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Combinatorial Algebra |

# Description

The purpose of this conference is to encourage unique collaborations between combinatorial algebraists and algebraic combinatorics. This is both a regional meeting and an opportunity for researchers with similar mathematical interests but variations of focus to develop and exchange ideas.Links to the previous workshops can be found

**here**.

# Dates

The meeting will take place starting in the afternoon of Friday, January 25th, 2013 and end early in the afternoon on Sunday, January 27th, 2013.

# Online registration and application for support

The on-line registration and application for funding for the Combinatorial Algebra meets Algebraic Combinatorics Meeting January 25-27, 2013 is now open on the Fields site at:

http://www.fields.utoronto.ca/programs/scientific/12-13/caac2013/

Registration will be on-line to Jan. 20 and on site Jan. 25. The application for travel support has a deadline of Dec. 9.

# Schedule and Location

The conference will take place at the Fields Institute in Toronto, Ontario. The talks on Saturday, January 26 will take place at Room 1170 at Bahen center behind the Fields Institute.

#### Schedule

Date and Time Speaker Title (click titles for abstract) 25 January, 1pm-2pm Kiumars Kaveh Multiplicities of ideals and volume of convex diagrams The notion of (Samuel) multiplicity of an ideal I in a local ring R generalizes the fundamental notion of intersection multiplicity of subvarieties in a variety. Focus of much research in commutative algebra has been on its definitions/extensions, properties and computation. In this talk we make a connection between the notion of multiplicity and convex geometry. More precisely to each (primary) ideal in a (large class of) local rings of dim n, we associate a convex set (diagram) in the positive orthant R_+^n such that the volume of the complement of the diagram gives the multiplicity of the ideal. This can be considered as the local version of the theory of Okounkov bodies (of linear systems) and extends the notion of Newton diagram of a monomial ideal (in a polynomial algebra). For the most part the talk is accessible to anyone with just basic knowledge in algebra and geometry. This is joint work with Askold Khovanskii.

25 January, 2:30pm-3:15pm Yong-Su Shin Some applications of star-configurations in $P^n$

25 January, 3:30pm-4:15pm Uwe Nagel Enumerations deciding the Weak Lefschetz Property For proving one implication of the g-theorem on simplicial polytopes, Stanley established the Weak Lefschetz Property of their Stanley-Reisner ring. Deciding the presence of this property is often a difficult problem. We propose an approach for studying monomial ideals in three variables via lozenge tilings of certain planar regions. It provides combinatorial interpretations for the weak Lefschetz property. As a consequence, the presence of the weak Lefschetz property can be decided in new cases. Attention is also given to ground fields of positive characteristic

26 January, 9:30am-10:30am Frank Sottile Galois groups of Schubert problems Building on work of Jordan from 1870, in 1979 Harris showed that a geometric monodromy group associated to a problem in enumerative geometry is equal to the Galois group of an associated field extension. Vakil gave a geometric-combinatorial criterion that implies a Galois group contains the alternating group. With Brooks and Martin del Campo, we used Vakil's criterion to show that all Schubert problems involving lines have at least alternating Galois group.

My talk will describe this background and sketch a current project to systematically determine Galois groups of all Schubert problems of moderate size on all small classical flag manifolds, investigating at least several million problems. This will use supercomputers employing several overlapping methods, including combinatorial criteria, symbolic computation, and numerical homotopy continuation, and require the development of new algorithms and software.

26 January, 11:00am-11:45am Hugh Thomas Higher-dimensional analogues of tropical cluster structures Various tropical aspects of cluster structures (exchange relations, g-vectors) generalize naturally beyond the setting of cluster algebras. I will explain this, based on arXiv:1001.5437 joint with Steffen Oppermann, and also subsequent joint work with Steffen Oppermann and Charles Paquette. This talk could also serve as an introduction to the tropical aspects of cluster algebras.

26 January, 1:30pm-2:15pm Liz Beazley Projections on Parabolic Quotients of Affine Weyl Groups Affine Weyl groups and their parabolic quotients are used extensively as indexing sets for objects in representation theory, algebraic geometry, and number theory. Moreover, we can conveniently realize the elements of certain quotients via intuitive geometric and combinatorial models such as abaci, alcoves, root lattice points, and core partitions. Berg, Jones, and Vazirani have described a bijection between n-cores with first part equal to k and (n-1)-cores with first part less than or equal to k. This bijection also has an interpretation in terms of the correspondence of Lapointe and Morse between n-cores and (n-1)-bounded partitions. This correspondence played a crucial role in the development of k-Schur functions, which are known to represent the Schubert basis in the homology of the affine Grassmannian. In this talk we discuss how to generalize this bijection of Berg, Jones, and Vazirani to parabolic quotients of affine Weyl groups in other classical Lie types. We have developed not only combinatorial techniques to describe this map, but also a visually explicit method utilizing the geometric properties of the alcove model coming from the root system associated to the affine Weyl group.

26 January, 3:00pm-4pm Sara Faridi Monomial Resolutions This talk is about combinatorial techniques to investigate minimal resolutions of monomial ideals. We focus in particular on the projective dimension (the length of a minimal free resolution) and how to bound or compute it using the combinatorics of the monomial ideal.

27 January, 9am-9:45am Ilias Kotsireas Hadamard and skew-Hadamard matrices of new orders Hadamard and skew-Hadamard matrices are supposed to exist for every order n such that n is a multiple of 4. Even though there are several infinite classes and constructions for Hadamard and skew-Hadamard matrices, these well-known conjectures are open for more than 100 years. We report on recent computational progress on these conjectures, namely the discovery of Hadamard and skew-Hadamard matrices of new orders. This progress is mainly due to a recent algorithmic breakthrough. This is joint work with D. Djokovic (U. Waterloo) and O. Golubitsky (Google Inc).

PLEASE NOTE: Ilias sends his regrets and will be unable to come, so this talk has been cancelled

27 January, 10:00am-10:45am Leila Khatami The poset of the nilpotent commutator of a nilpotent matrix

27 January, 11:30am-12:30pm Thomas Lam Positroid varieties The positroid stratification of the Grassmannian $G(k,n)$ is the intersection of the cyclic rotations of the Schubert stratification. This stratification was studied extensively in Postnikov's work on total positivity of the Grassmannian. It also appears in the study of Poisson geometry, Frobenius splitting, KP solitons, scattering amplitudes,... I will discuss some combinatorial, cohomological, and geometric properties of this stratification.

This talk will be based on joint work with Allen Knutson and David Speyer.

# Invited Speakers

The following people agreed to be our main speakers.

**Sara Faridi (Dalhousie Univeristy)****Kiumars Kaveh (University of Pittsburgh)****Thomas Lam (University of Michigan)****Frank Sottile (Texas A&M)**

Additional contributed talks will be presented by:

- Elizabeth Beazley (Haverford College)
- Leila Khatami (Union College)
- Uwe Nagel (University of Kentucky)
- Hugh Thomas (University of New Brunswick)
- Yong-Su Shin (Sungshin Women's University, Seoul)
- Ilias Kotsireas (Wilfrid Laurier)

We asked for additional speakers who wanted to present to submit an abstract by October 28th (deadline passed).

# Travel to Toronto

**by Train:**Toronto's train station is about 2.5 km from The Fields Institute. Amtrak and Via Rail both have routes between major cities.

**by Bus:**The bus terminal is 1.6km from the Fields Institute. Bus company Coach Canada/Megabus has regular service to Toronto from regional cities.

**by Air:**Toronto is serviced by Pearson International Airport and Billy Bishop Toronto City Airport. Both Porter Airlines and Air Canada have flights to Billy Bishop airport and all major airlines have service to Pearson Airport.

Some funds are available to assist participants, especially students and postdocs with travel and acommodation costs. Send inquiries to

**Carolina Benedetti**(caro.benedetti@gmail.com). Applications should be submitted through the Fields online application form by December 9, 2012.

# Contact Information

Please contact
**Carolina Benedetti** (caro.benedetti@gmail.com), **Nantel Bergeron** (bergeron@yorku.ca),
and **Mike Zabrocki** (zabrocki@mathstat.yorku.ca) if you have any questions
about the conference.

# Organizers

# Sponsors

We would like to thank the following organizations for their support of this conference.