Algecom Committee: Hal Schenck Alexander Yong David Speyer Uli Walther Saugata Basu Evgeny Mukhin Peter Tingley Chris Drupieski Date: April 29, 2017
Location: Department of Mathematics at University of Illinois at UrbanaChampaign Local Organizers (questions related to Algecom 15): Alex Yong ayong@illinois.edu
Laura Escobar lescobar@illinois.edu
Benjamin Wyser bwyser@illinois.edu
Registration is free.
To register, email the local organizer Alexander Yong by April 15.
Limited travel support is available for graduate students. To apply, email Alexander Yong by March 10 with: the name of your university, the name of your advisor, a summary of your research interests, a summary of your anticipated travel expenses, and whether you plan to participate in the poster session. Funding decisions will be made by March 15, 2017. For a map, click here.
All talks will be in Altgeld Hall.
There will be a poster session in Altgeld Hall.
Parking on campus lots is free on Saturdays. The closest one is lot C3 here on this map
Speakers:
Angelica Cueto (Ohio State University) Rachel Karpman (Ohio State University) Martha Precup (Northwestern) Stephanie van Willigenburg (U. British Columbia)
Schedule:
Coffee and pastries 99:30am (Math Lounge, Altgeld Hall)
Maria Angelica Cueto (Ohio State) 9:3010:25am Title: Anticanonical tropical del Pezzo cubic surfaces contain exactly 27 lines.
Abstract: Since the beginning of tropical geometry, a persistent challenge has been to emulate tropical versions of classical results in algebraic geometry. The wellknow statement ``any smooth surface of degree 3 in P^3 contains exactly 27 lines'' is known to be false tropically. Work of Vigeland from 2007 provides examples of cubic surfaces with infinitely many lines and gives a classification of tropical lines on general smooth tropical surfaces in TP^3.
In this talk I will explain how to correct this pathology. The novel idea is to consider the embedding of a smooth cubic surface in P^44 via its anticanonical bundle. The tropicalization induced by this embedding contains exactly 27 lines under a mild genericity assumption. More precisely, smooth cubic surfaces in P^3 are del Pezzos, and can be obtained by blowing up P^2 at six points in general position. We identify these points with six parameters over a field with nontrivial valuation. Our genericity assumption involves the valuations of 36 linear expressions in these parameters which give the positive roots of type E_6. Tropical convexity plays a central role in ruling out the existence of extra tropical lines on the tropical cubic surface.
This talk is based on an ongoing project joint with Anand Deopurkar.
Rachel Karpman (Ohio State) 11:00am11:55am
Title: Total Positivity and Planar Networks
Abstract: A matrix is totally nonnnegative if all of its minors are nonnegative real numbers. The Lindström Lemma gives a remarkable relationship between totally nonnegative matrices and planar networks. Given a weighted planar directed network which is sufficiently "nice," we may construct a corresponding totally nonnegative matrix, and every totally nonnegative matrix arises in this way. The concept of total nonnegativity extends naturally to the Grassmannian Gr(k,n), the algebraic variety whose points correspond to kdimensional subspaces of a fixed ndimensional complex vector space. Postnikov introduced a family of coordinate charts on the totally nonnegative Grassmannian, defined in terms of weighted planar networks; this may be viewed as a Grassmannian analogue of the Lindström Lemma. In this talk, we discuss the results of Lindström and Postnikov, and extend Postnikov's theory to the Lagrangian Grassmannian, a variety whose points correspond to isotropic subspaces of a fixed vector space with respect to a symplectic form.
Lunch (see below for details) 12:002:00pm
Martha Precup (Northwestern) 2:002:55pm
Title: Semisimple Hessenberg varieties
Abstract: Semisimple Hessenberg varieties are subvarieties of the flag variety with important connections to representation theory, algebraic geometry, and combinatorics. Like Schubert varieties, the structure of semisimple Hessenberg varieties can be studied using the combinatorics of the symmetric group. In this talk, we will define these varieties and compute their GKMgraphs. Then we’ll give a combinatorial criterion for identifying singular points in certain semisimple Hessenberg varieties. These results are joint work with Erik Insko.
Stephanie van Willigenburg (UBC) 3:304:30pm
Title: Quasisymmetric Schur functions
Abstract: In algebraic combinatorics a central area of study is Schur functions. These functions were introduced early in the last century with respect to representation theory, and since then have become important in other areas such as quantum physics and algebraic geometry.
These functions also form a basis for the algebra of symmetric functions, which in turn forms a subalgebra of the algebra of quasisymmetric functions that itself impacts areas from category theory to card shuffling. Despite this strong connection, the existence of a natural quasisymmetric refinement of Schur functions was considered unlikely for many years.
In this talk we will meet such a natural refinement of Schur functions, called quasisymmetric Schur functions. Furthermore, we will see how these quasisymmetric Schur functions refine many wellknown Schur function properties, with combinatorics that strongly reflects the classical case including diagrams, and pattern avoidance in permutations.Poster session and informal discussions: 4:305:45pm

Lunch: There are numerous lunch options within walking distance, the closest being the ????
 Dinner: ~5:15pm We will have a banquet for registered participants at the ???. We will walk over from Altgeld Hall at 6:00pm, just after the poster session ends.

Getting to Urbana:Parking: Lodging: A) The IIllini Union (5742776500) There are 10 queen bed rooms booked at the Illini Union for $104/night. Also booked are 4 studio queens and 4 studio kings for $109/night at Townplace Suites to give participants an option of having a larger room. The participants will need to book by March 28 in order to get a room in the block. To make reservations online at the Illini Union: 1. Go to http://union.illinois.edu/stay/illiniunionhotel 2. Click on “Book Now” located upper left of page 3. The below screen appears. 4. Fill in dates and other info. 5. Way at the bottom, click on the circle of the “Group Booking Code.” 6. Fill in your group booking code: ALGECOM 7. Click on “Check Availability”. To make reservations online at Townplace Suites: Click on ALGECOM 15.
Childcare: Parents attending the conference and looking for childcare may find care.com a useful reference.

