Past

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.