
Fri 2 Sep 2011, 9:00am
SPECIAL
One Time Event
Math 100

Qualifying Exams

Math 100
Fri 2 Sep 2011, 9:00am4:00pm
Details
hide

Mathematics, UBC

Tue 6 Sep 2011, 3:30pm
Stochastic Dynamics Working Group
IAM lounge (LSK 306)

Organizational Meeting & Noise in PiecewiseSmooth Systems with Sliding

IAM lounge (LSK 306)
Tue 6 Sep 2011, 3:30pm4:30pm
Abstract
hide

Dan Coombs, Michael Doebeli, Ailana Fraser, Greg Martin

Wed 7 Sep 2011, 3:00pm
SPECIAL
One Time Event
Math 204

Graduate Awards Info Session

Math 204
Wed 7 Sep 2011, 3:00pm4:00pm
Details
hide

Ericsson

Wed 7 Sep 2011, 3:00pm
PIMS Seminars and PDF Colloquiums / Probability Seminar
WMAX 110

Some Transmission and Reception Algorithms for Superimposed Radio Signals

WMAX 110
Wed 7 Sep 2011, 3:00pm4:00pm
Abstract
The electromagnetic radio frequency spectrum is a scarce and valuable resource. Its utilization can be improved by allowing multilayer communications, in which several signals are simultaneously transmitted and received in the same frequency band. In this talk I will describe some algorithms for the transmission and reception of multilayer signals. These algorithms are compatible with commonly used transmitter and receiver equipment that was not designed for multilayer communications. Such algorithms have great practical importance because they yield increased network capacity and at the same time allow telecom equipment manufacturers (including both handset and network vendors) and wireless operators to obtain additional returns on their multibillion dollar investments.
hide

Academy of Finland and University of Sydney

Thu 8 Sep 2011, 3:30pm
Diff. Geom, Math. Phys., PDE Seminar
WMAX 110 (PIMS) (Schedule and location subject to change)

The AharonovBohm effect and the Calderon problem for connection Laplacians

WMAX 110 (PIMS) (Schedule and location subject to change)
Thu 8 Sep 2011, 3:30pm4:30pm
Abstract
The AharonovBohm eﬀect is a quantum mechanical phenomenon where electrons passing through a region of vanishing magnetic ﬁeld gets scattered due to topological eﬀects. It turns
out that this phenomenon is closely related to the cohomology of forms with integer coeﬃcients. We study this relationship from the point of view of the Calder´n problem and see that it can be captured in how Cauchy data of the connection laplacian determines uniquely the holonomy representation of the connection.
The work was partially supported by Finnish Academy of Science and by NSF Grant No.DMS0807502.
hide

UBC

Fri 9 Sep 2011, 3:00pm
Department Colloquium
MATX 1100

Quasisymmetric refinements of Schur functions

MATX 1100
Fri 9 Sep 2011, 3:00pm4:00pm
Abstract
Schur functions were introduced early in the last century with respect to representation theory, and since then have become important functions in other areas such as combinatorics and algebraic geometry. They have a beautiful combinatorial description in terms of diagrams, which allows many of their properties to be determined.
These symmetric functions form a subalgebra of the algebra of quasisymmetric functions, which date from the 1980s. Despite this connection, the existence of a natural quasisymmetric refinement of Schur functions has been considered unlikely.
In this talk we introduce quasisymmetric Schur functions, which partition Schur functions in an intuitive way. Furthermore, we show how these quasisymmetric Schur functions refine many wellknown Schur function properties with combinatorics that strongly reflect the classical case. This is joint work with Christine Bessenrodt, Jim Haglund, Kurt Luoto and Sarah Mason.
The talk will require no prior knowledge of any of the above terms.
hide

UBC

Mon 12 Sep 2011, 3:00pm
Algebraic Geometry Seminar
WMAX 110

Motivic DonaldsonThomas invariants for the one loop quiver with potential

WMAX 110
Mon 12 Sep 2011, 3:00pm4:00pm
Abstract
In this talk I will give an introduction to DonaldsonThomas invariants, and then their motivic incarnation. I'll discuss motivic vanishing cycles and lambda rings, before moving to the main example of the talk  the one loop quiver with potential. It turns out that the motivic DT invariants in this simple example have a neat presentation, and in a break with other worked out examples these invariants really involve the mondromy of the motivic vanishing cycle.
hide


Tue 13 Sep 2011, 2:00pm
Stochastic Dynamics Working Group
IAM Lounge (LSK 306)

Noise in PiecewiseSmooth Systems with Sliding (continued)

IAM Lounge (LSK 306)
Tue 13 Sep 2011, 2:00pm3:00pm
Abstract
hide

UBC

Tue 13 Sep 2011, 3:30pm
Diff. Geom, Math. Phys., PDE Seminar
WMAX 110 (PIMS)

A Selfdual Polar Factorization for Vector Fields

WMAX 110 (PIMS)
Tue 13 Sep 2011, 3:30pm4:30pm
Abstract
We show that any nondegenerate vector field u in L^{\infty}(\Omega, \R^N), where \Omega is a bounded domain in \R^N, can be written as {equation} \hbox{u(x)= \nabla_1 H(S(x), x) for a.e. x \in \Omega}, {equation} where S is a measure preserving point transformation on \Omega such that S^2=I a.e (an involution), and H: \R^N \times \R^N \to \R is a globally Lipschitz antisymmetric convexconcave Hamiltonian. Moreover, u is a monotone map if and only if S can be taken to be the identity, which suggests that our result is a selfdual version of Brenier's polar decomposition for the vector field u as u(x)=\nabla \phi (S(x)), where \phi is convex and S is a measure preserving transformation. We also describe how our polar decomposition can be reformulated as a selfdual mass transport problem.
hide

UBC

Tue 13 Sep 2011, 4:00pm
Algebraic Groups and Related Structures
Math 126

Central Simple Algebras with Involutions

Math 126
Tue 13 Sep 2011, 4:00pm5:00pm
Abstract
We continue our series on Central Simple Algebras with involution, introducing various group scheme associated with them.
hide

University of Chicago

Wed 14 Sep 2011, 3:00pm
Topology and related seminars
WMAX 216 (PIMS)

Isometries of aspherical manifolds

WMAX 216 (PIMS)
Wed 14 Sep 2011, 3:00pm4:30pm
Abstract
We describe some recent results about isometry groups of aspherical Riemannian manifolds, and also isometry groups of their universal covers. For instance, we show that on an irreducible locally symmetric space of dimension > 2, no metric has more symmetry than the locally symmetric metric.
hide

TU Berlin

Wed 14 Sep 2011, 3:00pm
Probability Seminar
MATH 126

Markov chain approximations to nonsymmetric diffusions with bounded coefficients

MATH 126
Wed 14 Sep 2011, 3:00pm4:00pm
Abstract
We consider a certain class of nonsymmetric Markov chains and obtain heat kernel bounds and parabolic Harnack inequalities. Using the heat kernel estimates, we establish a sufficient condition for the family of Markov chains to converge to nonsymmetric diffusions. As an application, we approximate nonsymmetric divergence forms with bounded coefficients by nonsymmetric Markov chains. This extends the results by StroockZheng to the nonsymmetric divergence forms.
Joint work with Takashi Kumagai.
hide

UBC

Wed 14 Sep 2011, 4:00pm
Undergraduate Colloquium
MATH 225

Everyday mathematics: two modelling case studies

MATH 225
Wed 14 Sep 2011, 4:00pm5:00pm
Abstract
This is the academic year's first UBC Undergraduate Mathematics Colloquium talk. UBC/UMC talks are meant for undergraduate students interested in mathematics beyond the curriculum. They are accessible at all levels. Our speakers are dynamic professional mathematicians with a reputation for interesting research and strong teaching.
Eric Cytrynbaum is our first speaker. Professor Cytrynbaum will talk about mathematical modelling. He will present two case studies from everyday life, including one describing the rise and fall of an airbreathing mammal attempting to maintain neutral buoyancy in the water.
hide

UBC

Fri 16 Sep 2011, 3:00pm
Department Colloquium
MATX 1100

Unlikely intersections in Arithmetic Dynamics

MATX 1100
Fri 16 Sep 2011, 3:00pm4:00pm
Abstract
Given a polynomial f with complex coefficients, and a complex number z, we call the orbit of z under f the set of all images of z under the iterates of f. If the orbit of z under f is finite, we call the number z preperiodic for f. We study the following two basic questions regarding orbits of complex numbers under polynomials.
1) For two polynomials f and g, and for two complex numbers a and b, when does the orbit of a under f intersect the orbit of b under g in infinitely many points?
2) For two polynomials f and g, when there exists an infinite set of complex numbers z which are preperiodic both for f and for g?
hide

UBC

Mon 19 Sep 2011, 3:00pm
Harmonic Analysis Seminar
MATX 1101

Buffon's needle probability for rational product Cantor sets

MATX 1101
Mon 19 Sep 2011, 3:00pm4:00pm
Abstract
We investigate the probability that "Buffon's Needle" lands near a onedimensional selfsimilar product set in the complex plane, where the similarity maps have rational centers and identical scalings. If the factors A and B are defined by at most 6 similarities, then the likelihood that the needle intersects an e^{n}neighborhood of such a set is at most Cn^{p/\log\log n} for some p>0.
hide

UBC

Mon 19 Sep 2011, 3:10pm
Algebraic Geometry Seminar
WMAX 110

Higher rank stable pairs and virtual localization

WMAX 110
Mon 19 Sep 2011, 3:10pm4:10pm
Abstract
We introduce a higher rank analog of the PandharipandeThomas theory of stable pairs on a CalabiYau threefold X. More precisely, we develop a moduli theory for frozen triples given by the data O_X^{r}(n)>F where F is a sheaf of pure dimension 1. The moduli space of such objects does not naturally determine an enumerative theory: that is, it does not naturally possess a perfect symmetric obstruction theory. Instead, we build a zerodimensional virtual fundamental class by hand, by truncating a deformationobstruction theory coming from the moduli of objects in the derived category of X. This yields the first deformationtheoretic construction of a higherrank enumerative theory for CalabiYau threefolds. We calculate this enumerative theory for local P^1 using the GraberPandharipande virtual localization technique. In a sequel to this project (arXiv:1101.2251), we show how to compute similar invariants associated to frozen triples using Kontsevich Soibelman, JoyceSong wallcrossing techniques.
hide

UBC

Tue 20 Sep 2011, 3:30pm
Diff. Geom, Math. Phys., PDE Seminar
WMAX 110 (PIMS)

Backward uniqueness for the heat equation in cones

WMAX 110 (PIMS)
Tue 20 Sep 2011, 3:30pm4:30pm
Abstract
I will talk about the backward uniqueness of the heat equation in unbounded domains. It is known that a bounded solution of the heat equation in a halfspace which becomes zero at some time must be identically zero, even though no assumptions are made on the boundary values of the solutions. In a recent example, Luis Escauriaza showed that this statement fails if the halfspace is replaced by cones with opening angle smaller than 90 degrees. In a joint work with Vladimir Sverak we show the result remains true for cones with opening angle larger than 110 degrees. Our proof covers heat equations having lowerorder terms with bounded measurable coefficients.
hide

UBC

Tue 20 Sep 2011, 3:30pm
Algebraic Groups and Related Structures
MATX 1101

Cohomology of Central Simple Algebra with involution

MATX 1101
Tue 20 Sep 2011, 3:30pm5:00pm
Abstract
We define and discuss representable functors and groups schemes, as a first step towards describing the various group schemes associated to central simple algebras with involution.
hide


Wed 21 Sep 2011, 2:00pm
Complex Fluids Seminar
Math 125

KelvinHelmholtz instabilities in sheared density stratified flows

Math 125
Wed 21 Sep 2011, 2:00pm3:00pm
hide

UBC

Wed 21 Sep 2011, 3:00pm
Topology and related seminars
WMAX 216 (PIMS)

Expander graphs, metrics and knots.

WMAX 216 (PIMS)
Wed 21 Sep 2011, 3:00pm4:00pm
Abstract
We prove that every closed, smooth manifold of at least dimension 3 admits a sequence of Riemannian metrics with pinched curvature, volume tending to infinity but whose first eigenvalue of the Laplacian remains bounded away from 0. As a consequence we construct sequences of hyperbolic knots whose complements have again volume tending to infinity and whose Cheeger constant is uniformly bounded away from 0. This is joint work with Marc Lackenby.
hide

UBC

Wed 21 Sep 2011, 3:00pm
Probability Seminar
MATH 126

Metastable densities for contact processes on random graphs

MATH 126
Wed 21 Sep 2011, 3:00pm4:00pm
Abstract
Joint work with Thomas Mountford and Qiang Yao. We consider
the contact process on a random graph chosen with a fixed degree,
power law distribution, according to a model proposed by Newman,
Strogatz and Watts (2001). We follow the work of Chatterjee and
Durrett (2009) who showed that for arbitrarily small infection
parameter $\lambda > 0$, the limiting metastable density does not tend
to zero as the graph size becomes large. We show three distinct
regimes for this density depending on the tail of the degree law.
References:
 Charterjee,S. and Durrett, R. (2009): Contact process on random
graphs with degree power law distribution have critical value zero.
Annals of Probability 37 (2009)
 Newman, M.E.J., Strogatz, S.H. and Watts, D.J. (2001): Random graphs
with arbitrary degree distributions and their applications. Physical
Review E. 64, paper 026118
hide

SFU/UBC

Thu 22 Sep 2011, 3:00pm
Number Theory Seminar
Room ASB 10900 (IRMACS  SFU Campus)

Messing with perfection

Room ASB 10900 (IRMACS  SFU Campus)
Thu 22 Sep 2011, 3:00pm3:50pm
Abstract
Let s(n) denote the sum of the proper divisors of n, so, e.g., s(4)=1+2=3. A natural number n is called *perfect* if s(n)=n and *amicable* if s(n) =/= n but s(s(n))=n. For example, 6 is a perfect number, and 220 is an amicable number. Questions about perfect and amicable numbers constitute some of the oldest unsolved problems in mathematics. I will talk about old and new theorems concerning these numbers and their generalizations. Some of this is joint work with Mits Kobayashi (Cal Poly Pomona), Florian Luca (Universidad Nacional Autónoma de México), and Carl Pomerance (Dartmouth College).
hide

UBC

Thu 22 Sep 2011, 3:30pm
Algebraic Groups and Related Structures

Group schemes associated to central simple algebras with involution

Thu 22 Sep 2011, 3:30pm4:30pm
Abstract
We continue to describe the foundation of group schemes theory to later discuss group schemes associated to central simple algebras with involution.
hide

SFU

Thu 22 Sep 2011, 4:10pm
Number Theory Seminar
Room ASB 10900 (IRMACS  SFU Campus)

Imaginary quadratic class numbers and Sha on congruent number curves

Room ASB 10900 (IRMACS  SFU Campus)
Thu 22 Sep 2011, 4:10pm5:00pm
Abstract
We consider two classical number theoretic problems that may seem quite
unrelated:
* What is the power of 2 dividing the class number of Q(sqrt(n))
* Which n are congruent numbers (n called congruent if it occurs as the
area of a rightangled triangle with rational length sides)
The second question is equivalent to determining whether the elliptic curve E_n: y^2=x^3n^2*x has positive rank. This observation suggest we might want to consider:
* What is the power of 2 in the order of Sha(E_n).
If we restrict to prime values n=p, it is already known that partial answers to these questions can be related to the splitting of p in the quartic number field Q(sqrt(1+i)).
In this talk we will discuss the next step in the classification.
hide

U. ParisSud, Orsay

Fri 23 Sep 2011, 3:00pm
Department Colloquium
MATX 1100 (PIMS/UBC Distinguished Colloquium)

The hypoelliptic Laplacian

MATX 1100 (PIMS/UBC Distinguished Colloquium)
Fri 23 Sep 2011, 3:00pm4:00pm
Abstract
If X is a Riemannian manifold, the Laplacian is a second order elliptic operator on X. The hypoelliptic Laplacian L_b is an operator acting on the total space of the tangent bundle of X, that is supposed to interpolate between the elliptic Laplacian (when b > 0) and the geodesic flow (when b > \infty). Up to lower order terms, L_b is a weighted sum of the harmonic oscillator along the fibre TX and of the generator of the geodesic flow. In the talk, we will explain the underlying algebraic, analytic and probabilistic aspects of its construction, and outline some of the applications obtained so far.
Other Information:JeanMichel Bismut was born in 1948. He is a Professor of Mathematics at University ParisSud (Orsay), and a member of the Academie des Sciences, of the Academia Europaea, and of the Deutsche Akademie Leopoldina. He received his 'Doctorat d'Etat' from Universite Paris VI in 1973 for his work in the control of stochastic processes. His interests in probability theory led him to study refinements of the index theorem of AtiyahSinger. Through his work on Quillen metrics, he participated to the proof of a RiemannRoch theorem in arithmetic geometry. He constructed an exotic Hodge theory, whose corresponding Laplacian is a hypoelliptic operator on the cotangent bundle of a Riemannian manifold. Recently, he used the hypoelliptic Laplacian to give a new approach to the evaluation of orbital integrals. JeanMichel Bismut was a plenary speaker at the International Congress of Mathematics in Berlin in 1998, and a vicepresident of International Mathematical Union (I.M.U.) from 2002 to 2006.
hide

UBC

Mon 26 Sep 2011, 3:10pm
Algebraic Geometry Seminar
WMAX 110

Cox rings and pseudoeffective cones of projectivized toric vector bundles

WMAX 110
Mon 26 Sep 2011, 3:10pm4:10pm
Abstract
Projectivized toric vector bundles are a large class of rational varieties that share some of the pleasant properties of toric varieties and other Mori dream spaces. Hering, Mustata and Payne proved that the Mori cones of these varieties are polyhedral and asked if their Cox rings are indeed finitely generated. We present the complete answer to this question. There are several proofs of a positive answer in the rank two case [HausenSuss, Gonzalez]. One of these proofs relies on the simple structure of the Okounkov body of these varieties with respect to a special flag of subvarieties. For higher ranks we study projectivizations of a special class of toric vector bundles that includes cotangent bundles, whose associated Klyachko filtrations are particularly simple. For these projectivized bundles, we give generators for the cone of effective divisors and a presentation of the Cox ring as a polynomial algebra over the Cox ring of a blowup of a projective space along a sequence of linear subspaces [GonzalezHeringPayneSuss]. As applications, we show that the projectivized cotangent bundles of some toric varieties are not Mori dream spaces and give examples of projectivized toric vector bundles whose Cox rings are isomorphic to that of M_{0,n}.
hide

U. ParisSud, Orsay

Mon 26 Sep 2011, 4:00pm
SPECIAL
Diff. Geom, Math. Phys., PDE Seminar / Probability Seminar
MATX 1100

The Langevin process and the trace formula

MATX 1100
Mon 26 Sep 2011, 4:00pm5:00pm
Abstract
I will explain the probabilistic interpretation of the hypoelliptic Laplacian L_b . To L_b, one can associate the diﬀusion on the manifold X that is a solution of the diﬀerential equation b^2 x'' = −x' + w'. For b = 0, we get x' = w', the equation of Brownian motion, and for b = +∞, we obtain the equation of geodesics x'' = 0. I will explain the rigorous results one can derive on the corresponding heat kernels via the Malliavin calculus. These will include uniform Gaussian decay of the hypoelliptic heat kernel over a symmetric space.
hide

UBC

Tue 27 Sep 2011, 2:00pm
Stochastic Dynamics Working Group
IAM Lounge

A Stochastic Dynamics Approach to Some Neuron Models

IAM Lounge
Tue 27 Sep 2011, 2:00pm3:00pm
Abstract
hide

UBC and Dalian University of Technology (Dalian, China)

Tue 27 Sep 2011, 2:30pm
Symmetries and Differential Equations Seminar
Math Annex 1118

Nonlocally related systems, nonlocal symmetries and new exact solutions of the nonlinear Kompaneets equation

Math Annex 1118
Tue 27 Sep 2011, 2:30pm3:30pm
Abstract
The Kompaneets equation describes the spectra of photons interacting with a rarefied electron gas and includes three parameters. In 2010, Ibragimov obtained some timedependent exact solutions for several restrictions of the parameters of this equation. In this talk, a tree of equivalent nonlocally related PDE systems is constructed for the nonlinear Kompaneets (NLK) equation. The tree includes some nonlocally related equivalent subsystems. For a twoparameter class of NLK equations, a point symmetry classification is given of these nonlocally related PDE systems and shown to yield previosuly unknown nontrivial nonlocal symmetries of the NLK equation.
Invariant solutions arising from these nonlocal symmetries are shown to yield wider classes of timedependent exact solutions for the NLK equation beyond those previously obtained by Ibragimov. In particular, for five classes of initial conditions, each involving two parameters, previously unknown explicit solutions are obtained. Interestingly, each of these solutions is expressed in terms of elementary functions. Three of the classes exhibit quiescent behaviour, and the other two classes exhibit blow up behaviour in finite time. As a consequence, it is shown that the corresponding nontrivial stationary solutions, obtained by Dubinov in 2009, are unstable. In particular, it is shown that only the stationary quiescent solution is stable.
hide

U. ParisSud, Orsay

Tue 27 Sep 2011, 3:30pm
Diff. Geom, Math. Phys., PDE Seminar
WMAX 110 (PIMS)

Orbital integrals and the hypoelliptic Laplacian

WMAX 110 (PIMS)
Tue 27 Sep 2011, 3:30pm5:00pm
Abstract
Third talk in the series. If G is a reductive Lie group with Lie algebra g, orbital integrals are key ingredient in Selberg’s trace formula. I will explain how one can think of the evaluation of orbital integrals as the computation of a Lefschetz trace. Using in particular the Dirac operator of Kostant, the standard Casimir operator of X = G/K is deformed to a hypoelliptic operator L_b acting on the total space of a canonically ﬂat vector bundle on X, that contains TX as a subbundle. The symbol of this hypoelliptic operator is exactly the one described in the previous talks. When descending the situation to a locally symmetric space, the spectrum of the original Casimir remains rigidly embedded in the spectrum of the hypoelliptic deformation. Making b → +∞ gives an explicit evaluation of semisimple orbital integrals.
hide

UBC

Tue 27 Sep 2011, 4:00pm
Discrete Math Seminar
MATX 1102

Rational distance sets

MATX 1102
Tue 27 Sep 2011, 4:00pm5:00pm
Abstract
A rational distance set is a subset of the real plane such that all
pairwise distances are rational numbers. It's not too hard to
construct an infinite rational distance set contained in a line or in
a circle, but if you do not allow 3 points on a line or 4 on a circle,
the current record is a set of 7 points, found a few years ago with a
computer. On the other hand, no one knows if a rational distance set,
no 3 points on a line or 4 on a circle, could be infinite. Erdős
conjectured that it would have to have a very special form, like an
algebraic curve.
In a paper with Jozsef Solymosi we showed that the only algebraic
curves that contain infinite rational distance sets are lines and
circles. In my talk I will explain the ideas involved and give an
outline of our proof.
hide

UBC

Wed 28 Sep 2011, 3:00pm
Probability Seminar
MATH 126

Stochastic geometric representations of the quantum CurieWeiss model

MATH 126
Wed 28 Sep 2011, 3:00pm4:00pm
Abstract
We develop path integral representations for Quantum Ising models. To
connect with classical FortuinKasteleyn (FK) representation, we begin by
presenting the FK representation of the classical CurieWeiss model (the
Ising model on complete graph) via the language of Poisson Point Processes.
We then show how to derive a general FK representation for Quantum Ising
model. This representation was originally developed by M. Campanino, A.
Klein, J.F. Perez (1991) and M. Aizenman, A. Klein, C.M. Newman (1993).
We apply the above to the quantum CurieWeiss model in transversal field.
First, we present the full FK representation of this model. Examining the
form of the resulting measure and dropping the weight component from it
leads to the natural extension of the Erd\"os  Rényi random graphs. Finally,
we consider the ground state of the quantum CurieWeiss model via partial FK
representation. We prove the existence of a phase transition in the ground
state when the strength of the transversal field equals one.
hide

UBC

Fri 30 Sep 2011, 3:00pm
Department Colloquium
MATX 1100

Bounds on pointline incidences

MATX 1100
Fri 30 Sep 2011, 3:00pm4:00pm
Abstract
A central result in discrete geometry is the SzemerediTrotter theorem which gives a sharp bound on the number of pointline incidences in the Euclidean plane. The result has various generalizations and applications.
In this talk we prove an extension of the SzemerediTrotter theorem and we show some new applications. For example, using incidence bounds, we show that if M is an nelement set of k x k matrices with real coefficients such that det(A  B) is not zero for any distinct A,B elements of M and V,W are nelement sets of k dimensional vectors, then V +W + MW >> n^{5/4}.
Part of the talk is based on joint work with Terry Tao.
hide

Note for Attendees
Tea and cookies will be served in the Math 125 Lounge at approximately 2:45 p.m. prior to the colloquium.