I will discuss the application of Zariski geometries to Mordell Lang, and review the main ideas which are used in the interpretation of a field, given the assumption of non local modularity. I consider some open problems in adapting Zilber's construction to the case of minimal types in separably closed fields.
If two L-functions are added together, the Euler product is destroyed.
Thus the linear combination is not an L-function, and hence we should
not expect a Riemann Hypothesis for it. This is indeed the case: Not
all the zeros of linear combinations of L-functions lie on the
critical line.
However, if the two L-functions have the same functional equation then
almost all the zeros do lie on the critical line. This is not seen
when they have different functional equations.
We will discuss these results (which are due to Bombieri and Hejhal)
during the talk, and demonstrate them using characteristic polynomials
of random unitary matrices, where similar phenomena are observed. If
the two matrices have the same determinant, almost all the zeros of
linear combinations of characteristic polynomials lie on the unit
circle, whereas if they have different determinants all the zeros lie
off the unit circle.
Systems with delays frequently appear in engineering. The presence of delays makes system analysis and control design very complicated. In this talk, the standard H-infinity control problem of time-delay systems will be discussed. The emphasis will be on systems having an input or output delay. The problem is solved in the frequency domain via reduction to a one-block problem and then further to an extended Nehari problem using a simple and intuitive method. After solving the extended Nehari problem, the original problem is solved. The solvability of the extended Nehari problem (or the one-block problem) is equivalent to the nonsingularity of a delay-dependent matrix and the solvability conditions of the standard H-infinity control problem with a delay are then formulated in terms of the existence of solutions to two delay-independent algebraic Riccati equations and a delay-dependent nonsingular matrix.
In this talk, I will discuss various aspects of approximation by radial basis functions on spheres. After a short introduction to the subject of scattered data approximation on spheres and optimal recovery, I will particularly talk about error analysis, a hybrid approximation scheme involving polynomials and radial basis functions and, if time permits, solving nonlinear parabolic equations on spheres.
Discrete problems have a habit of being beautiful but difficult. This can be true even of discrete problems whose continuous analogues are easy. For example: computing the surface area of a sphere of radius N^{1/2} in k-dimensional Euclidean space (easy). Counting the number of representations of an integer N as a sum of k squares (historically hard). In this talk we'll survey a menagerie of discrete analogues of operators arising in harmonic analysis, including singular integral operators (such as the Hilbert transform), maximal functions, and fractional integral operators. In certain cases we can learn everything we want to know about the discrete operator immediately, from its continuous analogue. In other cases the discrete operator requires a completely new approach. We'll see what makes a discrete operator easy/hard to treat, and outline some of the methods that are breaking new ground, key aspects of which come from number theory. In particular, we will highlight the roles played by theta functions, exponential sums, Waring's problem, and the circle method of Hardy and Littlewood. No previous knowledge of singular integral operators or the circle method will be assumed.
In euclidean space there is a well-known parallelogram law relating the
length of vectors a, b, a+b and a-b. In the talk I give a similar formula
for translation lengths of isometries of CAT(0)-spaces. Given an action of
the automorphism group of a free product on a CAT(0)-space, I show that
certain elements can only act by zero translation length. In comparison to
other well-known actions this leads to restrictions about homomorphisms of
these groups to other groups, e.g. mapping class groups.
This talk will summarize some of the problems and conjectures that I haven't managed to solve (although I have tried to) while spending my three years in this job. It will cover the areas of group theory, representation theory, both of general finite groups and of symmetric groups, and fusion systems.
In joint work with Tim Perutz, we give a complete characterization of the Fukaya category of the punctured torus, denoted by $Fuk(T_0)$. This, in particular, means that one can write down an explicit minimal model for $Fuk(T_0)$ in the form of an A-infinity algebra, denoted by A, and classify A-infinity structures on the relevant algebra. A result that we will discuss is that no associative algebra is quasi-equivalent to the model A of the Fukaya category of the punctured torus, i.e., A is non-formal. $Fuk(T_0)$ will be connected to many topics of interest: 1) It is the boundary category that we associate to a 3-manifold with torus boundary in our extension of Heegaard Floer theory to manifolds with boundary, 2) It is quasi-equivalent to the category of perfect complexes on an irreducible rational curve with a double point, an instance of homological mirror symmetry.
(Note change in time and location)
The purpose of this talk is to give an introduction to the theory and
practice of integer factorization. More precisely, I plan to talk about the
p-1 method, the elliptic curve method, the quadratic sieve, and if time
permits the number field sieve.
Bacteria are ubiquitous on Earth and perform many vital roles in addition to being responsible for a variety of diseases. Locomotion allows the bacterium to explore the environment to find nutrient-rich locations and is also crucial in the formation of large colonies, known as biofilms, on solid surfaces immersed in the fluid. Many bacteria swim by turning corkscrew-shaped flagella. This can be studied computationally by considering hydrodynamic forces acting on the bacterium as the flagellum rotates. Using a boundary element method to solve the Stokes flow equations, it is found that details of the shape of the cell and flagellum affect both swimming efficiency and attraction of the swimmer towards flat no-slip surfaces. For example, simulations show that relatively small changes in cell elongation or flagellum length could make the difference between an affinity for swimming near surfaces and a repulsion. A new model is introduced for considering elastic behaviour in the bacterial hook that links the flagellum to the motor in the cell body. This model, based on Kirchhoff rod theory, predicts upper and lower bounds on the hook stiffness for effective swimming.
There have been significant progress in the calculation of scattering amplitudes in N=4 SYM. In this talk I will consider `form factors', which are defined not only with on-shell asymptotic states but also with one off-shell operator inserted. Such quantities are in some sense the hybrid of on-shell quantities (such
as scattering amplitudes) and off-shell quantities (such as correlation functions). We will see that form factors inherit many nice properties of scattering amplitudes, in particular we will consider their supersymmetrization and the dual picture.
In many applications it is of interest to compute minimizers of
a functional I(u) which is the of the form $J(u)=\Phi(u)+R(u)$,
with $R(u)$ quadratic. We describe a stochastic algorithm for
this problem which avoids explicit computation of gradients of $\Phi$;
it requires only the ability to sample from a Gaussian measure
with Cameron-Martin norm squared equal to $R(u)$, and the ability
to evaluate $\Phi$. We show that, in an appropriate parameter limit,
a piecewise linear interpolant of the algorithm converges weakly to a noisy
gradient flow. \\
Joint work with Natesh Pillai (Harvard) and Alex Thiery (Warwick).
In the 1990's Haagerup discovered a new subfactor, and hence a new topological quantum field theory, that has so far proved inaccessible by the methods of quantum groups and conformal field theory. It was the subfactor of smallest index beyond 4. This led to a classification project-classify all subfactors to as large an index as possible. So far we have gone as far as index 5. It is known that at index 6 wildness phenomena occur which preclude a simple listing of all subfactors of that index. It is possible that wildness occurs at a smaller index value, the main candidate being approximately 5.236.
Abstract: In the 1990's Haagerup discovered a new subfactor, and hence a new topological quantum field theory, that has so far proved inaccessible by the methods of quantum groups and conformal field theory. It was the subfactor of smallest index beyond 4. This led to a classification project-classify all subfactors to as large an index as possible. So far we have gone as far as index 5. It is known that at index 6 wildness phenomena occur which preclude a simple listing of all subfactors of that index. It is possible that wildness occurs at a smaller index value, the main candidate being approximately 5.236.
The moduli space of cubic surfaces is known to be isomorphic to a quotient of the unit ball in C^4 by an arithmetic
group. We review this construction, then explain how to construct
an explicit inverse to the period map by using suitable theta functions. This gives a new proof of the isomorphism between the two spaces.
Duality methods can be very powerful tools for the analysis of stochastic
processes. However, there seems to be no general theory available
yet. In this talk, I will discuss and aim to clarify various notions
of duality, give some recent rather striking examples (applied to
stochastic PDEs, interacting particle systems and combinatorial stochastic
processes)
and try to give some systematic insight into the type of questions
that can in principle be tackled. Finally, I will try to provide you
with some intuition for this fascinating technique.
Given a deterministically time-changed Brownian motion Z starting from 1, whose time-change V (t) satisfies $V (t) > t$ for all $t>=0$, we perform an explicit construction of a process X which is Brownian motion in its own filtration and that hits zero for the first time at V (s), where $s:= inf {t > 0 : Z_t = 0}$. We also provide the semimartingale decomposition of $X >$ under
the filtration jointly generated by X and Z. Our construction relies on a combination of enlargement of filtration and filtering techniques. The resulting process X may be viewed as the analogue of a 3-dimensional Bessel bridge starting from 1 at time 0 and ending at 0 at the random time $V (s)$.
We call this a dynamic Bessel bridge since V(s) is not known in advance. Our study is motivated by insider trading models with default risk.(this is a joint work with Luciano Campi and Umut Cetin)
We will give a quick and dirty introduction to Gromov-Witten theory and discuss some integrality properties of GW invariants. We will start by briefly recalling some basic properties of the Deligne Mumford moduli space of curves. We will then try to define GW invariants using both algebraic and symplectic geometry (both definitions will be rather sloppy, but hopefully the basic idea will become visible), talk a bit about the axiomatic definition due to to Kontsevich and Manin, and discuss some applications like quantum cohomology. Finally, we will talk a bit about integrality and the Gopakumar-Vafa conjecture. Just as a word of warning: this talk is intended as an introduction to the
subject and should give an overview, so we will perhaps be a bit sloppy here and there...
Emma Warneford: "Formation of Zonal Jets and the Quasigeostrophic Theory of the Thermodynamic Shallow Water Equations"
Georgios Anastasiades: "Quantile forecasting of wind power using variability indices"
After Sela and Kharlampovich-Myasnikov independently proved that non abelian free groups share the same common theory model theoretic interest for the subject arose.
In this talk I will present a survey of results around this theory starting with basic model theoretic properties mostly coming from the connectedness of the free group (Poizat).
Then I will sketch our proof with C.Perin for the homogeneity of non abelian free groups and I will give several applications, the most important being the description of forking independence.
In the last part I will discuss a list of open problems, that fit in the context of geometric stability theory, together with some ideas/partial answers to them.
The first half of this seminar will discuss the hedging problem faced by a large sports betting agent who has to risk-manage an unwanted position in a bet on the simultaneous outcome of multiple football matches, by trading in moderately liquid simple bets on individual results. The resulting mathematical framework is that of a coupled system of multi-dimensional HJB equations.
This leads to the wider question of the numerical approximation of such problems. Dynamic programming with PDEs, while very accurate in low dimensions, becomes practically intractable as the dimensionality increases. Monte Carlo methods, while robust for computing linear expectations in high dimensions, are not per se well suited to dynamic programming. This leaves high-dimensional stochastic control problems to be considered computationally infeasible in general.
In the second half of the seminar, we will outline ongoing work in this area by sparse grid techniques and asymptotic expansions, the former exploiting smoothness of the value function, the latter a fast decay in the importance of principal components. We hope to instigate a discussion of other possible approaches including e.g. BSDEs.
15th Biennial OXFORD / CAMBRIDGE MEETING
PROGRAMME FOR THE
‘WOOLLY OWL TROPHY’
Invited Judges
John Harper
(Victoria University of Wellington, NZ)
Arash Yavari
(Georgia Tech, Atlanta, USA)
Sharon Stephen
(University of Birmingham, UK)
10:45 Morning Coffee The Maths Inst Common Room
... for Torelli groups of surfaces.
I will discuss two of my papers that develop PDE methods for weak KAM theory, in the context of a singular variational problem that can be interpreted as a regularization of Mather's variational principle by an entropy term. This is, sort of, a statistical mechanics approach to the problem. I will show how the Euler-Lagrange PDE yield approximate changes to action-angle variables for the corresponding Hamiltonian dynamics.
Random matrices were introduced by E. Wigner to model the excitation spectrum of large nuclei. The central idea is based on the hypothesis that the local statistics of the excitation spectrum for a large complicated system is universal. Dyson Brownian motion is the flow of eigenvalues of random matrices when each matrix element performs independent Brownian motions. In this lecture, we will explain the connection between the universality of random matrices and the approach to local equilibrium of Dyson Brownian motion. The main tools in our approach are the logarithmic Sobolev inequality and entropy flow. The method will be applied to the adjacency matrices of Erdos-Renyi graphs.
Direct products of finitely generated free groups have a surprisingly rich subgroup structure. We will talk about how the finiteness properties of a subgroup of a direct product relate to the way it is embedded in the ambient product. Central to this connection is a conjecture on finiteness properties of fibre products, which we will present along with different approaches towards solving it.
It is known that generic and universal structures and Ramsey classes are related. We explain this connection and complement it by some new examples. Particularly we disscuss universal and Ramsey classes defined by existence and non-existence of homomorphisms.
The problem of isometric embedding of a Riemannian Manifold into
Euclidean space is a classical issue in differential geometry and
nonlinear PDE. In this talk, I will outline recent work my
co-workers and I have done, using ideas from continuum mechanics as a guide,
formulating the problem, and giving (we hope) some new insight
into the role of " entropy".
The main mechanism for crystal plasticity is the formation and motion of a special class of defects, the dislocations. These are topological defects in the crystalline structure that can be identify with lines on which energy concentrates. In recent years there has been a considerable effort for the mathematical derivation of models that describe these objects at different scales (from an energetic and a dynamical point of view). The results obtained mainly concern special geometries, as one dimensional models, reduction to straight dislocations, the activation of only one slip system, etc.
The description of the problem is indeed extremely complex in its generality.
In the presentation will be given an overview of the variational models for dislocations that can be obtained through an asymptotic analysis of systems of discrete dislocations.
Under suitable scales we study the ``variational limit'' (by means of Gamma-convergence) of a three dimensional (static) discrete model and deduce a line tension anisotropic energy. The characterization of the line tension energy density requires a relaxation result for energies defined on curves.
The talk will explain how a geometric principle
gave rise to a new variational description of information-theoretic entropy and
how this led to the solution of a problem dating back to the 50's: whether the
the central limit theorem is driven by an analogue of the second law of
thermodynamics.
This problem is classically addressed by the so-called Skorohod Embedding problem. We instead develop a stochastic control approach. Unlike the previous literature, our formulation seeks the optimal no arbitrage bounds given the knowledge of the distribution at some (or various) point in time. This problem is converted into a classical stochastic control problem by means of convex duality. We obtain a general characterization, and provide explicit optimal bounds in some examples beyond the known classical ones. In particular, we solve completely the case of finitely many given marginals.
In spirit with John's talk we will discuss how topological invariants can be defined within a purely algebraic framework. After having introduced étale fundamental groups, we will discuss conjectures of Gieseker, relating those to certain "flat bundles" in finite characteristic. If time remains we will comment on the recent proof of Esnault-Sun.