Seminars archive

Let A/F be a division algebra of degree 3, and X its Severi-Brauer variety which is a form of the projective plane. The linear system of cubic curves is defined on X, and so we can let C \subset X be one such. If C is a nonsingular such curve, then C is a genus one curve with Jacobian E, an elliptic curve. The question we address is the one asked by Asher Auel, namely, which E arise. We give an answer that depends on the structure of A.

Summary : Let G be a connected linear algebraic group over a field k. By definition, a groupic G-variety X over k is a smooth (left) G-variety with a dense open set isomorphic to G with its (left) action on itself. Let X be a groupic G-variety over a number field. Under a suitable noncompactness hypothesis for the simple factors of the semisimple part of G at the archimedean places, Cao and Xu show that the Brauer-Manin obstruction is the only obstruction to strong approximation for X off the archimedean places. The proof builds upon the case X=G (handled in earlier papers by Xu and the speaker, Harari, Demarche). The toric case (G is a torus) was already handled in a previous paper by Cao and Xu. For X projective, the statement is a weak approximation result and the theorem has been known for a long time (Sansuc).. The proof of the strong approximation result for an arbitrary groupic G-variety X involves novel arguments, both geometric and arithmetic.

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We describe several generalizations of the moduli space of marked rational curves, their combinatorial structure and construction methods. In particular, we report joint work with Noah Giansiracusa in which we revisit the smooth configuration space compactifying n distinct points in affine space up to translation and homothety. I will also report joint work with Kenny Ascher in understanding a good locus inside the moduli space of weighted line arrangements constructed by V. Alexeev.

I will discuss a matrix completion problem arising in combinatorial statistics and explain how we can use results in algebraic geometry to understand it better. The object linking the two different areas is the cone of sums of squares and its properties as a convex cone.

Problems in computational science often require interdisciplinary collaboration. Differences in the vocabulary, skills, and expectations of different fields can make these collaboration efforts extremely challenging. In this talk, I will discuss strategies for working on interdisciplinary teams that can contribute to the success (or failure) of a project. In particular, I will discuss some common pitfalls that I have seen mathematicians fall into when working with engineers and domain scientists.

Problems in computational science often require interdisciplinary collaboration. Differences in the vocabulary, skills, and expectations of different fields can make these collaboration efforts extremely challenging. In this talk, I will discuss strategies for working on interdisciplinary teams that can contribute to the success (or failure) of a project. In particular, I will discuss some common pitfalls that I have seen mathematicians fall into when working with engineers and domain scientists.

Problems in computational science often require interdisciplinary collaboration. Differences in the vocabulary, skills, and expectations of different fields can make these collaboration efforts extremely challenging. In this talk, I will discuss strategies for working on interdisciplinary teams that can contribute to the success (or failure) of a project. In particular, I will discuss some common pitfalls that I have seen mathematicians fall into when working with engineers and domain scientists.

The chromatic polynomial of a graph counts its proper colourings. This polynomial's coefficients were conjectured to form a unimodal sequence by Read in 1968. This conjecture was extended by Rota in his 1970 address to assert the log-concavity of the characteristic polynomial of matroids which are the common generalizations of graphs and linear subspaces. We discuss the resolution of this conjecture which is joint work with Karim Adiprasito and June Huh. The solution draws on ideas from the theory of algebraic varieties, specifically Hodge theory, showing how a question about graph theory leads to a solution involving Grothendieck's standard conjectures.

Twenty years ago David J. Gross presented, at the National Academy of Sciences, a paper entitled "The role of symmetry in fundamental physics", which was later published in PNAS. I shall use that talk as a thread in order to illustrate my recent work related to the following main themes: (A) going from Classical to Quantum Mechanics by preserving Noether symmetries; (B) finding hidden linearity of maximally superintegrable systems; (C) determining Lagrangians (and Noether symmetries) for systems without Lagrangians. I shall provide several examples for each theme, including models in population dynamics and the Lorenz system in meteorology