### Schubert Galois Groups

- Series
- Algebra Seminar
- Time
- Friday, March 15, 2019 - 14:00 for 1 hour (actually 50 minutes)
- Location
- Skiles 005
- Speaker
- Frank Sottile – Texas A&amp;M

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- Series
- Algebra Seminar
- Time
- Friday, March 15, 2019 - 14:00 for 1 hour (actually 50 minutes)
- Location
- Skiles 005
- Speaker
- Frank Sottile – Texas A&amp;M

Problems from enumerative geometry have Galois groups. Like those from field extensions, these Galois groups reflect the internal structure of the original problem. The Schubert calculus is a class of problems in enumerative geometry that is very well understood, and may be used as a laboratory to study new phenomena in enumerative geometry.I will discuss this background, and sketch a picture that is emerging from a sustained study of Schubert problems from the perspective of Galois theory. This includes a conjecture concerning the possible Schubert Galois groups, a partial solution of the inverse Galois problem, as well as glimpses of the outline of a possible classification of Schubert problems for their Galois groups.

- Series
- Algebra Seminar
- Time
- Monday, March 11, 2019 - 12:50 for 1 hour (actually 50 minutes)
- Location
- Skiles 005
- Speaker
- Borys Kadets – MIT – bkadets@mit.edu

Let X be a degree d curve in the projective space P^r.

A general hyperplane H intersects X at d distinct points; varying H defines a monodromy action on X∩H. The resulting permutation group G is the sectional monodromy group of X. When the ground field has characteristic zero the group G is known to be the full symmetric group.

By work of Harris, if G contains the alternating group, then X satisfies a strengthened Castelnuovo's inequality (relating the degree and the genus of X).

The talk is concerned with sectional monodromy groups in positive characteristic. I will describe all non-strange non-degenerate curves in projective spaces of dimension r>2 for which G is not symmetric or alternating. For a particular family of plane curves, I will compute the sectional monodromy groups and thus answer an old question on Galois groups of generic trinomials.

- Series
- Algebra Seminar
- Time
- Monday, March 4, 2019 - 12:50 for 1 hour (actually 50 minutes)
- Location
- Skiles 005
- Speaker
- Chris Eur – University of California, Berkeley – chrisweur@berkeley.edu

We introduce a certain nef generating set for the Chow ring of the wonderful compactification of a hyperplane arrangement complement. This presentation yields a monomial basis of the Chow ring that admits a geometric and combinatorial interpretation with several applications. Geometrically, one can recover Poincare duality, compute the volume polynomial, and identify a portion of a polyhedral boundary of the nef cone. Combinatorially, one can generalize Postnikov's result on volumes of generalized permutohedra, prove Mason's conjecture on the log-concavity of independent sets for certain matroids, and define a new valuative invariant of a matroid that measures its closeness to uniform matroids. This is an on-going joint work with Connor Simpson and Spencer Backman.

- Series
- Algebra Seminar
- Time
- Monday, February 18, 2019 - 13:00 for 1 hour (actually 50 minutes)
- Location
- Skiles 005
- Speaker
- Robert Walker – U Michigan

We survey dissertation work of my academic sister Sarah Mayes-Tang (2013 Ph.D.). As time allows, we aim towards two objectives. First, in terms of combinatorial algebraic geometry we weave a narrative from linear star configurations in projective spaces to matroid configurations therein, the latter being a recent development investigated by the quartet of Geramita -- Harbourne -- Migliore -- Nagel. Second, we pitch a prospectus for further work in follow-up to both Sarah's work and the matroid configuration investigation.

- Series
- Algebra Seminar
- Time
- Monday, February 11, 2019 - 13:00 for 1 hour (actually 50 minutes)
- Location
- Skiles 005
- Speaker
- Andrew Obus – Baruch College, CUNY – Andrew.Obus@baruch.cuny.edu

Mac Lane's technique of "inductive valuations" is over 80 years old, but has only recently been used to attack problems about arithmetic surfaces. We will give an explicit, hands-on introduction to the theory, requiring little background beyond the definition of a non-archimedean valuation. We will then outline how this theory is helpful for resolving "weak wild" quotient singularities of arithmetic surfaces, as well as for proving conductor-discriminant inequalities for higher genus curves. The first project is joint work with Stefan Wewers, and the second is joint work with Padmavathi Srinivasan.

- Series
- Algebra Seminar
- Time
- Monday, February 4, 2019 - 12:50 for 1 hour (actually 50 minutes)
- Location
- Skiles 005
- Speaker
- Botong Wang – University of Wisconsin-Madison – wang@math.wisc.edu

Matroids are basic combinatorial objects arising from graphs and vector configurations. Given a vector configuration, I will introduce a “matroid Schubert variety” which shares various similarities with classical Schubert varieties. I will discuss how the Hodge theory of such matroid Schubert varieties can be used to prove a purely combinatorial conjecture, the “top-heavy” conjecture of Dowling-Wilson. I will also report an on-going project joint with Tom Braden, June Huh, Jacob Matherne, Nick Proudfoot on the cohomology theory of non-realizable matroids.

- Series
- Algebra Seminar
- Time
- Monday, January 28, 2019 - 12:50 for 1 hour (actually 50 minutes)
- Location
- Skiles 005
- Speaker
- Jackson Morrow – Emory university – jmorrow4692@gmail.com

The conjectures of Green—Griffths—Lang predict the precise interplay between different notions of hyperbolicity: Brody hyperbolic, arithmetically hyperbolic, Kobayashi hyperbolic, algebraically hyperbolic, groupless, and more. In his thesis (1993), W.~Cherry defined a notion of non-Archimedean hyperbolicity; however, his definition does not seem to be the "correct" version, as it does not mirror complex hyperbolicity. In recent work, A.~Javanpeykar and A.~Vezzani introduced a new non-Archimedean notion of hyperbolicity, which ameliorates this issue, and also stated a non-Archimedean variant of the Green—Griffths—Lang conjecture. In this talk, I will discuss complex and non-Archimedean notions of hyperbolicity as well as some recent progress on the non-Archimedean Green—Griffths—Lang conjecture. This is joint work with Ariyan Javanpeykar (Mainz) and Alberto Vezzani (Paris 13).

- Series
- Algebra Seminar
- Time
- Friday, January 25, 2019 - 14:00 for 1 hour (actually 50 minutes)
- Location
- Skiles 005
- Speaker
- Chi Ho Yuen – University of Bern – chi.yuen@math.unibe.ch

An amoeba is the image of a subvariety X of an algebraic torus under the logarithmic moment map. Nisse and Sottile conjectured that the (real) dimension of an amoeba is smaller than the expected one, namely, two times the complex dimension of X, precisely when X has certain symmetry with respect to toric actions. We prove their conjecture and derive a formula for the dimension of an amoeba. We also provide a connection with tropical geometry. This is joint work with Jan Draisma and Johannes Rau.

- Series
- Algebra Seminar
- Time
- Wednesday, December 5, 2018 - 14:30 for 1 hour (actually 50 minutes)
- Location
- Skiles 249
- Speaker
- Farbod Shokrieh – University of Copenhagen – farbod@math.ku.dk

Classical Kazhdan's theorem for Riemann surfaces describes the limiting behavior of canonical (Arakelov) measures on finite covers in relation to the hyperbolic measure. I will present a generalized version of this theorem for metric graphs. (Joint work with Chenxi Wu.)

- Series
- Algebra Seminar
- Time
- Monday, December 3, 2018 - 15:00 for 1 hour (actually 50 minutes)
- Location
- Skiles 006
- Speaker
- Bruce Reznick – University of Illinois, Urbana Champaign

One variation of the Waring problem is to ask for the shortest non-trivial equations of the form f_1^d + ... + f_r^d = 0, under various conditions on r, d and where f_j is a binary form. In this talk I'll limit myself to quadratic forms, and show all solutions for r=4 and d=3,4,5. I'll also give tools for you to find such equations on your own. The talk will touch on topics from algebra, analysis, number theory, combinatorics and algebraic geometry and name-check such notables as Euler, Sylvester and Ramanujan, but be basically self-contained. To whet your appetite: (x^2 + xy - y^2)^3 + (x^2 - xy - y^2)^3 = 2x^6 - 2y^6.

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