Past

This talk will discuss the notion of a Nahm transform in differential geometry, as a way of relating solutions to one differential equation on a manifold, to solutions of another differential equation on a different manifold. The guiding example is the correspondence between solutions to the Bogomolny equations on $\mathbb{R}^3$ and Nahm equations on $\mathbb{R}$. We extract the key features from this example to create a general framework.

Nematic elastomers are rubbery elastic solids made of cross-linked polymeric chains with embedded nematic mesogens. Their mechanical behaviour results from the interaction of electro-optical effects typical of nematic liquid crystals with the elasticity of a rubbery matrix. We show that the geometrically linear counterpart of some compressible models for these materials can be justified via Gamma-convergence. A similar analysis on other compressible models leads to the question whether linearised elasticity can be derived from finite elasticity via Gamma-convergence under weak conditions of growth (from below) of the energy density. We answer to this question for the case of single well energy densities.

We discuss Ogden-type extensions of the energy density currently used to model nematic elastomers, which provide a suitable framework to study the stiffening response at high imposed stretches.

Finally, we present some results concerning the attainment of minimal energy for both the geometrically linear and the nonlinear model.

The "de Rham-Witt complex" of Deligne and Illusie is a functorial complex of sheaves $W^*(X)$ on a smooth algebraic variety $X$ over a finite field, computing the cristalline cohomology of $X$. I am going to present a non-commutative generalization of this: even for a non-commutative ring $A$, one can define a functorial "Hochschild-Witt complex" with homology $WHH^*(A)$; if $A$ is commutative, then $WHH^i(A)=W^i(X)$, $X = Spec A$ (this is analogous to the isomorphism $HH^i(A)=H^i(X)$ discovered by Hochschild, Kostant and Rosenberg). Moreover, the construction of the Hochschild-Witt complex is actually simpler than the Deligne-Illusie construction, and it allows to clarify the structure of the de Rham-Witt complex.

I shall present a geometric property valid in many Hrushovski
amalgamation constructions, relative CM-triviality, and derive
consequences on definable groups: modulo their centre they are already
products of groups interpretable in the initial theories used for the
construction. For the bad field constructed in this way, I shall
moreover classify all interpretable groups up to isogeny.

The study of free groups and their automorphisms has a long pedigree, going back to the work of Nielsen and Dehn in the early 20th century, but in many ways the subject only truly reached maturity with the introduction of Outer Space by Culler and Vogtmann in the “Big Bang” of 1986. In this (non-expert) talk, I will walk us through the construction of Outer Space and some related complexes, and survey some group-theoretic applications.

We prove that the rank gradient vanishes for mapping class groups, Aut(Fn) for all n, Out(Fn), n > 2 and any Artin group whose underlying graph is connected. We compute the rank gradient and verify that it is equal to the first L2-Betti number for some classes of Coxeter groups.

We study the problem of covering R^d by overlapping translates of a convex polytope, such that almost every point of R^d is covered exactly k times. Such a covering of Euclidean space by a discrete set of translations is called a k-tiling. The investigation of simple tilings by translations (which we call 1-tilings in this context) began with the work of Fedorov and Minkowski, and was later extended by Venkov and McMullen to give a complete characterization of all convex objects that 1-tile R^d. By contrast, for k ≥ 2, the collection of polytopes that k-tile is much wider than the collection of polytopes that 1-tile, and there is currently no known analogous characterization for the polytopes that k-tile. Here we first give the necessary conditions for polytopes P that k-tile, by proving that if P k-tiles R^d by translations, then it is centrally symmetric, and its facets are also centrally symmetric. These are the analogues of Minkowski’s conditions for 1-tiling polytopes, but it turns out that very new methods are necessary for the development of the theory. In the case that P has rational vertices, we also prove that the converse is true; that is, if P is a rational, centrally symmetric polytope, and if P has centrally symmetric facets, then P must k-tile R^d for some positive integer k.

I will explain an approach to study DT theory of toric CY 3-folds using $L_\infty$ algebras. Based on the construction of strong exceptional collection of line bundles on Fano toric stack of dimension two, we realize any bounded families of sheaves on local surfaces support on zero section as critical sets of the Chern-Simons functions. As a consequence of this construction, several interesting properties of DT invariants on local surfaces can be checked.

Abstract: We consider an implicit finite difference
scheme on uniform grids in time and space for the Cauchy problem for a second
order parabolic stochastic partial differential equation where the parabolicity
condition is allowed to degenerate. Such equations arise in the nonlinear
filtering theory of partially observable diffusion processes. We show that the
convergence of the spatial approximation can be accelerated to an arbitrarily
high order, under suitable regularity assumptions, by applying an extrapolation
technique.

Suppose we have a collection of blocks, which gradually split apart as time goes on. Each block waits an exponential amount
of time with parameter given by its size to some power alpha, independently of the other blocks. Every block then splits randomly,but according to the same distribution. In this talk, I will focus on the case where alpha is negative, which
means that smaller blocks split faster than larger ones. This gives rise to the phenomenon of loss of mass, whereby the smaller blocks split faster and faster until they are reduced to ``dust''. Indeed, it turns out that the whole state is reduced to dust in a finite time, almost surely (we call this the extinction time). A natural question is then: how do the block sizes behave as the process approaches its extinction time? The answer turns out to involve a somewhat unusual ``spine'' decomposition for the fragmentation, and Markov renewal theory.

This is joint work with Bénédicte Haas (Paris-Dauphine).

The melting of ice in Greenland and Antarctica would, of course, be by far the major contributor any possible sea level rise. Thus, to make science-based predictions about sea-level rise, it is crucial that the ice sheets covering those land masses be accurately mathematically modeled and computationally simulated. In fact, the 2007 IPCC report on the state of the climate did not include predictions about sea level rise because it was concluded there that the science of ice sheets was not developed to a sufficient degree. As a result, predictions could not be rationally and

confidently made. In recent years, there has been much activity in trying to improve the state-of-the-art of ice sheet modeling and simulation. In

this lecture, we review a hierarchy of mathematical models for the flow of ice, pointing out the relative merits and demerits of each, showing how

they are coupled to other climate system components (ocean and atmosphere), and discussing where further modeling work is needed. We then discuss algorithmic approaches for the approximate solution of ice sheet flow models and present and compare results obtained from simulations using the different mathematical models.

Motivated by a conjecture of Alday, Gaiotto and Tachikawa (AGT

conjecture), we construct an action of

a suitable $W$-algebra on the equivariant cohomology of the moduli

space $M_r$ of rank r instantons on $A^2$ (i.e.

on the moduli space of rank $r$ torsion free sheaves on $P^2$,

trivialized at the line at infinity). We show that

the resulting $W$-module is identified with a Verma module, and the

characteristic class of $M_r$ is the Whittaker vector

of that Verma module. One of the main ingredients of our construction

is the so-called cohomological Hall algebra of the

commuting variety, which is a certain associative algebra structure on

the direct sum of equivariant cohomology spaces

of the commuting varieties of $gl(r)$, for all $r$. Joint work with E. Vasserot.

I will present an efficient algorithm for computing certain special values of Rankin triple product $p$-adic L-functions and give an application of this to the explicit construction of rational points on elliptic curves.

The $p$-adic Gross-Zagier formula for diagonal cycles and the $p$-adic Beilinson formulae described in the lectures of Rotger and Bertolini respectively suggest a connection between certain  {\em $p$-adic iterated integrals} attached to modular forms and rational points on elliptic curves. I will describe an ongoing project (in collaboration with Alan Lauder and Victor Rotger) whose goal is to explore these relationships numerically, with the goal of better understanding the notion of {\em Stark-Heegner points}. It is hoped that these experiments might suggest new perspectives on Stark-Heegner points based on suitable {\em $p$-adic deformations} of the global objects--diagonal cycles, Beilinson-Kato and Beilinson-Flach elements-- described in the lectures of Rotger, Bertolini, Dasgupta, and Loeffler, following  the influential approach to $p$-adic $L$-functions pioneered by Coates-Wiles, Kato, and Perrin-Riou.

We define an integral version of Sczech cocycle on ${\rm GL}_n(\mathbf{Z})$ by raising the level at a prime $\ell$.As a result, we obtain a new construction of the $p$-adic L-functions of Barsky/Cassou-Nogu\`es/Deligne-Ribet. This cohomological construction further allows for a study of the leading term of these L-functions at $s=0$:\\1) we obtain a new proof that the order of vanishing is at least the oneconjectured by Gross. This was already known as result of Wiles.\\2) we deduce an analog of the modular symbol algorithm for ${\rm GL}_n$ from the cocyclerelation and LLL. It enables for the efficient computation of the special values of these $p$-adic L-functions.\\When combined  with a refinement of the Gross-Stark conjecture, we obtain some examples of numerical construction of $\mathfrak p$-units in class fields of totally real (cubic) fields.This is joint work with Samit Dasgupta.

I will describe a construction of special cohomology classes over the cyclotomic tower for the product of the Galois representations attached to two modular forms, which $p$-adically interpolate the "Beilinson--Flach elements" of Bertolini, Darmon and Rotger. I will also describe some applications to the Iwasawa theory of modular forms over imaginary quadratic fields.

We show that the $p$-adic L-function associated to the tensor square of a $p$-ordinary eigenform factors as the product of the symmetric square $p$-adic L-function of the form with a Kubota-Leopoldt $p$-adic L-function.  Our method of proof follows that of Gross, who proved a factorization for Katz's $ p$-adic L-function for a character arising as the restriction of a Dirichlet character.  We prove certain special value formulae for classical and $p$-adicRankin L-series at non-critical points.  The formula of Bertolini, Darmon, and Rotger in the $p$-adic setting is a key element of our proof.  As demonstrated by Citro, we obtain as a corollary of our main result a proof of the exceptional zero conjecture of Greenberg for the symmetric square.

I will report on $p$-adic Beilinson's formulas, relating the values of certain Rankin $p$-adic L-functions outside their range of classical interpolation, to $p$-adic syntomic regulators of Beilinson-Kato and Beilinson-Flach elements. Applications to the theory of Euler systems and to the Birch and Swinnerton-Dyer conjecture will also be discussed. This is joint work with Henri Darmon and Victor Rotger.

In this lecture I shall introduce certain generalised Gross-Kudla-Schoen diagonal cycles in the product of three Kuga-Sato varieties and a $p$-adic formula of Gross-Zagier type which relates the images of these diagonal cycles under the $p$-adic Abel-Jacobi map to special values of the $p$-adic L-function attached to the Garrett triple convolution of three  Hida families of  modular forms. This formula has applications to the Birch--Swinnerton-Dyer conjecture and the theory of Stark-Heegner points. (Joint work with Henri Darmon.)

The main result of the talk is that two curves over a finite field are isomorphic, up to automorphisms of the ground field, if and only if there is an isomorphism of groups of Dirichlet characters such that the corresponding L-series are all equal. This can be shown by combining Uchida's proof of the anabelian theorem for global function fields with methods from (noncommutative) dynamical systems. I will also discuss how to turn this theorem into a theoretical algorithm that, given a listing of L-functions, determines an equation for the corresponding curve(s).

I will present an overview of a series of joint works with Akio Tamagawa about l-adic representations of etale fundamental group of curves (to simplify, over finitely generated fields of characteristic 0).
More precisely, when the generic representation is GLP (geometrically Lie perfect) i.e. the Lie algebra of the geometric etale fundamental group is perfect, we show that the associated local $\ell$-adic Galois representations satisfies a strong uniform open image theorem (ouside a `small' exceptional locus). Representations on l-adic cohohomology provide an important example of GLP representations. In that case, one can even provethat the exceptional loci that appear in the statement of our stronguniform open image theorem are independent of $\ell$, which was predicted by motivic conjectures.
Without the GLP assumption, we prove that the  associated local l-adic Galois representations still satisfy remarkable rigidity properties: the codimension of the image at the special fibre in the image at the generic fibre is at most 2 (outside a 'small' exceptional locus) and its Lie algebra is controlled by the first terms of the derived series of the Lie algebra of the image at the generic fibre.
I will state the results precisely, mention a few applications/open questions and draw a general picture of the proof in the GLP case (which,in particular, intertwins via the formalism of Galois categories, arithmetico-geometric properties of curves and $\ell$-adic geometry). If time allows, I will also give a few hints about the $\ell$-independency of the exceptional loci or the non GLP case.

We use Schlage-Puchta's concept of p-deficiency and Lackenby's property of p-largeness to show that a group having a finite presentation with p-deficiency greater than 1 is large. What about when p-deficiency is exactly one? We also generalise a result of Grigorchuk on Coxeter groups to odd primes.

In this talk, I will describe a construction of a $p$-adic L-function attached to a triple of $p$-adic Coleman families of cusp forms.  This function interpolates algebraic parts of special values of Garrett triple product L-functions at balanced triples of weights.  Our construction is complementary to that of Harris and Tilouine which treats the case of unbalanced weights.

Given a $p$-adic weight and a finite slope we describe a Hecke and Galois equivariant geometric map relating elliptic overconvergent modular symbols and overconvergent modular forms of that slope, appropriate weights and $\mathbf{C}_p$-coefficients. We show that for a fixed slope, with the possible exception of a discrete family of weights, this map is an isomorphism.

Using Faltings' theory of the Hodge-Tate sequence of an abelian scheme we construct certain sheaves $\Omega^\kappa$, where $\kappa$ is a not-necessarily integral weight, over formal subschemes of modular varieties over which the canonical subgroup exists.   These sheaves generalize the integral powers, $\omega^k$, of the sheaf $\omega$ of relative differentials on a modular curve.   Global sections of $\Omega^\kappa$ provide geometric realizations of overconvergent automorphic forms of non-integral weight.  Applications of this approach to the theory of $p$-adic Hilbert modular forms will be given.   This is joint work with Fabrizio Andreotti and Adrian Iovita.

We apply the theory of the radius of convergence of a $p$-adic connection to the special case of the direct image of the constant connection via a finite morphism of compact $p$-adic curves, smooth in the sense of rigid geometry. We show that a trivial lower bound for that radius implies a global form of Robert's $p$-adic Rolle theorem. The proof is based on a widely believed, although unpublished, result of simultaneous semistable reduction for finite morphisms of smooth $p$-adic curves. We also clarify the relation between the notion of radius of convergence used in our previous work and the more intrinsic one used by Kedlaya. (The paper is available athttp://arxiv.org/abs/1209.0081)

For a proper semistable curve over a DVR of mixed characteristics we re prove the ``invariant cycles theorem'' with trivial coefficients  by Chiarellotto i.e. that the group of elements annihilated by the monodromy operator on the first de Rham cohomology group of the generic fiber coincides with the first rigid cohomology group of the special fiber, without the hypothesis that the residue field is finite. This is done using the explicit description of the monodromy operator on the de Rham cohomology of the generic fiber with coefficients convergent F-isocrystals given in a work of Coleman and Iovita. We apply these ideas to the case where the coefficients are unipotent convergent F-isocrystals defined on the special fiber: we show that the invariant cycles theorem does not hold in general in this setting. Moreover we give a sufficient condition for the non exactness. It is a joint work with B. Chiarellotto, R. Coleman and A. Iovita.

In 1986 Mazur, Tate and Teitelbaum came up with a $p$-adic analogue of the conjecture of Birch and Swinnerton-Dyer for elliptic curves over the rationals. In this talk I will report on joint work with Jennifer Balakrishnan and William Stein on a generalization of this conjecture to the case of modular abelian varieties and primes $p$ of good ordinary reduction. I will discuss the theoretical background that led us to the formulation of the conjecture, as well as numerical evidence supporting it in the case of modular abelian surfaces and the algorithms that we used to gather this evidence.

We extend the following theorem of H. Imai in several ways: If  $A$  is an abelian variety with potentially good reduction over a finite extension  $K$  of  $\mathbf{Q}_p$, then it has only finitely many rational torsion points over the maximal $p$-cyclotomic extension of  $K$. In particular, we prove the finiteness over $K(K^{1/p^\infty})$.

The overconvergence of the canonical subgroup of the universal abelian variety is one of the key ingredients of the theory of overconvergent modular forms. In this talk, I will show the overconvergence of the canonical subgroup with expected properties via the Breuil--Kisin classification, including the case of $p=2$.

I will explain recent joint work with Matthew Emerton and David Savitt, in which we relate the geometry of various tamely potentially Barsotti--Tate deformation rings for two-dimensional Galois representations to the integral structure of the cohomology of Shimura curves. As a consequence, we establish some conjectures of Breuil regarding this integral structure.
The key technique is the Taylor--–Wiles--–Kisin patching argument, which,when combined with a new, geometric perspective on the Breuil–--Mezard conjecture, forges a tight link between the structure of cohomology (a global automorphic invariant) and local deformation rings (local Galois-theoretic invariants).

Composite membranes comprised of an ultra-thin coating film formed over a porous support membrane are the basis for state-of-the-art reverse osmosis (RO) and nanofiltration (NF) membranes, offering the possibility to independently optimize the support membrane and the coating film. However, limited information exists on transport through such composite membrane structures. Numerical calculations have been carried out in order to probe the impacts of the support membrane skin-layer pore size and porosity, support membrane bulk micro-porosity, and coating film thickness and morphology (i.e. surface roughness) on solvent and solute transport through composite membranes. Results suggest that the flux and rejection of a composite membrane may be fine-tuned, by adjusting support membrane skin layer porosity and pore size, independent of the properties of the coating film. Further, the water flux over the membrane surface is unevenly distributed, creating local ‘hot spots’ of high flux that may govern initial stages of membrane fouling and scaling. The analysis provides important insight on how the non-trivial interaction of support properties and film roughness may result in widely varying transport properties of the composite structure. In particular, the simulations reveal inherent trade-offs between flux, rejection and fouling propensity (the latter due to ‘hot spots’), which are purely consequences of geometrical factors, irrespective of materials chemistry.

I will be concerned with initial-boundary value problems for systems of conservation laws in one space variable. First, I will go over some of the most relevant features of these problems. In particular, I will stress that different viscous approximation lead, in general, to different limits.

Next, I will discuss possible ways of characterizing the limit of a given viscous approximation. Also, I will establish a uniqueness criterion that allows to conclude that the limit of a self-similar approximation introduced by Dafermos et al. actually coincide with the limit of the physical viscous approximation. Finally, if time allows I will mention consequences on the design of numerical schemes. The talk will be based on joint works with S. Bianchini, C. Christoforou and S. Mishra.

• Alfonso Bueno - Recent advances in mathematical modelling of cardiac tissue: A fractional step forward
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Russian mathematician V.I.Arnold conjectured that convex, homogeneous bodies with less than four equilibria (also called mono-monostatic) may exist. Not only did his conjecture turn out to be true, the newly discovered objects show various interesting features. Our goal is to give an overview of these findings as well as to present some new results. We will point out that mono-monostatic bodies are neither flat, nor thin, they are not similar to typical objects with more equilibria and they are hard to approximate by polyhedra. Despite these "negative" traits, there seems to be strong indication that these forms appear in Nature due to their special mechanical properties.

Topological defects (TDs) are unavoidable consequence of continuous symmetry breaking phase transitions. They exhibit several universal features and often span apparently completely different systems. Particularly convenient testing ground to study basic physics of TDs are liquid crystals (LCs) due to their softness, liquid character and optical anisotropy. In the lecture I will present our recent theoretical studies of TDs in nematic LCs, which are of interest also to other branches of physics.

I will first focus on coarsening dynamics of TDs following the isotropic-nematic phase transition. Among others we have tested the validity of the Kibble-Zurek [1,2] prediction on the size of the so called protodomains, which was originally derived to estimate density of TDs as a function of inflation time in the early universe. Next I will consider nematic LC shells [3]. These systems are of interest because they could pave path to mm sized scaled crystals exhibiting different symmetries. Particular attention will be paid to curvature induced unbinding of pairs of topological defects. This process might play important role in membrane fission processes.  

[1] W.H. Zurek, Nature 317, 505 (1985).

[2] Z. Bradac et al.,  J.Chem.Phys 135, 024506 (2011)

[3] S. Kralj et al.,  Soft Matter 7, 670 (2011); 8, 2460  (2012).

Telomeres, non-coding terminal structures of DNA strands, consist of repetitive long tandem repeats of a specific length. An absence of an enzyme, telomerase, in certain cellular structures requires an alternative telomerase-independent pathway for telomeric sequence length regulation. Besides linear telomeres other configurations such as telomeric circles and telomeric loops were experimentally observed. They are suspected to play an important role in a universal mechanism for stabilization of the ends of linear DNA that possibly dates back to pre-telomerase ages. We propose a mathematical model that captures biophysical interactions of various telomeric structures on a short time scale and that is able to reproduce experimental measurements in mtDNA of yeast. Moreover, the model opens up a couple of interesting mathematical problems such as validity of a quasi-steady state approximation and dynamic properties of discrete coagulation-fragmentation systems. We also identify and estimate key factors influencing the length distribution of telomeric circles, loops and strand invasions using numerical simulations.

For a given positive measure on a fixed domain, Gaussian quadrature routines can be defined via their property of integrating an arbitrary polynomial of degree $2n+1$ exactly using only $n+1$ quadrature nodes. In the special case of Gauss--Jacobi quadrature, this means that $$\int_{-1}^1 (1+x)^\alpha(1-x)^\beta f(x) dx = \sum_{j=0}^{n} w_j f(x_j), \quad \alpha, \beta > -1, $$ whenever $f(x)$ is a polynomial of degree at most $2n+1$. When $f$ is not a polynomial, but a function analytic in a neighbourhood of $[-1,1]$, the above is not an equality but an approximation that converges exponentially fast as $n$ is increased.

An undergraduate mathematician taking a numerical analysis course could tell you that the nodes $x_j$ are roots of the Jacobi polynomial $P^{\alpha,\beta}_{n+1}(x)$, the degree $n+1$ polynomial orthogonal with respect to the given weight, and that the quadrature weight at each node is related to the derivative $P'^{\alpha,\beta}_{n+1}(x_j)$. However, these values are not generally known in closed form, and we must compute them numerically... but how?

Traditional approaches involve applying the recurrence relation satisfied by the orthogonal polynomials, or solving the Jacobi matrix eigenvalue problem in the algorithm of Golub and Welsch, but these methods are inherently limited by a minimal complexity of $O(n^2)$. The current state-of-the-art is the $O(n)$ algorithm developed by Glasier, Liu, and Rokhlin, which hops from root to root using a Newton iteration evaluated with a Taylor series defined by the ODE satisfied by $P^{\alpha,\beta}_{n+1}$.

We propose an alternative approach, whereby the function and derivative evaluations required in the Newton iteration are computed independently in $O(1)$ operations per point using certain well-known asymptotic expansions. We shall review these expansions, outline the new algorithm, and demonstrate improvements in both accuracy and efficiency. 

Organisers: Hilary Priestley, Drew Moshier and Leo Cabrer.

This will be devoted to the applications of dualities to logic and algebra, focusing on general techniques. Thus it will seek to complement the specialised coverage in meetings devoted to, for example, modal logic, residuated structures and many-valued logics, or coalgebras. The featured topics for the Workshop will be drawn from completions of ordered structures, and applications; admissible rules, unification theory, interpolation and amalgamation; aspects of many-valued and substructural logics and ordered algebraic structures. Keynote speakers will be Leo Cabrer and Mai Gehrke.