ of $X$ at random by including each element of $X$ independently with probability $p$. If $\mathcal{F}$ is a family of subsets of $X$, one can consider the $p$-biased measure of $\mathcal{F}$, denoted by $\mu_p(\mathcal{F})$, as a function of $p$. If $\mathcal{F}$ is closed under taking supersets, then this function is an increasing function of $p$. Seminal results of Friedgut and Friedgut-Kalai give criteria under which this function has a 'sharp threshold'. Perhaps surprisingly, a careful analysis of the behaviour of this function also yields some rather strong results in extremal combinatorics which do not explicitly mention the $p$-biased measure - in particular, in the field of Erdős-Ko-Rado type problems. We will discuss some of these, including a recent proof of an old conjecture of Frankl that a symmetric three-wise intersecting family of subsets of $\{1,2,\ldots,n\}$ has size $o(2^n)$, and some 'stability' results characterizing the structure of 'large' $t$-intersecting families of $k$-element subsets of $\{1,2,\ldots,n\}$. Based on joint work with (subsets of) Nathan Keller, Noam Lifschitz and Bhargav Narayanan.

We define categorical matrix factorizations in a suspended additive category, 
with respect to a central element. Such a factorization is a sequence of maps 
which is two-periodic up to suspension, and whose composition equals the 
corresponding coordinate map of the central element. When the category in 
question is that of free modules over a commutative ring, together with the 
identity suspension, then these factorizations are just the classical matrix 
factorizations. We show that the homotopy category of categorical matrix 
factorizations is triangulated, and discuss some possible future directions. 
This is joint work with Dave Jorgensen.

The big phase space is an infinite dimensional manifold which is the arena
for topological quantum field theories and quantum cohomology (or
equivalently, dispersive integrable systems). tt*-geometry was introduced by
Cecotti and Vafa and is a way to introduce an Hermitian structure on what
would be naturally complex objects, and the theory has many links with
singularity theory, variation of Hodge structures, Higgs bundles, integrable
systems etc.. In this talk the two ideas will be combined to give a
tt*-geometry on the big phase space.

(joint work with Liana David)

The large sieve is a powerful analytic tool in number theory, with many beautiful and diverse applications. In its most general form it resembles an approximate Bessel's inequality, and this clear modern theory rests on the combined effort of countless mathematicians in the mid-twentieth century -- Linnik, Roth, Selberg, Montgomery, Vaughan, and Bombieri, to name a few. However, it is hardly obvious to the beginner why this rather abstract inequality should be called 'large', or 'sieve'. In this introductory talk, aimed particularly at new graduate students, we discuss the rudimentary theory of the large sieve, some particular applications to sieving problems, and (at least one) proof. 

We will give the results on the models of thin plates and rods in nonlinear elasticity by doing simultaneous homogenization and dimensional reduction. In the case of bending plate we are able to obtain the models only under periodicity assumption and assuming some special relation between the periodicity of the material and thickness of the body. In the von K\'arm\'an regime of rods and plates and in the bending regime of rods we are able to obtain the models in the general non-periodic setting. In this talk we will focus on the derivation of the rod model in the bending regime without any assumption on periodicity.

 
The dimension of a finite-dimensional vector space V can be computed as the trace of the identity endomorphism id_V.  This dimension is also the value F_V(S^1) of the circle in the 1-dimensional field theory F_V associated to the vector space.  The trace of any endomorphism f:V-->V can be interpreted as the value of that field theory on a circle with a defect point labeled by the endomorphism f.  This last invariant makes sense even when the vector space is infinite-dimensional, and gives the trace of a trace-class operator on Hilbert space.  We introduce a 2-dimensional analog of this invariant, the `2-trace'.  The 2-dimension of a finite-dimensional separable k-algebra A is the dimension of the center of the algebra.  This 2-dimension is also the value F_A(S^1 x S^1) of the torus in the 2-dimensional field theory F_A associated to the algebra. Given a 2-endomorphism p of the algebra (that is an element of the center), the 2-trace of p is the value of the field theory on a torus with a defect point labeled by p.  Generalizations of this invariant to other defect configurations make sense even when the algebra is not finite-dimensional or separable, and this leads to a general notion of 2-trace class and 2-trace in any 2-category.  This is joint work with Andre Henriques.

We study the small-time fluctuations for diffusion processes which are conditioned by their initial and final positions and whose diffusivity has a sub-Riemannian structure. In the case where the endpoints agree, we discuss the convergence of the suitably rescaled fluctuations to a limiting diffusion loop, which is equal in law to the loop we obtain by taking the limiting process of the unconditioned rescaled diffusion processes and condition it to return to its starting point. The generator of the unconditioned limiting rescaled diffusion process can be described in terms of the original generator.

We review the concept of solitons in the Ricci flow, and describe various methods for generating examples, including some where the equations

may be solved in closed form

Malliavin calculus provides a framework to differentiate functionals defined on a Gaussian probability space with respect to the underlying noise. This allows to develop analysis on path space with infinite-dimensional generalisations of Fourier analysis, Sobolev spaces, etc from R^d. In this talk, we attempt to build a Lipschitz à la E. M. Stein (as opposed to Sobolev) function theory on rough path space. This framework allows to pathwise differentiate functionals on rough paths with respect to the underlying rough path. Time permitting, we show how to obtain Feynman-Kac-type representations for solutions to some high-order (>2) linear parabolic equations on R^d.

TBC

Pure spinor space, a cone over orthogonal Grassmannian OGr(5,10), is a central concept in the Berkovits formulation of string theory. The space of states of the beta-gamma system on pure spinors is tensor factor in the Hilbert space of string theory . This is why it would be nice to have a good definition of this space of states. This is not a straightforward task because of the conical singularity of the target. In the talk I will explain a strategy for attacking  conical targets. In the case of pure spinors the method gives a formula for partition function of pure spinors.

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.

To book please email @email

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).

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.