Past

I present some recent work with Bruce Driver and Christian Litterer on rough paths 'constrained’ to lie in a d - dimensional submanifold of a Euclidean space E. We will present a natural definition for this class of rough paths and then describe the (second) order geometric calculus which arises out of this definition. The talk will conclude with more advanced applications, including a rough version of Cartan’s development map.

Trees are not just combinatorial structures: they are also

biological structures, both in the obvious way but also in the

study of evolution. Starting from DNA samples from living

species, biologists use increasingly sophisticated mathematical

techniques to reconstruct the most likely “phylogenetic tree”

describing how these species evolved from earlier ones. In their

work on this subject, they have encountered an interesting

example of an operad, which is obtained by applying a variant of

the Boardmann–Vogt “W construction” to the operad for

commutative monoids. The operations in this operad are labelled

trees of a certain sort, and it plays a universal role in the

study of stochastic processes that involve branching. It also

shows up in tropical algebra. This talk is based on work in

progress with Nina Otter [www.fair-fish.ch].

The splitting-up method is a powerful tool to solve (SP)DEs by dividing the equation into a set of simpler equations that are easier to handle. I will speak about how such splitting schemes can be derived and extended by insights from the theory of rough paths.

Finally, I will discuss numerics for real-world applications that appear in the management of risk and engineering applications like nonlinear filtering.

We explore the technique of elementary submodels to prove 
results in topology and set theory. We will in particular prove the 
delta system lemma, and Arhangelskii's result that a first countable 
Lindelof space has cardinality not exceeding continuum.

Nature and the world of human technology are full of
networks. People like to draw diagrams of networks: flow charts,
electrical circuit diagrams, signal flow diagrams, Bayesian networks,
Feynman diagrams and the like. Mathematically-minded people know that
in principle these diagrams fit into a common framework: category
theory. But we are still far from a unified theory of networks.

The aim of this lecture is to give a general introduction to

the interacting particle system and applications in finance, especially

in the pricing of American options. We survey the main techniques and

results on Snell envelope, and provide a general framework to analyse

these numerical methods. New algorithms are introduced and analysed

theoretically and numerically.

This talk will give an introduction to generalized complex geometry, where complex and symplectic structures are particular cases of the same structure, namely, a generalized complex structure. We will also talk about a sister theory, generalized complex geometry of type Bn, where generalized complex structures are defined for odd-dimensional manifolds as well as even-dimensional ones.

In this work, we want to construct the solution $(Y,Z,K)$ to the following BSDE

$$\begin{array}{l}

Y_t=\xi+\int_t^Tf(s,Y_s,Z_s)ds-\int_t^TZ_sdB_s+K_T-K_t, \quad 0\le t\le T, \\

{\mathbf E}[l(t, Y_t)]\ge 0, \quad 0\le t\le T,\\

\int_0^T{\mathbf E}[l(t, Y_t)]dK_t=0, \\

\end{array}

$$

where $x\mapsto l(t, x)$ is non-decreasing and the terminal condition $\xi$

is such that ${\mathbf E}[l(T,\xi)]\ge 0$.

This equation is different from the (classical) reflected BSDE. In particular, for a solution $(Y,Z,K)$,

we require that $K$ is deterministic. We will first study the case when $l$ is linear, and then general cases.

We also give some application to mathematical finance. This is a joint work with Philippe Briand and Romuald Elie.

In Parkinson’s disease, increased power of oscillations in firing rate has been observed throughout the cortico-basal-ganglia circuit. In

particular, the excessive oscillations in the beta range (13-30Hz) have been shown to be associated with difficulty of movement initiation. However, on the basis of experimental data alone it is difficult to determine where these oscillations are generated, due to complex and recurrent structure of the cortico-basal-ganglia-thalamic circuit. This talk will describe a mathematical model of a subset of basal-ganglia that is able to reproduce experimentally observed patterns of activity. The analysis of the model suggests where and under which conditions the beta oscillations are produced.

For Logic Seminar: Note change of time and location!

This talk will focus on extremizers for

a family of Fourier restriction inequalities on planar curves. It turns

out that, depending on whether or not a certain geometric condition

related to the curvature is satisfied, extremizing sequences of

nonnegative functions may or may not have a subsequence which converges

to an extremizer. We hope to describe the method of proof, which is of

concentration compactness flavor, in some detail. Tools include bilinear

estimates, a variational calculation, a modification of the usual

method of stationary phase and several explicit computations.

I will explain why one can symplectically embed closed symplectic manifolds (with integral symplectic form) into CPⁿ and compute the weak homotopy type of the space of all symplectic embeddings of such a symplectic manifold into CP^∞.

The classical small cancellation theory goes back to the 1950's and 1960's when the geometry of 2-complexes with a unique 0-cell was studied, i.e. the standard 2-complex of a finite presentation. D.T. Wise generalizes the Small Cancellation Theory to 2-complexes with arbitray 0-cells showing that certain classes of Small Cancellation Groups act properly discontinuously and cocompactly on CAT(0) Cube complexes and hence have codimesion 1-subgroups. To be more precise I will introduce "his" version of small Cancellation Theory and go roughly through the main ideas of his construction of the cube complex using Sageeve's famous construction. I'll try to make the ideas intuitively clear by using many pictures. The goal is to show that B(4)-T(4) and B(6)-C(7) groups act properly discontinuously and cocompactly on CAT(0) Cube complexes and if there is time to explain the difficulty of the B(6) case. The talk should be self contained. So don't worry if you have never had heard about "Small Cancellation".

In this talk, I will explain part of the programme of Gorenstein, Lyons

and Solomon (GLS) to provide a new proof of the CFSG. I will focus on

the difference between the initial notion of groups of characteristic

$2$-type (groups like Lie type groups of characteristic $2$) and the GLS

notion of groups of even type. I will then discuss work in progress

with Capdeboscq to study groups of even type and small $2$-local odd

rank. As a byproduct of the discussion, a picture of the structure of a

finite simple group of even type will emerge.

Compressed sensing and matrix completion are techniques by which simplicity in data can be exploited for more efficient data acquisition. For instance, if a matrix is known to be (approximately) low rank then it can be recovered from few of its entries. The design and analysis of computationally efficient algorithms for these problems has been extensively studies over the last 8 years. In this talk we present a new algorithm that balances low per iteration complexity with fast asymptotic convergence. This algorithm has been shown to have faster recovery time than any other known algorithm in the area, both for small scale problems and massively parallel GPU implementations. The new algorithm adapts the classical nonlinear conjugate gradient algorithm and shows the efficacy of a linear algebra perspective to compressed sensing and matrix completion.

Several objects can be associated to a list of vectors with integer coordinates: among others, a family of tori called toric arrangement, a convex polytope called zonotope, a function called vector partition function; these objects have been described in a recent book by De Concini and Procesi. The linear algebra of the list of vectors is axiomatized by the combinatorial notion of a matroid; but several properties of the objects above depend also on the arithmetics of the list. This can be encoded by the notion of a "matroid over Z". Similarly, applications to tropical geometry suggest the introduction of matroids over a discrete valuation ring.Motivated by the examples above, we introduce the more general notion of a "matroid over a commutative ring R". Such a matroid arises for example from a list of elements in a R-module. When R is a Dedekind domain, we can extend the usual properties and operations holding for matroids (e.g., duality). We can also compute the Tutte-Grothendieck ring of matroids over R; the class of a matroid in such a ring specializes to several invariants, such as the Tutte polynomial and the Tutte quasipolynomial. We will also outline other possible applications and open problems. (Joint work with Alex Fink).

In 1992 (or thereabouts) Bloch and Kriz gave the first explicit definition of the category of mixed Tate motives (MTM). Their definition relies heavily on the theory of algebraic cycles. Unfortunately, traditional methods of representing algebraic cycles (such as in terms of formal linear combinations of systems of polynomial equations) are notoriously difficult to work with, so progress in capitalizing on this description of the category to illuminate outstanding conjectures in the field has been slow. More recently, Gangl, Goncharov, and Levin suggested a simpler way to understand this category (and by extension, algebraic cycles more generally) by relating specific algebraic cycles to rooted, decorated, planar trees. In our talks, describing work in progress, we generalize this correspondence and attempt to systematize the connection between algebraic cycles and graphs. We will construct a Lie coalgebra L from a certain algebra of admissible graphs, discuss various properties that it satisfies (such as a well defined and simply described realization functor to the category of mixed Hodge structures), and relate the category of co-representations of L to the category MTM. One promising consequence of our investigations is the appearance of alternative bases of rational motives that have not previously appeared in the literature, suggesting a richer rational structure than had been previously suspected. In addition, our results give the first bounds on the complexity of computing admissibility of algebraic cycles, a previously unexplored topic.

When applied to an inequality constrained optimization problem, interior point methods generate iterates that belong to the interior of the set determined by the constraints, thus avoiding/ignoring the combinatorial aspect of the solution. This comes at the cost of difficulty in predicting the optimal active constraints that would enable termination.  We propose the use of controlled perturbations to address this challenge. Namely, in the context of linear programming with only nonnegativity constraints as the inequality constraints, we consider perturbing the nonnegativity constraints so as to enlarge the feasible set. Theoretically, we show that if the perturbations are chosen appropriately, the solution of the original problem lies on or close to the central path of the perturbed problem and that a primal-dual path-following algorithm applied to the perturbed problem is able to predict the optimal active-set of the original problem when the duality gap (for the perturbed problem) is not too small. Encouraging preliminary numerical experience is obtained when comparing the perturbed and unperturbed interior point algorithms' active-set predictions for the purpose of cross-over to simplex.