Past

An investor trades a safe and several risky assets with linear price impact to maximize expected utility from terminal wealth.

In the limit for small impact costs, we explicitly determine the optimal policy and welfare, in a general Markovian setting allowing for stochastic market,

cost, and preference parameters. These results shed light on the general structure of the problem at hand, and also unveil close connections to

optimal execution problems and to other market frictions such as proportional and fixed transaction costs.

Direct-search methods are a class of popular derivative-free

algorithms characterized by evaluating the objective function

using a step size and a number of (polling) directions.

When applied to the minimization of smooth functions, the

polling directions are typically taken from positive spanning sets

which in turn must have at least n+1 vectors in an n-dimensional variable space.

In addition, to ensure the global convergence of these algorithms,

the positive spanning sets used throughout the iterations

must be uniformly non-degenerate in the sense of having a positive

(cosine) measure bounded away from zero.

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However, recent numerical results indicated that randomly generating

the polling directions without imposing the positive spanning property

can improve the performance of these methods, especially when the number

of directions is chosen considerably less than n+1.

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In this talk, we analyze direct-search algorithms when the polling

directions are probabilistic descent, meaning that with a certain

probability at least one of them is of descent type. Such a framework

enjoys almost-sure global convergence. More interestingly, we will show

a global decaying rate of $1/\sqrt{k}$ for the gradient size, with

overwhelmingly high probability, matching the corresponding rate for

the deterministic versions of the gradient method or of direct search.

Our analysis helps to understand numerical behavior and the choice of

the number of polling directions.

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This is joint work with Clément Royer, Serge Gratton, and Zaikun Zhang.

In this talk we are concerned with the stability of steady transonic shocks in supersonic flow around a wedge. 2-D and M-D potential stability will be presented.

This talk is based on the joint works with Prof. G.-Q. Chen, and Prof. S.X. Chen.

Many interesting properties of groups are inherited by their subgroups examples of such are finiteness, residual finiteness and being free. People have asked whether hyperbolicity is inherited by subgroups, there are a few counterexamples in this area. I will be detailing the proof of some of these including a construction due to Rips of a finitely generated not finitely presented subgroup of a hyperbolic group and an example of a finitely presented subgroup which is not hyperbolic.

A major project in number theory runs as follows. Suppose some Diophantine equation has infinitely many integer solutions. One can then ask how common solutions are: roughly how many solutions are there in integers $\in [ -B, \, B ] $? And ideally one wants an answer in terms of the geometry of the original equation.

What if we ask the same question about Diophantine inequalities, instead of equations? This is surely a less deep question, but has the advantage that all the geometry we need is over $\mathbb{R}$. This makes the best-understood examples much easier to state and understand.

I will start by introducing Somos sequences, defined by innocent-looking quadratic recursions which, surprisingly, always return integer values. I will then explain how they can be viewed in a much larger context, that of the Laurent phenomenon in the theory of cluster algebras. Some further steps take us to the the quantum cluster positivity conjecture of Berenstein and Zelevinski. I will finally explain how, following Nagao and Efimov, cohomological Donaldson-Thomas theory leads to a proof of this conjecture in some, perhaps all, cases. This is joint work with Davison, Maulik, Schuermann.

We introduce and develop a theory of limits for sequences of sparse graphs based on $L^p$ graphons, which generalizes both the existing $L^\infty$ theory of dense graph limits and its extension by Bollob\'as and Riordan to sparse graphs without dense spots. In doing so, we replace the no dense spots hypothesis with weaker assumptions, which allow us to analyze graphs with power law degree distributions. This gives the first broadly applicable limit theory for sparse graphs with unbounded average degrees.

Joint work with Christian Borgs, Jennifer T. Chayes, and Henry Cohn.

The modular invariant partition functions for SU(2) and SU(3)

conformal field theories have been classified. The SU(2) theory is closely

related to the preprojective algebras of Coxeter-Dynkin quivers. The

analogous finite dimensional superpotential algebras, which we call almost

Calabi-Yau algebras, associated to the SU(3) invariants will be discussed.

The periodic KdV equation $u_t=u_{xxx}+\beta uu_x$ arises from a Hamiltonian system with infinite-dimensional phase space $L^2({\bf T})$. Bourgain has shown that there exists a Gibbs probability measure $\nu$ on balls $\{\phi :\Vert \phi\Vert^2_{L^2}\leq N\}$ in the phase space such that the Cauchy problem for KdV is well posed on the support of $\nu$, and $\nu$ is invariant under the KdV flow. This talk will show that $\nu$ satisfies a logarithmic Sobolev inequality. The seminar presents logarithmic Sobolev inequalities for the modified periodic KdV equation and the cubic nonlinear Schr\"odinger equation. There will also be recent results from Blower, Brett and Doust regarding spectral concentration phenomena for Hill's equation.

This talk will discuss the discovery of Heegner points from a historic perspective. They are a beautiful application of analytic techniques to the study of rational points on elliptic curves, which is now a ubiquitous theme in number theory. We will start with a historical account of elliptic curves in the 60's and 70's, and a correspondence between Birch and Gross, culminating in the Gross-Zagier formula in the 80's. Time permitting, we will discuss certain applications and ramifications of these ideas in modern number theory. 

We shall discuss the statistics of the eigenvalues of large random Hermitian matrices when the temperature is very high. In particular we shall focus on the transition from Wigner/Airy to Poisson regime.

We produce new families of steady and expanding Ricci solitons
that are not of Kahler type. In the steady case, the asymptotics are
a mixture of the Hamilton cigar and the Bryant soliton paraboloid
asymptotics. We obtain some examples of Ricci solitons on homeomorphic
but non-diffeomorphic spaces. We also find numerical evidence of solitons
with more complicated topology.

 We'll discuss the connection between irreducibilty, D- and 
aD-spaces.

In elasticity theory, one naturally requires that the Jacobian determinant of the deformation is positive or even a-priori prescribed (for example incompressibility). However, such strongly non-linear and non-convex constraints are difficult to deal with in mathematical models. In this talk, which is based on joint work with K. Koumatos (Oxford) and E. Wiedemann (UBC/PIMS), I will present various recent results on how this constraint can be manipulated in subcritical Sobolev spaces, where the integrability exponent is less than the dimension.

In particular, I will give a characterization theorem for Young measures under this side constraint, which are widely used in the Calculus of Variations to model limits of nonlinear functions of weakly converging "generating" sequences. This is in the spirit of the celebrated Kinderlehrer--Pedregal Theorem and based on convex integration and "geometry" in matrix space.

Finally, applications to the minimization of integral functionals, the theory of semiconvex hulls, incompressible extensions, and approximation of weakly orientation-preserving maps by strictly orientation-preserving ones in Sobolev spaces are given.

Attempting to extend the methods of critical point theory (e.g., those of Morse theory and Lusternik-Schnirelman theory) to the study of strong local minimizers of integral functionals of the calculus of variations I will describe how the obstruction method of algebraic topology can be successfully used to tackle the enumeration problem for various homotopy classes of maps in Sobolev spaces and that how this will result in precise lower bounds on the number of such local minimizers in terms of convenient topological invariants of the underlying spaces. I will then move on to dicussing variants as well as applications of the result to some classes of geometric nonlinear PDEs in particular problems in nonlinear elasticity.

In the theory of dislocations one is naturally led to consider energies of “line tension” type concentrated on lines. These lines may have a local vector-valued multiplicity, and the energy may depend on this multiplicity and on the orientation of the line. In the two-dimensional case this problem reduces to the classical problem of energies defined on partitions which arises in the sharp-interface models for phase transitions. 

I will introduce the main results concerning functionals in the calculus of variations that are defined on partitions. Such partitions are nicely characterized as level sets of function with bounded variations with a discrete set of values.  In this setting I will recall the characterization of the lower semicontinuity and the relaxation formula, which gives rise to the notion of BV-ellipticity. The case of dislocations in a three-dimensional crystal requires a formulation in the setting of 1-rectifiable currents with multiplicity in a lattice. In this context I will describe the main results and some examples of interest, in which relaxation is necessary and can be characterized.

The Maxwell equation is an intermediate linear model for

Einstein's equation lying between the scalar wave equation and the

linearised Einstein equation. This talk will present the 5 key

estimates necessary to prove a uniform bound on an energy and a

Morawetz integrated local energy decay estimate for the nonradiating

part.

The major obstacles, relative to the scalar wave equation are: that a

scalar equation must be found for at least one of the components,

since there is no known decay estimate directly at the tensor level;

that the scalar equation has a complex potential; and that there are

stationary solutions and, in the nonzero $a$ Kerr case, it is more

difficult to project away from these stationary solutions.

If time permits, some discussion of a geometric proof using the hidden

symmetries will be given.

This is joint work with L. Andersson and is arXiv:1310.2664.

We will present recent results regarding conservation laws for the wave equation on null hypersurfaces.  We will show that an important example of a null hypersurface admitting such conserved quantities is the event horizon of extremal black holes. We will also show that a global analysis of the wave equation on such backgrounds implies that certain derivatives of solutions to the wave equation asymptotically blow up along the event horizon of such backgrounds. In the second part of the talk we will present a complete characterization of null hypersurfaces admitting conservation laws. For this, we will introduce and study the gluing problem for characteristic initial data and show that the only obstruction to gluing is in fact the existence of such conservation laws.

Joint Work with Philippe G. LeFloch. We consider vacuum
spacetimes with two spatial Killing vectors and with initial data
prescribed on $T^3$. The main results that we will present concern the
future asymptotic behaviour of the so-called polarized solutions. Under
a smallness assumption, we derive a full set of asymptotics for these
solutions. Within this symetry class, the Einstein equations reduce to a
system of wave equations coupled to a system of ordinary differential
equations. The main difficulty, not present in previous study of similar
systems, is that, even in the limit of large times, the two systems do
not directly decouple. We overcome this problem by the introduction of a
new system of ordinary differential equations, whose unknown are
renormalized variables with renormalization depending on the solution of
the non-linear wave equations.

In this talk I will discuss the Cauchy problem for bounded

self-gravitating elastic bodies in Einstein gravity. One of the main

difficulties is caused by the fact that the spacetime curvature must be

discontinuous at the boundary of the body. In order to treat the Cauchy

problem, one must show that the jump in the curvature propagates along

the timelike boundary of the spacetime track of the body. I will discuss

a proof of local well-posedness which takes this behavior into account.

I describe recent unique continuation results for linear wave equations obtained jointly with Spyros Alexakis and Arick Shao. They state, informally speaking, that solutions to the linear wave equation on asymptotically flat spacetimes are completely determined, in a neighbourhood of infinity, from their radiation towards infinity, understood in a suitable sense. We find that the mass of the spacetime plays a decisive role in the analysis.

We consider spacetimes arising from perturbations of the interior of Kerr

black holes. These spacetimes have a null boundary in the future such that

the metric extends continuously beyond. However, the Christoffel symbols

may fail to be square integrable in a neighborhood of any point on the

boundary. This is joint work with M. Dafermos

We present a mechanism of shock formation for a class of quasilinear wave equations. The solutions are stable and no symmetry assumption is assumed. The proof is based on the energy estimates and on the study of Lorentzian geometry defined by the solution.

 When given an explicit solution to an evolutionary partial differential equation, it is natural to ask whether the solution is stable, and if yes, what is the mechanism for stability and whether this mechanism survives under perturbations of the equation itself. Many familiar linear equations enjoy some notion of stability for the zero solution: solutions of the heat equation dissipate and decay uniformly and exponentially to zero, solutions of the Schrödinger equations disperse at a polynomial rate in time depending on spatial dimension, while solutions of the wave equation enjoy radiative decay (in the presence of at least two spatial dimensions) also at polynomial rates.

For this set of short course sessions, we will focus on the wave equation and its nonlinear perturbations. As mentioned above, the stability mechanism for the linear wave equation is that of radiative decay. Radiative decay depends on the number of spatial dimensions, and hence so does the stability of the zero solution for nonlinear wave equations. By the mid-1980s it was well understood that the stability mechanism survives generally (for “smooth nonlinearities”) when the spatial dimension is at least four, but for lower dimensions (two and three specifically; in dimension one there is no linear stability mechanism to start with) obstructions can arise when the nonlinearities are “stronger” than can be controlled by radiative decay. This led to the discovery of the null condition as a structural condition on the nonlinearities preventing the aforementioned obstructions. But what happens when the null condition is violated? This development spanning a quarter of a century, from F. John’s qualitative analysis of the spherically symmetric case, though S. Alinhac’s sharp control of the asymptotic lifespan, and culminating in D. Christodoulou’s full description of the null geometry, is the subject of this short course.

(1) We will start by reviewing the radiative decay mechanism for wave equations, and indicate the nonlinear stability results for high spatial dimensions. We then turn our attention to the case of three spatial dimensions: after a quick discussion of the null condition for quasilinear wave equations, we sketch, at the semilinear level, what happens when the null condition fails (in particular the asymptotic approximation of the solution by a Riccati equation).

(2) The semilinear picture is built up using a version of the method of characteristics associated with the standard wave operator. Turning to the quasilinear problem we will hence need to understand the characteristic geometry for a variable coefficient wave operator. This leads us to introduce the optical/acoustical function and its associated null structure equations.

(3) From this modern geometric perspective we next discuss, in some detail, the blow-up results obtained in the mid-1980s by F. John for quasilinear wave equations assuming radial symmetry.

(4) Finally, we indicate the main difficulties in extending the analysis to the non-radially-symmetric case, and how they can be resolved à la the recent tour de force of D. Christodoulou. While some knowledge of Lorentzian geometry and dynamics of wave equations will be helpful, this short course should be accessible to also graduate students with training in partial differential equations.

Imperial College London, United Kingdom E-mail address: @email

École Polytechnique Fédérale de Lausanne, Switzerland E-mail address: @email

Question: Is it a realistic proposition for a mathematician to use his/her skills to make a living from sports betting? The introduction of betting exchanges have fundamentally changed the potential profitability of gambling, and a professional mathematician's arsenal of numerical and theoretical weapons ought to give them a huge natural advantage over most "punters", so what might be realistically possible and what potential risks are involved? This talk will give some idea of the sort of plan that might be required to realise this ambition, and what might be further required to attain the aim of sustainable gambling profitability.

We show that for every $n\ge 2$ there exists a torsion-free one-ended word-hyperbolic group $G$ of rank $n$ admitting generating $n$-tuples $(a_1,\ldots ,a_n)$ and $(b_1,\ldots ,b_n)$ such that the $(2n-1)$-tuples $$(a_1,\ldots ,a_n, \underbrace{1,\ldots ,1}_{n-1 \text{ times}})\hbox{ and }(b_1,\ldots, b_n, \underbrace{1,\ldots ,1}_{n-1 \text{ times}} )$$ are not Nielsen-equivalent in $G$. The group $G$ is produced via a probabilistic construction (joint work with Ilya Kapovich).

TQFTs have received widespread attention in recent years. In mathematics

for example due to Lurie's proof of the cobordism hypothesis. In physics

they are used as toy models to understand structure, especially

boundaries and defects.

I will present a lattice construction of 2d Spin TFT. This mostly

motivated as both a toy model and stepping stone for a mathematical

construction of rational conformal field theories with fermions.

I will first describe a combinatorial model for spin surfaces that

consists of a triangulation and a finte set of extra data. This model is

then used to construct TFT correlators as morphisms in a symmetric

monoidal category, given a Frobenius algebra as input. The result is

shown to be independent of the triangulation used, and one obtains thus

a 2dTFT.

All results and constructions can be generalised to framed surfaces in a

relatively straightforward way.

 This lecture will cover two recent works on the mock modular
forms of Ramanujan.

I. Solution of Ramanujan's original conjectures about these functions.
(Joint work with Folsom and Rhoades)

II. A new theorem that mock modular forms are "generating functions" for
central L-values and derivatives of quadratic twist L-functions.
(Joint work with Alfes, Griffin, Rolen).

Worst-case portfolio optimization has been introduced in Korn and Wilmott

(2002) and is based on distinguishing between random stock price

fluctuations and market crashes which are subject to Knightian

uncertainty. Due to the absence of full probabilistic information, a

worst-case portfolio problem is considered that will be solved completely.

The corresponding optimal strategy is of a multi-part type and makes an

investor indifferent between the occurrence of the worst possible crash

and no crash at all.

We will consider various generalizations of this setting and - as a very

recent result - will in particular answer the question "Is it good to save

for bad times or should one consume more as long as one is still rich?"

Stick-slip behavior is a distinguishing characteristic of the flow of Whillans Ice Stream. Distinct from stick-slip on northern hemisphere glaciers, which is generally attributed to supraglacial melt, the behavior is thought be be controlled by fast processes at the bed and by tidally-induced stress. Modelling approaches to studying this phenomenon typically consider ice to be an elastically-deforming solid (e.g. Winberry et al, 2008; Sergienko et al, 2009). However, there remains a question of whether irreversible, i.e. viscous, deformation is important to the stick-slip process; and furthermore whether the details of stick-slip oscillations are important to ice stream evolution on longer time scales (years to decades).

To address this question I use two viscoelastic models of varying complexity. The first is a modification to the simple block-and-slider models traditionally used to examine earthquake processes on a very simplistic fashion. Results show that the role of viscosity in stick-slip depends on the dominant stress balance. These results are then considered in the context of a continuum description of a viscoelastic ice stream with a rate-weakening base capable of exhibiting stick-slip behavior. With the continuum model we examine the spatial and temporal aspects of stick-slip, their dependence on viscous effects, and how this behavior impacts the mean flow. Different models for the evolution of basal shear stress are examined in the experiments, with qualitatively similar results. A surprising outcome is that tidal effects, while greatly affecting the spectrum of the stick-slip cycle, may have relatively little effect on the mean flow.