### Seminar on NonLinear Algebra:

BERLIN, Germany, June 26, 2015
This is a special meeting of our "Seminar on Nonlinear Algebra" part of Bernd Sturmfel's project on NonLinear Algebra that is funded for 2015-2018 by the Einstein Foundation.

## Speakers and Schedule

The talks will be at TU Berlin in MA 621.

### Friday, June 26

 9:00-9:45 Mateusz Michalek (FU Berlin and UC Berkeley) From topology to algebraic geometry and back again 9:45-10:30 Michal Lason (FU Berlin) My favorite matroid conjectures 11:00-11:45 Emanuele Delucchi (University of Fribourg, Switzerland) Toric arrangements and G-seminatroids 14:00-14:45 Olga Holtz (TU Berlin and UC Berkeley) TBA 14:45-15:30 Matthias Lenz (University of Oxford) Splines, lattice points, and arithmetic matroids 16:00-16:45 Christian Haase (FU Berlin) A unimodular triangulation for the hives matrix

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## Abstracts

From topology to algebraic geometry and back again (Mateusz Michalek)

Secant varieties are known to play an important role in complexity theory, representation theory and algebraic geometry, relating to ranks of tensors. In my talk I would like to present applications of secants in topology through k-regular embeddings. An embedding of a variety in an affine space is called k-regular if any k points are mapped to linearly independent points. Numeric conditions for the existence of such maps are an object of intensive studies of algebraic topologists dating back to the problem posed by Borsuk in the fifties. Very recent results were obtained by Pavle Blagojevic, Wolfgang Lueck and Guenter Ziegler. Our results relate k-regular maps to punctual versions of secant varieties. These allows us to prove existence of such maps in special cases. The main new ingredient is providing relations to the geometry of the punctual Hilbert scheme and its Gorenstein locus. This is joint work with Jarosław Buczynski, Tadeusz Januszkiewicz and Joachim Jelisiejew.

My favorite matroid conjectures (Michal Lason)

Toric arrangements and G-seminatroids (Emanuele Delucchi)

Recent work of De Concini, Procesi and Vergne on vector partition functions gave a new impulse to the study of toric arrangements from an algebraic, topological and combinatorial point of view. One of the challenges here is to extend the successful combinatorial toolbox available for linear arrangements to this “nonlinear” case.

I will give a quick overview of the state of the art and, taking inspiration from the techniques used to obtain some recent topological results, I will introduce G-semimatroids as a possible approach to meeting this challenge, with a view towards providing a unified framework for many of the new discrete structures which have appeared recently in this context.

TBA (Olga Holtz)

Splines, lattice points, and arithmetic matroids (Matthias Lenz)

Formulas of Khovanskii-Pukhlikov, Brion-Vergne, and De Concini-Procesi-Vergne relate the volume and the number of integer points in a convex polytope. In this talk I will refine these formulas and talk about graded vector spaces that appear naturally in this context, namely Dahmen-Micchelli spaces and their duals, the so-called P-spaces. It will turn out that the combinatorics of these spaces is determined by the underlying arithmetic matroid.

A unimodular triangulation for the hives matrix (Christian Haase)

In their breakthrough paper, Knutson and Tao proved the saturation conjecture about the representation theory of GL$_n(\C)$: for partitions $\lambda, \mu, \nu$ and an integer $N$ the irreducible representation $V_\nu$ occurs as a subrepresentation of $V_\lambda \otimes V_\mu$ if $V_{N\nu}$ occurs inside $V_{N\lambda} \otimes V_{N\mu}$. They reformulated the conjecture in terms of the hives polytope $H(\lambda,\mu,\nu)$, and show that $H(\lambda,\mu,\nu)$ is non-empty only if it contains a point with all coordinates integral. De Loera and McAllister observe that this would follow if the defining matrix of $H(\lambda,\mu,\nu)$, considered as a vector configuration, had a unimodular cover. Based on computer experiments they conjecture that this homogenized hives matrix even has a unimodular triangulation. In this talk, I will argue that the original proof by Knutson and Tao already implies the existence of a regular unimodular triangulation.

st Updated:

June, 2015