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Series: Algebra Seminar

Duality is an important feature in convexity and in projective algebraic
geometry. We will discuss the interplay of these two dualities for the
cone of sums of squares of ternary forms and its dual cone, the Hankel
spectrahedron.

Series: Algebra Seminar

We will introduce, through examples, the philosophy of Delignethat "in characteristic zero, a deformation problem is controlled by adifferential graded (or "dg-") Lie algebra." Focusing on the deformationtheory of representations of a group, we will give an extension of thisphilosophy to positive characteristic. This will be justified by thepresence of a dg-algebra controlling the deformations, and the fact thatthe cohomology of the dg-algebra has an A-infinity algebra structureexplicitly presenting the deformation problem. This structure can bethought of as "higher cup products" on group cohomology, extending theusual cup product and often computable as Massey products. We will writedown concrete, representation-theoretic questions that are answered bythese higher cup products. To conclude, we will show that the cup productstructure on Galois cohomology, which is the subject of e.g. the motivicBloch-Kato conjecture and its proofs, is enriched by these higher cupproducts, and that this enrichment reflects properties of the Galois group.Familiarity with dg-algebras and infinity-algebras will not be presumed.

Series: Algebra Seminar

After reminding everyone why the symmetric powers Sym^n X of a scheme arise and are interesting from the point of view of the Weil conjectures, I'll recall the Dold-Thom theorem of algebraic topology, which governs the behavior of symmetric powers of a topological space. I'll then explain how the notion of étale homotopy allows us to compare these two realms of arithmetic geometry and algebraic topology, providing a homotopical refinement of a small part of the Weil conjectures.

Series: Algebra Seminar

A reciprocal linear space is the image of a linear space under
coordinate-wise inversion. This nice algebraic variety appears in many
contexts and its structure is governed by the combinatorics of the
underlying hyperplane arrangement. A reciprocal linear space is also an
example of a hyperbolic variety, meaning that there is a family of
linear spaces all of whose intersections with it are real. This special
real structure is witnessed by a determinantal representation of its
Chow form in the Grassmannian. In this talk, I will introduce reciprocal
linear spaces and discuss the relation of their algebraic properties to
their combinatorial and real structure.

Series: Algebra Seminar

Part II of last week's talk.

Series: Algebra Seminar

Primary decomposition is a fundamental operation in commutative
algebra. Although there are several algorithms to perform it, this remains
a very difficult undertaking in general. In cases with additional
combinatorial structure, it may be possible to do primary decomposition "by
hand". The goal of this talk is to explain in detail one such example.
This is joint work with Zekiye Eser; no prerequisites are assumed beyond
knowing the definitions of "polynomial ring" and "ideal".

Series: Algebra Seminar

The talk involves an explicit formula for the Chern class on K_3(F), F=number field, givenin terms of the cyclic quantum dilogarithm on the Bloch group of F. Such a formula constructsexcplicitly units in number fields, given a complete hyperbolic 3-manifold, and a complex root ofunity, and those units fit in the asymptotic expansion of quantum knot invariants. The existence ofsuch a formula was conjectured 4 years ago by Zagier (and abstractly follows from Voevodsky's work),and the final solution to the problem was given in recent joint work of the speaker with FrankCalegari and Don Zagier. The key ingredient to the concrete formula is a special function, thecyclic quantum dilogarithm, from a physics 1993 paper of Kashaev and others. The connection of thisformula with physics, and with the Quantum Modular Form Conjecture of Zagier continues with jointwork with Tudor Dimofte. But this is the topic of another talk.

Series: Algebra Seminar

We consider functions on finite abelian groups that are nonnegative and also sparse in the Fourier basis. We investigate conditions under which such functions admit sparse sum-of-certificates certificates of nonnegativity, i.e., certificates where the functions in the sum of squares decomposition have a small common sparsity pattern. Our conditions are purely combinatorial in nature, and are based on finding particularly nice chordal covers of a certain Cayley graph. These techniques allow us to show that any nonnegative quadratic function in binary variables is a sum of squares of functions of degree at mostceil(n/2), resolving a conjecture of Laurent. After discussing the connection with semidefinite programming lifts of polytopes, we also see how our techniques provide an example of separation between sizes ofsemidefinite programming lifts and linear programming lifts. This is joint work with James Saunderson and Pablo Parrilo.

Series: Algebra Seminar

Useful background:The paper I’m discussing: <a href="http://arxiv.org/abs/1502.03736" title="http://arxiv.org/abs/1502.03736">http://arxiv.org/abs/1502.03736</a>Terry Tao’s blog post on Dvir’s theorem: <a href="https://terrytao.wordpress.com/2008/03/24/dvirs-proof-of-the-finite-fiel... title="https://terrytao.wordpress.com/2008/03/24/dvirs-proof-of-the-finite-fiel... earlier paper with Terry and Richard Oberlin about Kakeya restriction over finite fields: <a href="http://arxiv.org/abs/0903.1879" title="http://arxiv.org/abs/0903.1879">http://arxiv.org/abs/0903.1879</a>

The study of extremal configurations of points and subspaces sits at the boundary between combinatorics, harmonic analysis, and number theory; since Dvir’s 2008 resolution of the Kakeya conjecture over finite fields, it has been clear that algebraic geometry is also part of the story.We prove a theorem of Kakeya type for the intersection of subsets of n-space over a finite field with k-planes. Let S be a subset of F_q^n with the "k-plane Furstenberg property": for every k-plane V, there is a k-plane W parallel to V which intersects S in at least q^c points. We prove that such a set has size at least a constant multiple of q^{cn/k}. The novelty is the method; we prove that the theorem holds, not only for subsets of the plane, but arbitrary 0-dimensional subschemes, and reduce the problem by Grobner methods to a simpler one about G_m-invariant non-reduced subschemes supported at a point. The talk will not assume that everyone in the room is an algebraic geometer. It will, however, try to convince everyone in the room that it can be useful to be an algebraic geometer.This is joint work with Daniel Erman.

Series: Algebra Seminar

The nearest point map of a real algebraic variety with respect to Euclidean distance is an algebraic function. The Euclidean distance degree is the number of critical points for this optimization problem. We focus on projective varieties seen in engineering applications, and we discuss tools for exact computation. Our running example is the Eckart-Young Theorem which relates the nearest point map for low rank matrices with the singular value decomposition. This is joint work with Jan Draisma, Emil Horobet, Giorgio Ottaviani, Rekha Thomas.