Presented By: Combinatorics Seminar - Department of Mathematics
ALGECOM -- conference in Algebra, Geometry and Combinatorics
This is a one day conference that travels around the Midwest; this Fall, we are hosting it in Ann Arbor.
Here is our schedule, title and abstracts. All talks are in East Hall 1360; the poster fair is in the East Hall Lower Atrium.
9:00-9:30 Coffee and registration.
9:30-10:30 Chris Eur (Carnegie Mellon University)
10:30-11:00 Break
11:00-12:00 Matt Larson (Princeton / Institute for Advanced Study)
12:00-1:30 Lunch
1:30-2:30 Patricia Klein (Texas A&M University)
2:30-3:00 Break
3:00-4:00 PM Jianping Pan (Arizona State University)
4:00-5:00 PM Poster fair
5:00 PM-??? Pizza
To register for ALGECOM, please fill out the google poll at
https://forms.gle/BVe3MHfckc8kXTkn9
All financial support has been allocated, but please still fill out the poll if you plan to attend.
The ALGECOM organizers are
David Speyer (U Michigan)
Peter Tingley (Loyola)
Alex Yong (UIUC)
Titles and abstracts:
Speaker: Chris Eur (Carnegie Mellon University)
Title: How do matroids behave like projective toric varieties?
Abstract: A tropical model of a matroid, called the Bergman fan, provides a toric variety whose Chow ring behaves like a smooth toric projective variety. What about sheaf cohomologies of line bundles or vector bundles? How should one make sense of such notions for a matroid? Even for linear matroids, many geometric questions remain. We discuss various developments surrounding these questions.
Speaker: Matt Larson (Princeton / Institute for Advanced Study)
Title: Fine multidegrees and Grobner degenerations
Abstract: Grobner degenerations are a method for studying the geometry of a subvariety X of C^n using a scaling action of the multiplicative group C*, but they often lose a lot of information. This information loss can be minimized by compactifying affine space. We show that the possible Grobner degenerations of X are controlled by the homology class of the closure of X in (P^1)^n. This gives a simple criterion for a collection of squarefree polynomials to be a universal Grobner basis, which can be used to recover many known results on universal Grobner bases. We apply this result to a family of matrix Schubert varieties. Joint work with Daoji Huang.
Speaker: Patricia Klein (Texas A&M University)
Title: Gröbner degeneration in Schubert calculus
Abstract: Roughly speaking, enumerative geometry is a field whose goal is to count the "typical" number of solutions to certain types of families of polynomial equations, particularly when that number is finite. With a great deal of effort, especially in the wake of the work of Hermann Schubert around the turn of the 20th century, mathematicians made rigorous the notion of a "typical" answer and also made rigorous certain simplifying strategies Schubert had suggested. Indeed, making Schubert's arguments precise was the topic of Hilbert's 15th problem, and the field born from this study is now called Schubert calculus. The simplifications Schubert had suggested entail sliding or deforming the geometric objects to be studied while preserving the total number of whatever it is one wants to count. These strategies are what are now called degeneration techniques. In this talk, we will review some classical results in Schubert calculus, describe some modern questions in the area, and then explain how these questions are studied via Gröbner degeneration in particular.
Speaker: Jianping Pan (Arizona State University)
Title: Variations of uncrowning algorithms and stable Grothendieck polynomials
Abstract: In this talk, I will provide the motivation, basic definitions, and abundant examples of tableaux related to variations of stable Grothendieck polynomials. The classic uncrowding algorithm defined by Buch on set-valued tableaux gives a bijective proof of Lenart’s Schur expansion for symmetric Grothendieck polynomials. Hook-valued tableaux are associated with stable canonical Grothendieck polynomials and we introduce an uncrowding algorithm on them. Recently Hwang, Jang, Kim, Song and Song gave a Schur expansion for the refined canonical stable Grothendieck polynomials (using exquisite tableaux). We discover a novel connection between the exquisite tableaux model and hook-valued tableaux via the uncrowding and jeu de taquin algorithms, using a classic result of Benkart, Sottile and Stroomer. This connection reveals a hidden symmetry of the hook-valued tableaux and the uncrowding algorithm defined on them. This talk is based on joint work with Pappe, Poh and Schilling, and with Jang, Kim, Pappe and Schilling.
Here is our schedule, title and abstracts. All talks are in East Hall 1360; the poster fair is in the East Hall Lower Atrium.
9:00-9:30 Coffee and registration.
9:30-10:30 Chris Eur (Carnegie Mellon University)
10:30-11:00 Break
11:00-12:00 Matt Larson (Princeton / Institute for Advanced Study)
12:00-1:30 Lunch
1:30-2:30 Patricia Klein (Texas A&M University)
2:30-3:00 Break
3:00-4:00 PM Jianping Pan (Arizona State University)
4:00-5:00 PM Poster fair
5:00 PM-??? Pizza
To register for ALGECOM, please fill out the google poll at
https://forms.gle/BVe3MHfckc8kXTkn9
All financial support has been allocated, but please still fill out the poll if you plan to attend.
The ALGECOM organizers are
David Speyer (U Michigan)
Peter Tingley (Loyola)
Alex Yong (UIUC)
Titles and abstracts:
Speaker: Chris Eur (Carnegie Mellon University)
Title: How do matroids behave like projective toric varieties?
Abstract: A tropical model of a matroid, called the Bergman fan, provides a toric variety whose Chow ring behaves like a smooth toric projective variety. What about sheaf cohomologies of line bundles or vector bundles? How should one make sense of such notions for a matroid? Even for linear matroids, many geometric questions remain. We discuss various developments surrounding these questions.
Speaker: Matt Larson (Princeton / Institute for Advanced Study)
Title: Fine multidegrees and Grobner degenerations
Abstract: Grobner degenerations are a method for studying the geometry of a subvariety X of C^n using a scaling action of the multiplicative group C*, but they often lose a lot of information. This information loss can be minimized by compactifying affine space. We show that the possible Grobner degenerations of X are controlled by the homology class of the closure of X in (P^1)^n. This gives a simple criterion for a collection of squarefree polynomials to be a universal Grobner basis, which can be used to recover many known results on universal Grobner bases. We apply this result to a family of matrix Schubert varieties. Joint work with Daoji Huang.
Speaker: Patricia Klein (Texas A&M University)
Title: Gröbner degeneration in Schubert calculus
Abstract: Roughly speaking, enumerative geometry is a field whose goal is to count the "typical" number of solutions to certain types of families of polynomial equations, particularly when that number is finite. With a great deal of effort, especially in the wake of the work of Hermann Schubert around the turn of the 20th century, mathematicians made rigorous the notion of a "typical" answer and also made rigorous certain simplifying strategies Schubert had suggested. Indeed, making Schubert's arguments precise was the topic of Hilbert's 15th problem, and the field born from this study is now called Schubert calculus. The simplifications Schubert had suggested entail sliding or deforming the geometric objects to be studied while preserving the total number of whatever it is one wants to count. These strategies are what are now called degeneration techniques. In this talk, we will review some classical results in Schubert calculus, describe some modern questions in the area, and then explain how these questions are studied via Gröbner degeneration in particular.
Speaker: Jianping Pan (Arizona State University)
Title: Variations of uncrowning algorithms and stable Grothendieck polynomials
Abstract: In this talk, I will provide the motivation, basic definitions, and abundant examples of tableaux related to variations of stable Grothendieck polynomials. The classic uncrowding algorithm defined by Buch on set-valued tableaux gives a bijective proof of Lenart’s Schur expansion for symmetric Grothendieck polynomials. Hook-valued tableaux are associated with stable canonical Grothendieck polynomials and we introduce an uncrowding algorithm on them. Recently Hwang, Jang, Kim, Song and Song gave a Schur expansion for the refined canonical stable Grothendieck polynomials (using exquisite tableaux). We discover a novel connection between the exquisite tableaux model and hook-valued tableaux via the uncrowding and jeu de taquin algorithms, using a classic result of Benkart, Sottile and Stroomer. This connection reveals a hidden symmetry of the hook-valued tableaux and the uncrowding algorithm defined on them. This talk is based on joint work with Pappe, Poh and Schilling, and with Jang, Kim, Pappe and Schilling.
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