Past

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.