Talks by N.S. Ermolaeva and B.B. Lurie have been presented.

CH is approached as a problem about the cardinality of the second number class ç. For the purpose, the theory of constituents is extended to the countably infinite case where the nodes of a constituent tree are sequences of finite constituents. Certain branches ("perfect" ones) specify the structures of which a model of a countably infinite constituent consists. In the case of ç, these branches keep on splitting indefinitely and hence have the cardinality of the continuum. Since ç is maximal, they are all satisfied in it.

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Notions of entropy and information, pioneered by Shannon, have been very powerful tools in coding theory. Coding theory aims to solve the problem of one-way communication: sending a message from Alice to Bob using as little communication as possible, sometimes over a noisy channel.

We will discuss several extensions of information-theoretic notions to the two-way communication setting. We use them to prove a direct sum theorem for randomized communication complexity, showing that implementing k copies of a functionality requires substantially more communication than just one copy.

More generally, we will show that information cost I(f) can be defined as a natural fundamental property of a functionality f. We will describe several new tight connections between I(f), direct sum theorems, interactive compression schemes, and amortized communication complexity.

Questions on counting usually involve finding integer solutions x,y of equations of the form f(x)=g(y) for polynomials f,g with integer or rational coefficients. For instance, if a,b,c are rational and ab is non-zero, then for m>n>2, the Diophantine equation a C^m_x + b Cⁿ_y = c has only finitely many integer solutions x,y. Another natural example comes from counting lattice points in generalized octahedra. The number of integral points on the n-dimensional octahedron |x₁| + |x₂| + ... + |x_n| ≤ r is given by the polynomial expression p_n(r) = ∑_i=0ⁿ 2ⁱ Cⁱ_nCⁱ_r and the question of whether two octahedra of different dimensions m,n can contain the same number of integral points becomes equivalent to the solvability of p_m(x)=p_n(y) in integers x,y. It turns out that when m>n>1, the above equation has only finitely many integral solutions.

The above results and many others appearing in the last decade have been made possible by a beautiful theorem of Yuri Bilu & Robert Tichy from 2000. For an absolutely irreducible polynomial F(x,y) in ℤ[x,y], a celebrated 1929 theorem due to Siegel shows the finiteness of the number of integer solutions of F(x,y)=0 except when the (projective completion of the) curve defined by F(x,y)=0 has genus 0 and has at most 2 points at infinity. This theorem generalizes also to S-integers in algebraic number fields. But, Siegel's theorem is ineffective. Bilu & Tichy obtained for the equation f(x)=g(y) with f,g polynomials over ℚ, a remarkable theorem which makes Siegel's theorem much more explicit (although still ineffective). Here, we discuss some Diophantine equations where Bilu-Tichy theorem applies and some others where elementary methods or methods of the Chebotarev density theorem apply.

The governing equations for the theories of conformal geometry and nonlinear materials science are basically the same and have been much studied for centuries. Despite some significant advances in the theory of these nonlinear elliptic equations there is still much to be considered. Here we discuss a little of what is known about these equations leading to a fascinating connection with Hilbert's fifth problem on Lie groups.

This talk will cover a broad sweep of modern mathematical ideas and connections between analysis, PDE, geometry, algebra and group theory painted with a broad brush.

Slides