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## Pick My Brain Seminar

Fall 2017, Northeastern

 Meeting weekly on Tuesday 4:00-5:00pm in 544 Nightingale Hall at Northeastern. This is an informal colloquium style seminar which provides the opportunity to learn about others areas of interest. Questions encouraged! Organizers: Ivan Martino, Pablo Soberón, Robin Walters. If you have a question, would like receive announcements, or would like to speak at seminar, email This seminar is supported in part by a Research and Training Grant (RTG) from the NSF.

### Schedule

 Date Speaker Title Sep 12 Ivan Martino Algebraic Structures related to Matroid Theory (Winter is Coming) abstract± In this talk I will describe certain algebraic structures related to Matroid Theory. After defining a matroid over a ring, I will focus on matroids over a field $k$ and matroids over the integers $\mathbb{Z}$. Poster. Following seminar, we will go out for food and refreshment. Sep 19 Laure Flapan At 3:30, before seminar, we will have tea, coffee and cookies in the lounge. Hodge groups, Hodge structures, and the Hodge Conjecture. abstract± I will introduce the Hodge group and discuss its usefulness in the context of the Hodge conjecture as well as, more generally, in other connections between topology and algebraic geometry. Poster. Sep 26 Emily Barnard At 3:30, before seminar, we will have Ivan's cake in the lounge. Counting and the Canonical Join Representation abstract± We will start the talk by considering some apparently unrelated counting problems. Each of family of objects that we want to count has some extra structure: a partial order that turns out to be a lattice. I will explain how a certain factorization’’ of the elements in a lattice can help us count. This talk will be heavy on examples, and accessible to all. Poster. Oct 3 Emanuele Macri The Genus of Space Curves and Stability in the Derived Category. abstract± In the first part of the talk, I will introduce a Conjecture by Hartshorne and Hirschiwitz on bounds for the genus space curves which are not contained in a surface of a given degree. For example, for curves not contained in a quadratic surface, this is nothing but the well-known Castelnuovo bound. In the second part of the talk, I will present how to approach this Conjecture by using derived categories (in a joint work in progress with Benjamin Schmidt), and where new ideas are needed to hopefully(!) complete this plan. Poster. Following seminar, we will go out for food and refreshment. Oct 10 Benjamin Sung There will be tea and cookies before the talk in the department lounge. Sheaf cohomology on Calabi-Yau hypersurfaces in toric varieties and D-brane instantons abstract± Sheaf cohomology of divisors on a Calabi-Yau threefold yields information about the zero mode spectrum of wrapped ED3 branes and hence the non-perturbative superpotential, which has direct applications for moduli stabilization and axion inflation. I will present a new technique and explicit formulas for these computations based on joint work with Andreas Braun, Cody Long, Liam McAllister, and Mike Stillman. Poster. Oct 17 Chris McDaniel (Endicott College) From Watanabe's Bold Conjecture to Soergel's Categorification Theorem. abstract± In commutative algebra, Watanabe's Bold Conjecture asserts that every complete intersection can always embed into some quadratic complete intersection with the same socle degree. We will show that this Bold Conjecture holds for a class of complete intersections called coinvariant rings, with Bott-Samelson rings acting as the quadratic ones. Bott-Samelson rings seem to have been introduced by Soergel who succeeded in uncovering some of their remarkable properties, including his celebrated Categorification Theorem. In fact, Soergel's Categorification Theorem and the conjectures stemming from it are what eventually led Elias and Williamson to a proof of the notorious Kazhdan-Lusztig positivity conjecture in representation theory. We will highlight some of these beautiful results of Soergel and Elias-Williamson, and of course describe their connection to Watanabe's Bold Conjecture. Poster. Oct 24 Andrew Laurie (MIT) Solitons, bubbling, and blow up for geometric PDE's. abstract± Solitons (coherent solitary waves) are the building blocks of the global-in-time dynamics and singularity formation for dispersive PDE. In the case of a globally defined solution, the soliton resolution conjecture asserts that as a solution evolves, it decomposes into a finite number of weakly interacting solitons plus a remainder exhibiting linear dynamics. In the case of a solution that develops a singularity by concentrating mass or energy, solitons often play the role of universal blow-up profiles -- zooming in on the solution near the singularity, the shape of a soliton comes into view. In this talk, we’ll mostly discuss the latter phenomena (which is often called bubbling) in the context of a model geometric PDE called the wave maps equation, which is a generalization of the free wave equation to manifold valued maps. Poster. Oct 31 Valerio Toledano Laredo Pick Valerio's Brain abstract± Valerio Toledano Laredo works in representation theory, particularly loop groups and quantum groups. More recently, he has explored the semiclassical aspect of quantum groups and uncovered, in collaboration with Tom Bridgeland of Sheffield University, a novel [and fascinating] dictionary between wall-crossing in Algebraic Geometry and Stokes phenomena for differential equations with irregular singularities in the complex plane. Poster. Following seminar, we will go out for food and refreshment. Nov 7 Bena Tshishiku (Harvard) The "Dark Matter" Problem. abstract± The dark matter problem in low-dimensional topology is about surface bundles, mapping class groups, moduli spaces, and cohomology (and has nothing to do with cosmology). I will explain this problem and the circle of ideas around surrounding it. Poster. Nov 14 Asilata Bapat (UGA) Examples of Compactifications of Quiver Varieties. abstract± I will start with a recent construction of McGerty and Nevins, which systematically produces compactifications of quiver varieties. I will explain some variants of this construction, as well as work in progress on computing specific examples, including the Hilbert scheme of points on a plane. Poster. Nov 21 Thanksgiving Break Nov 28 Jonathan Mboyo Esole Representations and anomalies in presence of a non-trivial Mordell-Weil group abstract± In this talk, I will explore how the presence of a non-trivial Mordell-Weil group on an elliptic fibration affects its relative Mori cone. I will also discuss how the Mordell-Weil group of the elliptic fibration modifies the structure of representations and anomalies in the dual gauge theory associated with the elliptic fibration. Poster. Dec 5 Mustazee Rahman (MIT) On largest eigenvalues of bounded degree graphs abstract± Classical Alon-Boppana theorem gives a sharp lower bound on the second largest eigenvalue of a regular graph. Obtaining such bounds for non-regular graphs is more complicated. I will explain some combinatorial and topological ideas that allows for Alon-Boppana type bounds for large, bounded degree graphs. These bounds also apply to unimodular networks, which are a stochastic generalization of finite graphs. In fact, the stochastic generalization plays a central role in deducing such bounds for finite graphs. Poster. End of Semester (Seminar Resumes Spring 2018)