ALGECOM-15

 

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http://csh.depaul.edu/departments/mathematical-sciences/Pages/default.aspx


https://sites.google.com/site/algecomday/algecom-13-1/notredame.JPG


Algebra, Geometry and Combinatorics Day (AlGeCom) is a one day, informal meeting of mathematicians from the University of Illinois, Purdue University, IUPUI, Loyola University Chicago , DePaul University, University of Notre Dame, the University of Michigan and nearby universities, with interests in algebra, geometry and combinatorics (widely interpreted).

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 Urbana-Champaign

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          9-9:30am  (Math Lounge, Altgeld Hall)

 

Maria Angelica Cueto (Ohio State) 9:30-10:25am

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 well-know 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) 11:00am-11:55am

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 k-dimensional subspaces of
 a fixed n-dimensional 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)   12:00-2:00pm


Martha Precup (Northwestern) 2:00-2:55pm

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 GKM-graphs.  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) 3:30-4:30pm

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
well-known Schur function properties,  with combinatorics that strongly reflects the classical
case including diagrams, and pattern avoidance in permutations.

Poster session and informal discussions:   4:30-5:45pm


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Lunch: There are numerous lunch options within walking distance,
the closest being the
????
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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.



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Getting to Urbana:



Parking:


Lodging:
A) The IIllini Union (574-277-6500)
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/illini-union-hotel

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.

 
B) Other possibilities are:

Hampton Inn (574-968-4737) and Downtown Champaign Hyatt (574-277-6500).

Childcare:  Parents attending the conference and looking for childcare may
find care.com a useful reference.