Define $e_{n}(t)=\{t/n\}$. Let $d_N$ denote the distance in $L^2(0,\infty ; t^{-2}dt)$ between the indicator function of $[1,\infty[$ and the vector space generated by $e_1, \dots, e_N$. A theorem of B\'aez-Duarte (2003) states that the Riemann hypothesis (RH) holds if and only if $d_N \rightarrow 0$ when $N \rightarrow \infty$. Assuming RH, we prove the estimate $$d_N^2 \leq (\log \log N)^{5/2+o(1)}(\log N)^{-1/2}.$$ I shall put this result in its historical context, from Nyman's criterion (1950) and its beautiful proof to a sketch of our proof. I shall focus on the main ingredient we used, a method of Maier and Montgomery, recently sharpened by Soundararajan, to get some upper bound for partial sums of the M\"obius function. (joint work with Michel Balazard)

In this hastily prepared talk, I will describe some preliminary results of Nathan Ng and myself that concern linear dependencies (with integer coefficients) among zeros of Dirichlet L-functions. We can show, for example, that given a Dirichlet L-function and an arithmetic progression of points on the critical line Re(s) = 1/2, a large number of points in the arithmetic progression are not zeros of the L-function. Furthermore, given a fixed linear form F in n variables, we show (assuming the Riemann hypothesis) that a large number of points of the form 1/2 + iF(gamma_1, ..., gamma_n) are not zeros of the Riemann zeta function, where the gamma_j are imaginary parts of such zeros. We also describe a theorem about prime number races that is linked to linear (in)dependence of zeros of Dirichlet L-functions.

In mean curvature flow (or MCF), a surface evolves to minimize its surface area as quickly as possible. One of the challenges of MCF is that the flow starting from a closed surface (like a sphere) always becomes singular and one of the most important problems is understanding these singularities. The simplest example comes from a round sphere, which evolves by staying round but having the radius shrink until it hits zero and then just disappears. Matt Grayson proved that this is the only type of singularity that occurs for simple closed curves in the plane. However, many other examples were discovered in higher dimensions (most of them by applied mathematicians doing numerical simulations). I will describe recent work with Toby Colding, MIT, where we classified the generic singularities of MCF of closed embedded hypersurfaces. The thrust of our result is that, in all dimensions, every singularity other than shrinking spheres and cylinders can be perturbed away.

In this talk I will overview some results about derived categories of toric stacks. In particular the problem of existense of strong exceptional collections of line bundles. Some connections of this problem to Mirror symmetry and combinatorics of polytopes will be mentioned.

We address the task of reconstructing images corrupted by Poisson noise, which is important in various applications, such as fluorescence microscopy, positronemission- tomography (PET) or astronomical imaging. We focus on reconstruction strategies, combining Bregman concepts, expectation maximization (EM) and total variation (TV) based regularization, and present analytical as well as numerical achievements. Recently extensions of the well-known EM/Richardson-Lucy algorithm received increasing attention for inverse problems with Poisson data. However, most algorithms for regularizations like TV produce images suffering from blurred edges due to lagged diffusivity, and neither can guarantee positivity nor provide analytical investigations including convergence. The first goal of this talk is to provide an accurate, robust and fast EM-TV method for computing cartoon reconstructions facilitating post-segmentation. The method can be reinterpreted as a modified forward-backward (FB) splitting strategy known from convex optimization. We establish the well-posedness of the basic variational problem and can prove the positivity preserving property of our method. A damped variant of the FB-EM-TV algorithm with modified time steps, is the key step towards convergence. Typically, TV-based reconstruction methods deliver reconstructions suffering from contrast reduction. Hence, as the second goal of this talk, we propose extensions to EM-TV, based on Bregman iterations and inverse scale space methods, in order to obtain improved imaging results by simultaneous contrast enhancement. We illustrate the performance of our techniques by 2D and 3D synthetic and real-world results in microscopy and tomography. Proceeding to 4D video reconstruction yields interesting challenges. Due to natural patient motion in medical imaging (e.g. heart or lung) or cell migration in microscopy, naive reconstructions can suffer from undesired blurring effects at object boundaries. Finally, we touch on combinations of reconstruction techniques and optimal transport strategies.

Subcellular oscillations of Min proteins within individual cells of E. coli serve to localize division to midcell. While significant progress has been made to understand the Min oscillation both experimentally and in modeling, I will present three outstanding Min mysteries. I will also present our ongoing work to develop generic submodels of the Min oscillation, and to systematically manipulate the Min oscillation experimentally. In particular, we find that the period of the Min oscillation responds dramatically to temperature and to the concentration of extracellular multivalent cations (including antimicrobial peptides).

The minimal heat kernel on a Riemannian manifold is conservative if it integrates to 1. If this is the case, the manifold is said to be stochastically complete. Since the heat kernel is the transition density function of Brownian motion, a manifold is stochastically complete if and only if Brownian motion does not explode. This interpretation opens a way for investigating conservation of the heat kernel by probability theory. To find a proper geometric condition for heat kernel conservation is an old geometric problem. The first result in this direction was due to S. T. Yau, who proved that a Riemannian manifold is stochastically complete if its Ricci curvature is bounded from below by a constant. However, it has been known for quite some time that the heat kernel conservation property is intimately related to the volume growth of a Riemannian manifold. We study this problem by looking at the more refined question of how fast Brownian motion escapes to infinity, for the existence of a deterministic upper bound for the escaping rate implies heat kernel conservation. We show how the Neumann heat kernel, time reversal of reflecting Brownian motion, and volumes of geodesic balls all come together in this problem and give an elegant and often sharp upper bound of the escaping rate solely in terms of the volume growth function without any extra geometric restriction besides geodesic completeness. The talk should be interesting and accessible to differential geometers, people in partial differential equations (pde-ers), and probabilists.

Let G be a nilpotent group and let g be its Lie algebra. I
will explain how Kirillov's orbit method associates an isomorphism
class of an irreducible representation of G to a point f in the dual
of g. I will then outline my joint work with T. Thomas, in which we
sharpen the Orbit Method and associate an irreducible representation
to f.
The arXiv code for our paper is: 0909.5670

We introduce the notion of a fat staircase and define when a skew diagram D is a sum of fat staircases. We give a collection of Schur-positivity results that may be obtained from each sum of fat staircases. Further, we determine conditions on when a diagram may be a sums of fat staircases.

The next talk for UBC/UMC, the undergraduate mathematics colloquium, will be given by George Bluman.

Title: Systematic Methods for Solving Ordinary Differential Equations (ODEs)

Abstract:

In the latter part of the 19^{th} century, Sophus Lie initiated his studies on continuous groups (“Lie” groups) in order to put order to, and thereby extend systematically, the hodgepodge of heuristic techniques for solving ODEs. Lie showed how to find one-parameter Lie groups of transformations (point symmetries) of a given ODE and showed how a point symmetry reduces the order of an ODE. If an nth order ODE has n point symmetries that yield an n-parameter solvable group of point transformations (with good reason, solvable groups were initially called integrable groups), then the ODE is completely integrable, i.e., its solution reduces to n integrations (n quadratures).

Another systematic method for solving ODEs is based on finding and using integrating factors (seeking “first integrals” or “conservation laws”). This technique was fully developed in the 1990s. Lie showed that the symmetry and integrating factor methods are directly related for first order ODEs. This turns out not to be the case for higher-order ODEs..It turns out that, in general, the integrating factor method is complementary to Lie’s symmetry reduction method.

Symmetry and integrating factor methods are highly algorithmic and hence amenable to symbolic computation. These methods systematically unify and extend well-known ad-hoc techniques, learned in undergraduate ODE courses, to construct explicit solutions for differential equations, especially for nonlinear ODEs. The interplay between these two systematic methods is especially interesting.

This lecture will give an elementary presentation of symmetry and integration factor methods for solving ODEs (which today are the basis for the MAPLE software package DSOLVE as well as other software packages for solving ODEs).

The three dimensional incompressible Navier-Stokes system has two different structures: The antisymmetry of the nonlinearity, which provides an a priori bound for L^2 norm, and its natural scaling, which preserves L^3 norm. After a brief survey, I will talk about two problems. First the regularity problem, with a focus on current results assuming scaling invariant bounds (Type I singularity). Second the asymptotics problem, both temporal and spatial.

In this talk, I will introduce a new approach to rationality problem of Fano varieites using derive category, proposed by Kuznetsov. The idea is to construct a subcategory of the Fano variety, which is unchanged under birational tranforms. I will focus on the example where the Fano varieity is complete intersection of quadrics and explain the link between this new approach with some classical approaches of rationality problem.

In *classical mechanics of fluids* which is based on classical continuum field theories, the effect of forces is only considered on an infinitesimal fluid element. Hence, distribution and transmission of forces in the medium is described by the concept of *stress*. Such a simple model, which obtained with neglecting some physical aspects of system, can predict flows with small disturbances and smooth variations. These models fail when used for description of some complex phenomena like *turbulence*.

One of the ways to attack to the problems like turbulence is the use of non-classical continuum theories. We can use a non-local continuum model and the equations of gradient materials or a Cosserat continuum model. Another way is to use statistical mechanics and thermodynamics.

The aim of this work is to generalize the concept of continua by use of principles of *rational mechanics* such as *determinism* and *material objectivity*.

After derivation of equations of motion of fluids of second and third grade and of a Cosserat fluid, they have been solved for some simple problems of turbulent flows and the results have been compared with experimental data and the comparison showed that our model can describe turbulence with a acceptable precision.

Abstract: We study and determine a homotopy type of
the moduli space of all generalized Morse functions on d-manifolds for given d.
This moduli space is closely connected to the moduli space of all Morse functions
studied by Madsen and Weiss and classifying space of the corresponding cobordism
category.

This talk will survey recent progress on clarifying the connection between enumerative combinatorics and cluster expansions. The combinatorics side concerns species of combinatorial structures and the associated exponential generating functions. Cluster expansions, on the other hand, are supposed to give convergent expressions for measures on infinite dimensional spaces, such as those that occur in statistical mechanics. There is a kind of dictionary between these two subjects that sheds light on each of them. In particular, it gives insight into new convergence results for cluster expansions.

This is the second part of ''New classification techniques for ordinary differential equations" in which I describe how one can use Cartan's equivalence method to compute the change of coordinates that maps two differential equations. In this talk I will present a new ordinary differential equation solver.

The Liouville function is defined by $\lambda(n):=(-1)^{\Omega(n)}$ where $\Omega(n)$ is the number of prime divisors of the positive integer $n$ counting multiplicity. Let $m \geq 2$ be an integer and $\zeta_m$ be a primitive $m$--th root of unity. As a generalization of Liouville's function, we study the function $\lambda_m (n):=\zeta_m^{\Omega(n)}$. Using properties of this function, we will show how, for any integer $j$, properties about the set of all positive integers $n$ with $\Omega(n) \equiv j \pmod{m}$ can be obtained. In particular, we will show that this set has (natural) density $1/m$. In fact, we will also obtain much information about error terms and will illustrate how the case $m=2$ is very different from the case $m>2$. This is joint work with Michael Coons.

Note for Attendees

Cookies and tea will be served between the two talks.

Abstract:Equivariant Lefschetz invariants have already appeared in algebraic topology. Here I will show how to approach them using the so-called equivariant KK-theory of Kasparov — the main tool of the new field of noncommutative geometry. I will sketch the construction of Lefschetz invariants for equivariant self-maps of a G-space, where G is a
discrete group, and then define them for more general objects than just
self-maps, called correspondences. There are always many interesting
equivariant self-correspondences of a space with a group action, even if the
group is not discrete. The case of compact connected groups seems in particular quite interesting. We state an equivariant version of the Lefschetz fixed-point formula for this situation. In the resulting formula, the geometric side is based on equivariant index theory of elliptic operators, while the global algebraic side involves the module trace of Hattori and Stallings. Computing the relevant traces seems to be a problem belonging properly to algebraic geometry

Apart from sharing some interesting touristic and climatological observations, I will report on the computational number theoretic improvements I have included in Magma in June 2009.

While explicit p-adic analytic methods for solving diophantine equations based on Chabauty's ideas have been available for around 10 years now, there has been a recent shift to concentrate computational effort on an additional phase that can combine p-adic information at several primes. Heuristically, the method commonly referred to as "Mordell-Weil sieving" should yield arbitrarily detailed information on the location of possible solutions. In practice, however, there are severe combinatorial obstructions to exploiting that information.

In joint work with Michael Stoll, we have developed good ways of avoiding the intermediate combinatorial explosion. This strategy has now been implemented in Magma and yields an almost completely automatic procedure to determine the rational points on a considerable class of algebraic curves.

Studying non-smooth geometric objects is a very important
and modern research topic in differential geometry and geometric
analysis. In particular it is interesting to know to which extent
these objects can be approximated by smooth ones.
In this talk I want to indicate how this problem can be related to the
study of systems of parabolic equations with irregular initial data.
Moreover I want to discuss various situations in which existence results
for these initial value problems have been obtained.

If H is a variation of Hodge structure over a variety S, then there is a family of complex tori J(H) over S associated to H. Admissible normal functions are certain sections of J(H) over S. Roughly speaking, they are the ones that have the possibility of coming from algebraic geometry.

I will explain recent work with Gregory Pearlstein proving that the locus where a section of J(H) vanishes is an algebraic subvariety of S. This answers a conjecture of Griffiths and Green.

Gauss-Lobatto spectral elements, based on hexahedral meshes, provide a very efficient way to solve transient wave equations in terms of storage and of computational time. Unfortunately, it is very difficult and almost impossible in some cases to produce pure hexahedral meshes for complex geometries. Until now, we have remedied this shortcoming by using tetrahedral meshes in which tetrahedral were split into four hexahedra, but this technique provides very distorted meshes which imply tu use about three times more unknown than pure hexahedral meshes to get the same accuracy. For this reason, we developed a strategy of solvers based on hybrid meshes containing mainly hexahedra and some tetrahedral, pyramids and wedges. First results show a dramatical gain in performance versus split tetrahedra.

I will explain why formulas for compatible intertwiners for
representation of nilpotent groups require the notion of the
determinant of a finite abelian p-group. I will then outline an
approach of Deligne for defining determinant using K-theory.

This talk is a continuation of my previous talk, but I will try to
assume very little.

Superhydrophobicity and water repellency is important not only for outdoor apparel, but also for applications in the fields of corrosion resistance, micro-fluidics, anti-fouling, bio-compatibility, low drag and low friction surfaces. This talk will give an introduction to superhydrophobicity and the underlying physics. Also it will show a way of using a femtosecond laser to modify the wetting behavior of common engineering materials, namely metals, and their subsequent use as low friction materials.

Group field theory is the higher-dimensional generalization of random matrix models. As it has built-in scales and automatically sums over metrics and discretizations, it provides a combinatoric origin for space time. Its graphs facilitate a new approach to algebraic topology. I exemplify this approach by introducing a graph's cellular structure and associated homology.

The next talk for UBC/UMC, the undergraduate mathematics colloquium, will be given by Richard Anstee.

Title: If you can't square the circle, then at least you can square the square

Abstract:

I will talk partly about W.T. Tutte, a Professor at the University of Waterloo. I will partly talk about the problem of the squared square, which is a dissection of a square into smaller squares, all of different sizes.

There may even be a little linear algebra and graph theory in the talk.

This seminar continues with an overview of the topics of the forthcoming Springer book "Applications of Symmetry Methods to Partial Differential Equations" by Bluman, Cheviakov and Anco.. In Part II, it will be shown how to obtain systematically nonlocal symmetries of PDE systems. In turn this leads to the question of how to find systematically conservation laws of PDE systems. In particular, it will be shown how to generalize the classical Noether's theorem to find conservation laws of PDE systems that are not variational.

For millenia, many misguided mathematicians moved to prove that Euclid's parallel postulate was a consequence of his other axioms. In this talk I will discuss hyperbolic geometry, and draw a picture or two proving all of these valiant efforts wrong.

I will also discuss relationships between complex analysis and the classification of all two-dimensional geometries, why hyperbolic geometry can be described (somewhat tongue-in-cheek) as "God's Geometry", and given time, a discussion of extant three dimensional geometries.

Biological aggregations such as insect swarms, bird flocks, and fish schools are arguably some of the most common and least understood patterns in nature. In this talk, I will discuss recent work on swarming models, focusing on the connection between inter-organism social interactions and properties of macroscopic swarm patterns. The first model is a conservation-type partial integrodifferential equation (PIDE). Social interactions of incompressible form lead to vortex-like swarms. The second model is a high-dimensional ODE description of locust groups. The statistical-mechanical properties of the attractive-repulsive social interaction potential control whether or not individuals form a rolling migratory swarm pattern similar to those observed in nature. For the third model, we again return to a conservation-type PIDE and, via long- and short-wave analysis, determine general conditions that social interactions must satisfy for the population to asymptotically spread, contract, or reach steady state.

We review the idea of using lattice paths as models of phase boundaries rather than as models of polymers. We also present the solution of a particular model that has demonstrated some novel mathematical features. Additionally this model may describe the steady state of a non-equilibrium model of molecular motors.

Superlattice patterns and quasipatterns, while well-studied in waves on the surface of vertically vibrated viscous fluids (Faraday waves), have found little attention in forced oscillatory systems. We study such patterns, comprised of 4 or more Fourier modes at different orientations, by applying multi-frequency forcing to systems undergoing a Hopf bifurcation to spatially homogeneous oscillations. For weak forcing composed of 3 frequencies near the 1:2- and 1:3-resonance, such systems can be described by a suitably extended complex Ginzburg-Landau equation with time periodic coefficients. Using Floquet theory and weakly nonlinear analysis we obtain the amplitude equations for simple patterns (comprised of 1, 2, or 3 modes) and superlattice patterns. We stabilize these patterns via spatiotemporal resonance and find stable subharmonic 4- and 5-mode patterns through judicious choice of the forcing function. For system parameters reported for experiments on the oscillatory Belousov–Zhabotinsky reaction we explicitly show that the forcing parameters can be tuned such that 4-mode patterns are the preferred patterns. We confirm our analysis through numerical simulation. This work was done jointly with Dr. Hermann Riecke at Northwestern University.

I will discuss some deformation properties of Fano
varieties. The general methods rely on the investigation of the
variation of the cone of effective curves and, more generally, of the
Mori chamber decomposition, which, according to Mori theory, encode
information on the geometry of the variety. The talk is based on
joint work with C. Hacon.

Computer algebra systems like Maple and Mathematica spend most of their time doing either polynomial arithmetic or linear algebra. For polynomials in one variable of degree n, the Fast Fourier Transform gives us an O(n log n) multiplication algorithm. But for polynomials in several variables, which are usually sparse (most of the coefficients are zero) there are no O(n log n) algorithms. So what's the best way to multiply and divide them?

In the talk I will present

Johnson's algorithm from 1974 for multiplying two sparse polynomials using a heap.

Our division algorithm which also uses a heap.

Some benchmarks showing that these algorithms are 100 times faster than Maple and Mathematica, and

a parallel algorithm for multiplication using a heap.

I'd also like to present an optimization which we call "immediate monomials" where we reduce multiplication of monomials ( x^i y^j z^k ) to one machine instruction and describe a new project we are doing with Maple to redesign the basic polynomial representation in Maple to hopefully get a good overall speedup.

We present a general variational method, involving self-dual variational calculus, for recovering non-linearities from prescribed solutions for certain types of PDEs which are not necessarily of Euler-Lagrange type, including parabolic equations. The approach can also be used for optimal control problems. The topological aspects involved, for the space of self-dual Lagrangians, and the space of maximal monotone vector fields on a reflexive Bancah space will be discussed.

We give a characterization of all degree 6 del Pezzo surfaces over an arbitrary
field. This characterization is derived by studying the toric structure of
the surface. Such a surface is determined, up to isomorphism, by a pair of
separable algebras, subject to some constraints. These algebras determine
geometric, arithmetic, and topological information about the variety, which
we will discuss. Some of this work is joint with Paul Smith and Sue Sierra.

In symbolic dynamics, a Z^d shift of finite type (or SFT) is the set of all ways to assign elements from a finite alphabet A to all sites of Z^d, subject to local rules about which elements of A are allowed to appear next to each other. A fundamental number associated to any SFT is its (topological) entropy, which is, roughly speaking, the exponential growth rate of the number of allowed patterns of size n.

The entropy of any Z SFT is easily computable (it is the log of an algebraic number). However, for d > 1, the situation becomes more complex. There are in fact only a few nontrivial examples of Z^2 SFTs whose entropies have explicit closed forms.

It is natural then to try to at least estimate these entropies. We will discuss some of the difficulties involved in doing this, and present a way of approximating entropy for a class of Z^2 SFTs by way of some easier-to-compute entropies of associated Z SFTs.

As a corollary of this technique, we can show that the entropy of any Z^2 SFT in this class is computable in polynomial time.

The potential catastrophic failure of a dam and the resultant widespread downstream flooding and damage is a scenario that is of great concern. A brief overview of the numerical methods most commonly used in the simulation of dam-break floods, namely Godunov-type finite-volume schemes solving the two-dimensional shallow water equations, will be given. Smoothed particle hydrodynamics will then be introduced and its potential as an alternative numerical scheme will be addressed. Results of the simulation of an actual historic dam-break event using both methods will be presented.

Abstract: In the 1960s, Wall developed a theory of finiteness obstructions for CW-complexes. We extend this result to diagrams of spaces and use it to investigate which homotopy G-spheres can be realized on a finite complex. We will conclude with some new examples of finite (non-linear, non-free) G spheres for some small groups G

We will describe a current approach to understanding stationary states in non-equilibrium statistical mechanics. We will then consider two examples: a system of stochastic differential equations for coupled oscillators, and a stochastic wave equation. Stationary states for these examples exhibit steady state energy flow.

Micro-Electro-Mechanical Systems (MEMS) and Nano-Electro-Mechanical Systems (NEMS), which combine electronics with miniature-size mechanical devices, are basic ingredients of contemporary technology. A key component of such systems is the simple idealized electrostatic device consisting of a thin and deformable plate, consisting of a dielectric material with a negligibly thin conducting film on its lower surface, that is held fixed along its boundary in the two dimensional plane. Above the deformable plate lies a rigid grounded plate. As one applies a positive voltage to the thin conducting film the deformable plate deflects upwards towards the ground plate. If the voltage is increased beyond a certain critical value then the deformable plate touches the ground plate, in finite time, and we have the so-called "pull in instability".

Unfortunately, models for electro-statically actuated micro-plates that account for moderately large deflections are quite complicated and not yet amenable to rigorous mathematical analysis. In the last 5 years, my students (Cowan, Esposito, Guo, Moradifam) and I, dealt with much simplified models that still lead to interesting second and fourth order nonlinear elliptic equations (in the stationary case) and to nonlinear parabolic equations (in the dynamic case). The non-linearity is of an inverse square type, which -- until recently – has not received much attention as a mathematical problem. It was therefore rewarding to see, besides the above practical considerations, that the model is actually a very rich source of interesting mathematical phenomena. Numerics and formal asymptotic analysis give lots of information and point to many conjectures, but even in the most simple idealized versions of electrostatic MEMS, one essentially needs the full available arsenal of modern nonlinear analysis and PDE techniques “to do" the required mathematics.

## Note for Attendees

Cookies and tea will be served between the two talks.