We present a change of variables under which the elliptic Calogero Hamiltonian (a two-dimensional Lame operator) becomes an algebraic operator with polynomial coefficients. We show that the model is equivalent to the sl(3) Euler-Arnold quantum top in a magnetic field the strength of which is defined by the coupling constant. For discrete values of the coupling constant a finite number of polynomial eigenfunctions occur. A three parameter pair of commuting differential operators in two variables of degree 2 and 3 is constructed.

# Seminar Physique Mathématique -- The three-body elliptic calogero model

# Seminar Logic, Category Theory, and Computation -- "By an easy induction..."

We discuss the countable lifting property. An error in an induction argument by Topping has raised interest in when the CLP holds. We review the case of maps between rings of continuous functions. The main content is on a counterexample due to Burgess to a suggestion made that a positive result holds for vector lattices over finite fields.

# Geometric Analysis -- Infinitely many solutions for the Schrodinger equations in RN with critical growth

# A Note on Efficiency Gains from Multiple Incomplete Subsamples

We demonstrate efficiency gains in estimation by optimally using multiple subsamples all but one of which are incomplete following a monotone pattern. The finite dimensional parameter of interest is defined by moment restrictions on a target population which is some arbitrary union of the possibly different subpopulations for the multiple subsamples. A form of the missing at random (MAR) assumption is made for identification. MAR also makes the information contained in each incomplete subsample usable and thus contributes to efficiency gains. We show that the characteristics and possibility of such efficiency gains can be very different from those in the two subsamples contexts that have been studied extensively in the literature. Implication of these results on possible sampling strategies is briefly noted. We also show that a set of unconditional and conditional moment restrictions exhausts all the relevant information in the subsamples and can be easily used in a Frisch-Waugh-Lovell type sequential way, by virtue of monotonicity, for efficient estimation of the parameter of interest.

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**Bio: **

Saraswata Chaudhuri is an assistant professor of Economics. His research focuses on econometrics with micro-level data. He received his PhD in Economics from University of Washington and was an assistant professor at University of North Carolina before moving to McGill.

# "By an easy induction..."

We discuss the countable lifting property. An error in an induction argument by Topping has raised interest in when the CLP holds. We review the case of maps between rings of continuous functions. The main content is on a counterexample due to Burgess to a suggestion made that a positive result holds for vector lattices over finite fields.

# CANCELLED Seminar Geometric Analysis -- Non-uniqueness results for the anisotropic Calderon problem with data measured

In this talk, we shall give some simple counterexamples to uniqueness for the Calderon problem on Riemannian manifolds with boundary when the Dirichlet and Neumann data are measured on disjoint sets of the boundary. We provide counterexamples in the case of two and three dimensional Riemannian manifolds with boundary having the topology of circular cylinders in dimension two and toric cylinders in dimension three. The construction could be easily extended to higher dimensional Riemannian manifolds. This is joint work with Niky Kamran (McGill University) and Francois Nicoleau (Universite de Nantes).

# Seminar Géométrie et topologie/Geometry-Topology -- A construction of slice knots via annulus modification

We define an operation on homology four ball which we call an n-twist annulus modification. We give a new construction of smoothly slice knots and exotically slice knots via n-twist annulus modifications. As an application, we present a new example of a smoothly slice knot with non-slice derivatives. Such examples were first discovered by Cochran and Davis. Also, we relate n-twist annulus modifications to n-fold annulus twists which was first introduced by Osoinach, then has been used by Abe and Tange to construct smoothly slice knots. Furthermore we consider n-twist annulus modifications in more general setting to show that any exotically slice knot can be obtained by the image of the unknot in the boundary of a smooth 4-manifold homeomorphic to B4 after an annulus modification.

# Seminar Probabilités-Probability -- Busemann functions and geodesics for the corner growth model

We consider the directed last-passage percolation model on the planar integer lattice with nearest-neighbor steps and general i.i.d. weights on the vertices, outside the class of exactly solvable models. Stationary cocycles are constructed for this percolation model from queueing fixed points. These cocycles define solutions to variational formulas that characterize limit shapes and yield new results for Busemann functions, geodesics and the competition interface. This is joint work with Nicos Georgiou and Timo Seppalainen.

# Working Seminar Mathematical Physics -- Enumerative geometry tau-functions and Virasoro group operators

In this talk I will describe relations between three enumerative geometry tau-functions: the Kontsevich--Witten, Hurwitz and Hodge tau-functions. The relations allow us to describe the tau-functions in terms of matrix integrals, Virasoro constraints and Kac--Schwarz operators. All constructed operators belong to the algebra (or group) of symmetries of the KP hierarchy.

# Colloquium DIRO -- Position-based cryptography

On 20 July 1969, millions of people held their breath as they watched, live on television, Neil Armstrong set foot on the Moon. Yet Fox Television has reported that a staggering 20% of Americans have had doubts about the Apollo 11 mission. Could it have been a hoax staged by Hollywood studios here on Earth? Position-based cryptography may offer a solution. This kind of cryptography uses the geographic position of a party as its sole credential. Normally digital keys or biometric features are used.

A central building block in position-based cryptography is that of position verification. The goal is to prove to a set of verifiers that one is at a certain geographical location. Protocols typically assume that messages cannot travel faster than the speed of light. By responding to a verifier in a timely manner one can guarantee that one is within a certain distance of that verifier. Quite recently it was shown that position-verification protocols only based on this relativistic principle can be broken by attackers who simulate being at the claimed position while physically residing elsewhere in space. Because of the no-cloning property of quantum information (qubits) it was believed that with the use of quantum messages one could devise protocols that were resistant to such collaborative attacks. Several schemes were proposed that later turned out to be insecure. Finally it was shown that also in the quantum case no unconditionally secure scheme is possible. We will review the field of position-based cryptography, classical as well as quantum, and highlight some of the research currently going on in order to develop, using reasonable assumptions on the capabilities of the attackers, protocols that are secure in practice. No prior knowledge of quantum information is necessary.