We give a short exposition on the zeta determinant for a Laplace - type operator on a closed Manifold as first described by Ray and Singer in their attempt to find an analytic counterpart to R-torsion.
We combine recent breakthroughs in modularity lifting with a
3-5-7 modularity switching argument to deduce modularity of elliptic curves over real
quadratic fields. We
discuss the implications for the Fermat equation. In particular we
show that if d is congruent
to 3 modulo 8, or congruent to 6 or 10 modulo 16, and $K=Q(\sqrt{d})$
then there is an
effectively computable constant B depending on K, such that if p>B is prime,
and $a^p+b^p+c^p=0$ with a,b,c in K, then abc=0. This is based on joint work with Nuno Freitas (Bayreuth) and Bao Le Hung (Harvard).
Combining classifiers into an ensemble aims at a more accurate and robust classification decision compared to that of a single classifier. For a successful ensemble, the individual classifiers must be as diverse and as accurate as possible. Achieving both simultaneously is impossible, hence compromises have been sought by a variety of ingenious ensemble creating methods. While diversity has been in the focus of the classifier ensemble research for a long time now, the importance of the combination rule has been often marginalised. Indeed, if the ensemble members are diverse, a simple majority (plurality) vote will suffice. However, engineering diversity is not a trivial problem. A bespoke (trainable) combination rule may compensate for the flaws in preparing the individual ensemble members. This talk will introduce classifier ensembles along with some combination rules, and will demonstrate the merit of choosing a suitable combination rule.
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.
\\
\\
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.
\\
\\
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.
\\
\\
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.