Cascade Lectures in Combinatorics
This is a series of one day conferences being organized in the Pacific Northwest region, with funding from the National Science Foundation. These will take place twice a year for at least five years, with each meeting on a Saturday. They are aimed at graduate students, postdocs, and faculty, bringing in four expert speakers for each meeting and aiming to create a sense of community with friendly and frequent meetings. Both meetings in 2021-22 will be on zoom, and then in subsequent years we plan to alternate between in-person meetings and online meetings.
Sixth meeting:
Date: May 18-19, 2023
Time: 9:30am May 18--2pm May 19 Pacific Daylight Savings Time
Location: Eugene, Oregon
Host: University of Oregon (UO)
How to register: email plhersh@uoregon.edu
Speakers:
Christian Gaetz, Cornell University
Rick Kenyon, Yale University
Allen Knutson, Cornell University
Bridget Tenner, Depaul University
Cynthia Vinzant, University of Washington
Poster session: There will also be a poster session for early career mathematicians to present their work.
Participant travel funding: There will be funding for some early career mathematicians travel and hotel -- check back later for information on how to apply.
Local organizing committee:
Patricia Hersh (University of Oregon)
John Machacek (University of Oregon)
Ben Young (University of Oregon)
Fifth meeting:
Date: November 4, 2023
Time: 10am-5pm Pacific Daylight Savings Time
Location: Zoom (meeting number will be sent to those who register)
Host: University of Oregon (UO)
How to register: email plhersh@uoregon.edu
Speakers:
Chris Eur (Harvard)
Sergey Fomin (University of Michigan)
Maria Monks Gillespie (Colorado State)
Sheila Sundaram (University of Minnesota)
Schedule (all in Pacific Daylight Savings Time):
9:30-10am: online ``coffee break''
10-11am: Sergey Fomin (U. Michigan)
11-11:30am: online ``coffee break''
11:30am-12:30pm: Sheila Sundaram (U. Minnesota)
12:30-1:30pm: online ``lunch break''
1:30-2:30pm: Chris Eur (Harvard)
2:30-3pm: online ``coffee break''
3-4pm: Maria Monks Gillespie (Colorado State)
Talk titles and abstracts:
Chris Eur: A Tale of two rings
Abstract: A complex projective manifold carries two well-studied rings, namely, the cohomology ring and the Grothendieck K-ring of vector bundles. For toric varieties, these have polyhedral descriptions, as the polytope algebra and the algebra of piecewise polynomials. For special toric varieties, we show an exceptional isomorphism between these two rings, different from the classical Hirzebruch-Riemann-Roch theorem, and discuss its utility in combinatorial contexts.
Joint works with Andrew Berget, Alex Fink, June Huh, Matt Larson, Hunter Spink, and Dennis Tseng.
Sergey Fomin: Incidences and tilings
Abstract: We show that various classical theorems of real/complex linear incidence geometry, such as the theorems of Pappus, Desargues, Möbius, and so on, can be interpreted as special cases of a single "master theorem" that involves an arbitrary tiling of a closed oriented surface by quadrilateral tiles. This yields a general mechanism for producing new incidence theorems and generalizing the known ones.
This is joint work with Pavlo Pylyavskyy.
Maria M. Gillespie: Battery-powered tableaux, Springer theory, and the Delta conjecture
Abstract: We present new formulas for the t=0 specialization of the polynomials involved in the Delta conjecture. One is a combinatorial Schur expansion in terms of "battery-powered tableaux", and its related formulation is simply an adjoint Schur operator applied to a Hall-Littlewood polynomial. This generalizes to give formulas for the Frobenius character of the cohomology ring of the "Delta-Springer varieties" defined by Griffin, Levinson, and Woo, which fit into the generalized Springer theory of Borho and MacPherson. We will describe how this generalized theory of partial resolutions of nilpotent varieties leads to our results, and state some more general conjectural formulas towards a Schur expansion for the Delta conjecture at the end.
This is joint work with Sean Griffin.
Sheila Sundaram: Stirling representations, supersolvable matroids and Koszul duality
Abstract: (Joint work with Ayah Almousa and Vic Reiner)
The unsigned Stirling numbers c(n,k) of the first kind give the Hilbert function for two algebras
associated to the hyperplane arrangement in type A, the Orlik-Solomon algebra and the graded Varchenko-Gelfand
algebra. Both algebras carry symmetric group actions with a rich structure, and have been well studied by topologists, algebraists and combinatorialists: the first coincides with the Whitney homology
of the partition lattice, and the second with a well known decomposition (Thrall's decomposition,
giving the higher Lie characters) of the universal enveloping algebra of the free Lie algebra. In each case the graded representations have dimension c(n,k).
Both these algebras are examples of Koszul algebras, for which the Priddy resolution defines a group-equivariant dual Koszul algebra.
Now the Hilbert function is given by the Stirling numbers S(n,k) of the second kind, and hence the Koszul duality relation defines representations of the symmetric group whose dimensions are the numbers S(n,k).
Investigating this observation led to the realisation that this situation generalises to all supersolvable matroids. The Koszul duality recurrence is shown to have interesting consequences.
For the resulting group representations, it implies the existence of branching rules which, in the case of the braid
arrangement, specialise by dimension to the classical enumerative recurrences satisfied by the Stirling numbers of both kinds.
It also implies representation stability in the sense of Church and Farb.
The associated Koszul dual representations appear to have other properties that are more mysterious;
for example, in the case of the braid arrangement, the Stirling representations of the Koszul dual
are sometimes tantalisingly close to being permutation modules. I will endeavour to give a flavour
of these phenomena in the talk.
Participants registered so far:
Ayah Almousa, U. Minnesota
Portia Anderson, Cornell
Olga Azenhas, U. Coimbra PORTUGAL
Marge Bayer, U. Kansas
Tatyana Benko, U. Oregon
Sara Billey, U. Washington
Kelsey Brown, Colorado State U.
Teressa Chambers, Brown University
Herman Chau, U. Washington
Sunita Chepuri, University of Puget Sound
Annika Christiansen, U. Oregon
Spencer Daugherty, North Carolina State University
Danai Deligeorgaki, KTH SWEDEN
Anton Dochtermann, Texas State
Daniel Dugger, U. Oregon
Sergi Elizalde, Dartmouth
Chris Eur, Harvard
Michael Feigen, U. Oregon
Sergey Fomin, U. Michigan
Leigh Foster, U. Oregon
Pavel Galashin, UCLA
Yibo Gao, Peking University CHINA
Terrence George, UCLA
Patricia Hersh, U. Oregon
Byung-Hak Hwang, SOUTH KOREA
Steven Karp, Notre Dame
Soyeon Kim, UC Davis
Juan Lanfranco, Notre Dame
Robert John Lentfer, UC Berkeley
Cordelia Yuqiao Li, U. Washington
Yifei Li, University of Illinois at Springfield
Jinting Liang, Michigan State
Hsin-Chieh Liao, U. Miami
Yuze Luan, UC Davis
John Machacek, U. Oregon
Olya Mandelshtam, U. Waterloo CANADA
Karola Meszaros, Cornell
Maria Monks Gillespie, Colorado State
Dania Morales, U. Kansas
Anastasia Nathanson, U. Minnesota
Jaeseong Oh, Yonsei University SOUTH KOREA
Jianping Pan, North Carolina State University
Joseph Pappe, UC Davis
David Perkinson, Reed College
Kyla Pohl, U. Oregon
Nick Proudfoot, U. Oregon
Anna Pun, Baruch College
Daniel Qin, UC Davis
Andrew Reimer-Berg, Colorado State U.
Tom Roby, U. Connecticut
Bruce Sagan, Michigan State
Kyle Salois, Colorado State
Anne Schilling, UC Davis
Jozsef Solymosi, UBC CANADA
Richard Stanley, MIT and University of Miami
Dennis Stanton, U. Minnesota
Yuxuan Sun, U. Minnesota
Sheila Sundaram, U. Minnesota
Josh Swanson, U. Southern California
Gabe Udell, Cornell
Yannic Vargas, TU Graz GERMANY
S. Venkitesh, U. Haifa, ISRAEL
Jeremy Wang, Brown
Anna Weigandt, U. Minnesota
Peter Winkler, Dartmouth
Weihong Xu, Virginia Tech
Ben Young, U. Oregon
Sergey Yuzvinsky, U. Oregon
Guilherme Zeus, Haverford College
Local organizing committee:
Patricia Hersh, University of Oregon
John Machacek, University of Oregon
Ben Young, University of Oregon
Fourth meeting:
Date: April 22, 2023
Time: 10am-5pm Pacific Daylight Savings Time
Location: Reed College, Portland, Oregon
IMPORTANT PARKING INFO: people who are driving should park in the North Parking Lot off of Steele Street. Those with accessibility issues should park in the West Lot off of 28th Ave.
Local organizers' conference web site (with further practical info): click here.
How to register: email plhersh@uoregon.edu by noon Pacific Daylight Savings Time on Thursday, April 20, 2023.
How to apply for participant travel funding: email plhersh@uoregon.edu asking for list of questions in relatively easy funding application form.
Speakers for fourth meeting:
Federico Ardila, San Francisco State University
Sara Billey, University of Washington
Brendon Rhoades, UCSD
Monica Vazirani, UC Davis
Talk titles and abstracts:
Federico Ardila: Combinatorial intersection theory
Abstract:
Intersection theory studies how subvarieties of an algebraic variety X intersect. Algebraically, this information is encoded in the Chow ring A(X), which is very difficult to describe in general. When X is the toric variety of a simplicial fan, there are several combinatorial and polyhedral descriptions of this ring due to Fulton-Sturmfels, Billera, and Danilov-Brion. These descriptions lead to interesting combinatorial problems, and in some cases, they are important ingredients in the proofs of long-standing conjectures. This talk will survey some examples of problems that arise in combinatorial intersection theory and a few approaches to solving them. It will feature joint work with Mont Cordero, Graham Denham, Chris Eur, June Huh, Nayeong Kim, Carly Klivans, and Raul Penaguiao, and will not assume previous familiarity with intersection theory.
Sara Billey: Combinatorial characterizations of smooth positroid varieties
Abstract: Positroids are certain representable matroids originally studied by
Postnikov in connection with the totally nonnegative Grassmannian and
now used widely in algebraic combinatorics. The positroids give rise
to determinantal equations defining positroid varieties as
subvarieties of the Grassmannian variety. Rietsch, Knutson-Lam-Speyer,
and Pawlowski studied geometric and cohomological properties of these
varieties. In this talk, we continue the study of the geometric
properties of positroid varieties by establishing several equivalent
conditions characterizing smooth positroid varieties using a variation
of pattern avoidance defined on decorated permutations, which are in
bijection with positroids. This allows us to give several formulas
for counting the number of smooth positroids according to natural
statistics on decorated permutations. Furthermore, we give a
combinatorial method for determining the dimension of the tangent
space of a positroid variety at the torus fixed points using an
induced subgraph of the Johnson graph. We will conclude with some
open problems in this area.
This talk is based on joint work with Jordan Weaver and Christian
Krattenthaler.
Brendon Rhoades: Zonotopal algebras, orbit harmonics, and Donaldson-Thomas invariants
Abstract: Kontsevich and Soibelman defined the cohomological Hall algebra $\mathcal{H}_Q$ attached to a quiver $Q$. When $Q$ is symmetric and has at least one loop at each vertex, Efimov proved that $\mathcal{H}_Q$ is freely generated as a supercommutative algebra by a certain multigraded vector space $V$. For any dimension vector $\gamma$, the dimension and Hilbert series of the $\gamma^{th}$ piece of $V$ are the {\em numerical} and {\em quantum Donaldson-Thomas invariants}. We give the first combinatorial interpretation of the numerical DT invariant as an orbit count of lattice points in a polytope of break divisors attached to $(Q, \gamma)$. We also interpret the quantum DT invariant in terms of graded rings studied by Postnikov-Shapiro and Ardila-Postnikov. Our central technique is the orbit harmonic method dating back to Kostant which has widespread application to combinatorial representation theory. This is joint work with Markus Reineke and Vasu Tewari.
Monica Vazirani: Parabolic Hilbert schemes and rational Cherednik algebras
Abstract: Young diagrams and standard tableaux on them parameterize irreducible representations of the symmetric group and their bases, respectively. There is a similar story for a nice family of representations of the double affine Hecke algebra (DAHA) or for the rational Cherednik algebra (a.k.a. rational DAHA) with appropriate modifications. This construction of the basis makes use of an alternate presentation of the rational DAHA and the basis diagonalizes the action of its Dunkl-Opdam subalgebra.
We can describe one such representation using the geometry of parabolic Hilbert schemes of points on plane curve singularities. The ``tableau" basis that diagonalizes the Dunkl-Opdam subalgebra is the basis of equivariant homology that comes from torus fixed points.
This is joint work with Eugene Gorsky and Jose Simental.
Participants registered so far:
Federico Ardila (SFSU)
Ava Bamforth (U. Oregon)
Tatyana Benko (U. Oregon)
Sara Billey (U. Washington)
Scott Blair (Reed College)
Megan Boes (U. Oregon)
John Caughman (Portland State University)
Herman Chau (U. Washington)
Tracy Chin (U. Washington)
Annika Christiansen (U. Oregon)
Natasha Crepeau (U. Washington)
Julie Curtis (U. Washington)
Zajj Daugherty (Reed College)
William Doerr (U. Oregon)
Daniel Dugger (U. Oregon)
Michael Feigen (U. Oregon)
Leigh Foster (U. Oregon)
Fern Gossow (U. Oregon)
Dan Guyer (U. Washington)
Junaid Hasan (U. Washington)
Patricia Hersh (U. Oregon)
Josh Hinman (U. Washington)
Ben Homan (U. Oregon)
Amzi Jeffs (Carnegie Mellon)
Jesse Kim (UCSD)
Stephen Lacina (U. Oregon)
Cordelia Li (U. Washington)
Sam Lindbloom-Airy (U. Washington)
Kevin Liu (U. Washington)
John Machacek (U. Oregon)
Alex Mason (U. Washington)
Alex Moll (Reed College)
Nancy Ann Neudauer (Pacific University)
Elena O'Grady (Reed College)
Soohyun Park (U. Chicago)
David Perkinson (Reed College)
Brendon Rhoades (UC San Diego)
Connor Sawaske (U. Washington alum)
Sanay Sehgal (Reed College)
Racher Snyder (U. Oregon)
Andrew Tawfeek (U. Washington)
Nicolas Jaramillo Torres (U. Oregon)
Monica Vazirani (UC Davis)
Elizabeth Xiao (UBC, CANADA)
Yirong Yang (U. Washington)
Ben Young (U. Oregon)
Local organizing committee:
Zajj Daugherty, Reed College
Patricia Hersh, University of Oregon
Alex Moll, Reed College
David Perkinson, Reed College
Third meeting:
Date: October 15, 2022
Time: 10am-5pm Pacific Daylight Savings Time
Locations: ZOOM (meeting number will be sent to those who register)
Host: University of Oregon (UO)
How to register: email plhersh@uoregon.edu
Schedule for third meeting (all in Pacific Daylight Savings Time):
9:30-10am: online ``coffee break''
10-11am: June Huh (Princeton)
11-11:30am: online ``coffee break''
11:30am-12:30pm: Jessica Striker (North Dakota State U.)
12:30-2:30pm: online ``lunch break''
2:30-3:30pm: Ricky Liu (U. Washington)
3:30-4pm: online ``coffee break''
4-5pm: Stephanie van Willigenburg (UBC) -- slides
Talk titles and abstracts for third meeting:
June Huh: Stellahedral geometry of matroids
Abstract: The main result is that valuative, homological, and numerical equivalence relations for matroids coincide. The central construction is the "augmented tautological classes of matroids," modeled after certain vector bundles on the stellahedral toric variety. Based on joint work with Chris Eur and Matt Larson.
Ricky Liu: Combinatorial mutations and birational maps
Abstract: A combinatorial mutation is a continuous, piecewise-linear, volume-preserving operation on rational polytopes. In this talk, we will present several examples of pairs of polytopes whose Ehrhart equivalence can be proven by constructing combinatorial mutations between them, most notably the chain polytopes of rectangular and trapezoidal posets. We will also show how these constructions can be viewed as tropicalizations of birational maps from dynamical algebraic combinatorics. This is based on joint work with Joseph Johnson.
Jessica Striker: Alternating sign matrix A B C (D)s
Abstract: In this talk, we will discuss my favorite combinatorial objects, alternating sign matrices, from Algebraic, Bijective, and Combinatorial perspectives. In addition, we'll peruse lurking Dynamics embedded in each of these perspectives.
Stephanie van Willigenburg: Schur functions in noncommuting variables
Abstract: In 2004 Rosas and Sagan asked whether there was a way to define a basis in the algebra of symmetric functions in noncommuting variables, NCSym, having properties analogous to the classical Schur functions, in the algebra of symmetric functions, Sym. Despite attempts, this question has remained open since then. In this talk we answer this question by introducing Schur functions in noncommuting variables, which naturally refine Rosas-Sagan Schur functions, in addition to having many analogous classical properties that we will discuss.
Pre-registered participants so far for third meeting:
Atiq Ali (LUMS University, PAKISTAN)
Ayah Almousa (U. Minnesota)
Moaaz Alqady (UO)
Robert Angarone (U. Minnesota)
Ahmed Ashraf (U. Toronto, CANADA)
Ava Bamforth (UO)
Tatyana Benko (UO)
Sarah Brauner (U. Minnesota)
Thomas Brown (Simon Fraser U., CANADA)
MacKenzie Carr (Simon Fraser U., CANADA)
Annika Christiansen (UO)
Elizabeth Dinkelman (MSTB, Annandale Campus)
Anton Dochtermann (Texas State U.)
Daniel Dugger (UO)
Christopher Eur (Harvard)
Michael Feigen (UO)
Leigh Foster (UO)
Jason Fulman (U. Southern California)
Yibo Gao, U. Michigan
Bennet Goeckner (U. of San Diego)
Fern Gossow (UO)
Curtis Greene (Haverford College)
Darij Grinberg (Drexel University)
Patricia Hersh (UO)
June Huh (Princeton)
Yibo Ji (Cornell)
Yuhan Jiang (Harvard)
Stephen Lacina (UO)
Shiyue Li (Brown)
Ricky Liu (U. Washington)
Guorui Ma (Tsinghua University, CHINA)
Yichen Ma (Cornell)
John Machacek (UO)
Karola Meszaros (Cornell)
Walter Morris (George Mason University)
Anastasia Nathanson (U.Minnesota)
Isabella Novik (U. Washington)
Chinwe Obi (Federal University of Technology Owerri, NIGERIA)
David Perkinson (Reed College)
Kyla Pohl (UO)
Nick Proudfoot (UO)
Caelan Ritter (U. Washington)
Anne Schilling (UC Davis)
Richard Stanley (MIT and U. Miami)
Dennis Stanton (U. Minnesota)
Tamon Stephen (Simon Fraser, CANADA)
Monroe Stephenson (Reed College)
Jessica Striker (North Dakota State U.)
Sheila Sundaram (Pierrepont School)
Andrew Tawfeek (U. Washington)
Vasu Tewari (U. Hawaii)
Lilla Tothmeresz (Eotvos University, HUNGARY)
Arkady Vaintrob (UO)
Stephanie van Willigenburg (UBC, CANADA)
Alex Vidinas (Cornell)
Jeremy Wang (UCLA)
Prairie Wentworth-Nice (Cornell)
John Whelan (Cornell)
Kaelyn Willingham (U. Minnesota)
Elizabeth Xiao (UBC, CANADA)
Alex Woo (U. Idaho)
Ben Young (UO)
Fiona Young (Cornell)
Sergey Yuzvinsky (UO)
Tom Zaslavsky (Binghamton U.)
Shaopeng Zhu (U. Maryland)
Organizing committee:
Patricia Hersh (U. Oregon)
John Machacek (U. Oregon)
Ben Young (U. Oregon)
Second meeting:
Date: April 2, 2022
Time: 10am-5pm Pacific Daylight Savings Time
Location: ZOOM (meeting number will be sent to those who register)
Host: University of Oregon (UO)
How to register: email plhersh@uoregon.edu by March 31.
Conference Schedule (in Pacific Daylight Savings Time):
9:30-10am: virtual coffee break
10-11am: Sylvie Corteel (UC Berkeley): ``Colored Vertex models coming from LLT polynomials'' slides
11-11:30am: virtual coffee break
11:30am-12:30pm: Olya Mandelshtam (U. Waterloo): ``Macdonald polynomials and the multispecies zero range process'' slides
12:30-2:30pm: virtual lunch break
2:30-3:30pm: Chris Fraser (Michigan State U.): ``Webs and canonical bases in degree two'' slides
3:30-4pm: virtual coffee break
4-5pm: Richard Stanley (MIT and U. Miami): ``Symmetric functions and permutation enumeration'' slides
Participants (registered so far):
Olga Azenhas, U. Coimbra
Adam Bamforth, U. Oregon
Esther Mae Banaian, Aarhus U., DENMARK
Tatyana Benko, U. Oregon
Sara Billey, U. Washington
Elijah Bodish, U. Oregon
Thomas Brown, Simon Fraser U., CANADA
Jesse Campion Loth, Simon Fraser U., CANADA
Elise Catania, U. Minnesota
Angel Chavez, U. Minnesota
Sunita Chepuri. U. Minnesota
Patricia Commins, U. Minnesota
Sylvie Corteel, UC Berkeley
Natasha Crepeau, U. Washington
Anton Dochtermann, Texas State University
Anne Dranowski, U. Southern California
Daniel Dugger, U. Oregon
Sen-Peng Eu, National Taiwan Normal U., TAIWAN
Chris Fraser, Michigan State University
Jason Fulman, U. Southern California
Jennifer Galovich, College of Saint Benedit Saint John's University
Andrey Glubokov, Purdue
Bennet Goeckner, U. Washington
Curtis Greene, Haverford College
Darij Grinberg, Drexel U.
Peter Tomas Gylys-Colwell, U. Washington
Joshua Hallam, Loyola Marymount U.
Andy Hardt, U. Minnesota
Junaid Hasan, U. Washington
Patricia Hersh, U. Oregon
Sam Hopkins, Howard University
Jonathan Jedwab, Simon Fraser U., CANADA
Elizabeth Kelley, UIUC
Stephen Lacina, U. Oregon
David Levin, U. Oregon
Shuxing Li, Simon Fraser U., CANADA
Yifei Li, U. Illinois Springfield
Hsin-Chieh Liao, U. Miami
Bingyan Liu, UIUC
John Machacek, U. Oregon
Olya Mandelshtam, U. Waterloo, CANADA
Jacob Matherne, U. Bonn, GERMANY
Everett Meike, NCSU
Karola Meszaros, Cornell
Kailash Misra, NCSU
George Nasr, U. Oregon
Anastasia Nathanson, U. Minnesota
Son Nguyen, U. Minnesota
Nicholas Ovenhouse, U. Minnesota
J.E. Paguyo, U. Southern California
Jianping Pan, NCSU
Joseph Pappe, UC Davis
David Perkinson, Reed College
Stephan Pfannerer-Mittas, TU Wien, AUSTRIA
Kyla Pohl, U. Oregon
Arun Ram, U. Melbourne, AUSTRALIA
Brendon Rhoades, UCSD
Tom Roby, U. Connecticut
Bruce Sagan, Michigan State U.
Giftson Santhosh, Simon Fraser U., CANADA
Anne Schilling, UC Davis
Jeanne Scott, U. Leeds, ENGLAND
Richard Stanley, MIT and U. Miami
Dennis Stanton, U. Minnesota
Carolyn Stephen, U. Minnesota
Tamon Stephen, Simon Fraser U., CANADA
Sheila Sundaram, Pierrepont School
Joshua Swanson, U. Southern California
Cynthia Vinzant, U. Washington
Shiyun Wang, U. Southern California
Lauren Williams, Harvard
Daniel Yaqubi, University of Torbat-e Jam
Ben Young, U. Oregon
Tianyi Yu, UCSD
Chenchen Zhao, U. Southern California
Titles and abstracts:
Sylvie Corteel: ``Colored vertex models coming from LLT polynomials''
Abstract: Integrable vertex models have recently been used to study various families of symmetric and nonsymmetric polynomials. In this talk, we will define Yang-Baxter integrable vertex models, from which we will construct a class of partition functions that equal the LLT polynomials of Lascoux, Leclerc, and Thibon and a class of partition functions that equal the super ribbon functions of Lam. Using the vertex model formalism, we can prove many properties of these polynomials, including (super)symmetry and Cauchy identities. Finally, we will discuss applications of our vertex models to domino tilings of the Aztec diamond. This is based on joint work with Andrew Gitlin, David Keating, and Jeremy Meza.
Chris Fraser: ``Webs and canonical bases in degree two''
Abstract: An ongoing problem in algebraic combinatorics is the construction of "good bases" for representations of simple Lie groups. I will give some background on some approaches to this problem, including an approach via web diagrams. Then I will explain a new result which fits into this framework: Lusztig's canonical basis for the degree two part of the Grassmannian coordinate ring consists of web diagrams.
Olya Mandelshtam: ``Macdonald polynomials and the multispecies zero range process''
Abstract:The connection of the Macdonald polynomials $P_{\lambda}$ to the well-studied particle model called the ASEP (asymmetric simple exclusion process) has been known for some time. Namely, the partition function of the ASEP is the specialization $P_{\lambda}(1,..1;1,t)$. Recently, an analogous connection has been found between the modified Macdonald polynomials and a multispecies totally asymmetric zero-range process (TAZRP), where $\widetilde{H}_{\lambda}(x_1,\ldots,x_n;1 t)$ specializes to the partition function of the latter. This link motivated a new formula for the modified Macdonald polynomials in terms of a statistic on tableaux called queue inversions. We present a Markov process on these tableaux that projects to the TAZRP, and derive formulas for stationary probabilities and certain observables such as densities and currents. This is joint work with Arvind Ayyer and James Martin.
Richard Stanley: ``Symmetric functions and permutation enumeration''
Abstract: We will survey some connections between permutations and
symmetric functions, focusing on the use of symmetric functions to
prove results on the enumeration of permutations. Topics will include
class multiplication, commutators, alternating permutations, Hurwitz
numbers, Lyndon symmetric functions, and generalized descent sets. The
talk will assume a basic knowledge of symmetric functions.
Organizing committee:
Patricia Hersh (UO)
John Machacek (UO)
Ben Young (UO)
First meeting:
Date: Nov. 6, 2021
Time: 10am-5pm Pacific Standard Time
Location: ZOOM (meeting number will be sent to those who register)
Host: University of Oregon (UO)
How to register: email plhersh@uoregon.edu
Speakers:
Victor Reiner (U. Minnesota) slides
Volkmar Welker (Phillipps U. Marburg) slides
Anna Weigandt (MIT)
Laura Escobar Vega (Washington U. in St. Louis)
Talk Schedule (all times are Pacific Daylight Savings Time):
9:45-10am, welcome
10-11am, Vic Reiner (U. Minnesota), ``Topology of augmented Bergman complexes''
11-11:30am, virtual coffee break at gather.town and zoom
11:30am-12:30pm, Volkmar Welker (Philipps U. Marburg, GERMANY, ``Matroid type complexes with orthogonality relations'')
12:30-2:30pm, virtual lunch break at gather.town and zoom
2:30-3:30pm, Anna Weigandt (MIT), ``The Castelnuovo-Mumford regularity of matrix Schubert varieties''
3:30-4pm, virtual coffee break at gather.town and zoom
4-5pm, Laura Escobar Vega (Washington U. in St. Louis), ``Which Schubert varieties are Hessenberg varieties?''
Registered participants so far:
Marcelo Aguiar, Cornell
Esther Banaian, U. Minnesota
Andrew Berget, Western Washington U.
Louis Billera, Cornell
Morgen Bills, U. Nebraska Lincoln
Elijah Bodish, UO
Sarah Brauner, U. Minnesota
Jon Brundan, UO
Herman Chau, U. Washington
Judy Chiang, UIUC
Anton Dochtermann, Texas State University
Daniel Dugger, UO
Ben Elias, UO
Laura Escobar Vega, Washington U. in St. Louis
Michael Feigen, UO
Federico Firoozi, Simon Fraser U. CANADA
Chris Fraser, Michigan State
Pavel Galashin, UCLA
Shiliang Gao, UIUC
Yibo Gao, MIT
Bennet Goeckner, U. Washington
Curtis Greene, Haverford College
Adam Daniel Gregory, U. Florida
Sean Griffin, UC Davis
Elena Hafner, Cornell
Joshua Hallam, Loyola Marymount University
Patricia Hersh, UO
Sam Hopkins, Howard University
Adeli Hutton, Washington U. in St. Louis
William Kantor, UO
Stephen Lacina, UO
Yifei Li, U. Illinois Springfield
Hsin-Chieh Liao, U. Miami
Haggai Liu, Simon Fraser, CANADA
Yichen Ma, Cornell
John Machacek, UO
Ricardo Mamede, U. Coimbra, PORTUGAL
Olya Mandelshtam, U. Waterloo, CANADA
Jacob Matherne, U. Bonn and Max Planck Institute for Mathematics
Alex McDonough, UC Davis
Jodi McWhirter, Washington U. in St. Louis
Karola Meszaros, Cornell
Marni Mishna, Simon Fraser, CANADA
George Nasr, UO
Isabella Novik, U. Washington
Gidon Orelowitz, UIUC
David Perkinson, Reed College
Martha Precup, Washington U. in St. Louis
Nick Proudfoot, UO
Arun Ram, U. Melbourne, AUSTRALIA
Eric Ramos, Bowdoin College
Nathan Reading, NCSU
Victor Reiner, U. Minnesota
Tom Roby, U. Connecticut
Christopher Ryba, UC Berkeley
Bruce Sagan, Michigan State
Linus Setiabrata, U. Chicago
Cliff Smyth, UNC Greensboro
Avery James St. Dizier, UIUC
Richard Stanley, MIT and U. Miami
Ada Stelzer, UIUC
Tamon Stephen, Simon Fraser, CANADA
Zachary Stier, UC Berkeley
Sheila Sundaram, Pierrepont School, Westport, CT
Vasu Tewari, U. Hawaii
Arkady Vaintrob, UO
Aswin Rangasamy Venkatesan, San Francisco State U.
Alex Vidinas, Cornell
Adam Volk, U. Nebraska Lincoln
Trung Vu, UIUC
Robert Marshawn Walker, U. Wisconsin
Anna Weigandt, MIT
Volkmar Welker, Philipps-Universitat Marburg, GERMANY
Prairie Wentworth-Nice, Cornell
Kaelyn Willingham, U. Minnesota
Peter Winkler, Dartmouth
Alex Woo, U. Idaho
Ben Young, UO
Fiona Young, Cornell
Shaopen Zhu, U. Maryland
Talk titles and abstracts:
Laura Escobar Vega: ``Which Schubert varieties are Hessenberg varieties?''
Abstract: Schubert varieties and Hessenberg varieties are subvarieties of the flag variety with connections to both algebraic combinatorics and representation theory. I will discuss joint work with Martha Precup and John Shareshian in which we investigate which Schubert varieties are Hessenberg varieties. In the process, we give a restriction on the Euler characteristic of codimension-one Hessenberg varieties, as well as a description of the singular loci of regular codimension-one Hessenberg varieties.
Victor Reiner: ``Topology of augmented Bergman complexes''
Abstract: (based on arxiv:2108.13394; joint with REU students E. Bullock, A. Kelley, K. Ren, G. Shemy, D. Shen, B. Sun, A. Tao, Z. Zhang)
The augmented Bergman complex of a matroid is a simplicial complex introduced recently in work of Braden, Huh, Matherne, Proudfoot and Wang. It may be viewed as a hybrid of two well-studied pure shellable simplicial complexes associated to matroids: the independent set complex and the order complex of the lattice of flats.
After recalling the relevance of the augmented Bergman complex in the B-H-M-P-W work, we show that it is shellable, via two different families of shelling orders. Both shellings determine the homotopy type, and comparing the two answers re-interprets a known convolution formula counting bases of the matroid. One of the shellings leads to a surprisingly simple description for how symmetries of the matroid act on the homology of the complex.
Anna Weigandt: ``The Castelnuovo-Mumford regularity of matrix Schubert varieties''
Abstract: The Castelnuovo-Mumford regularity of a graded module provides a measure of how complicated its minimal free resolution is. In work with Rajchogt, Ren, Robichaux, and St. Dizier, we noted that the CM-regularity of matrix Schubert varieties can be easily obtained by knowing the degree of the corresponding Grothendieck polynomial. Furthermore, we gave explicit, combinatorial formulas for the degrees of symmetric Grothendieck polynomials. In this talk, I will present a general degree formula for Grothendieck polynomials. This is joint work with Oliver Pechenik and David Speyer.
Volkmar Welker: ``Matroid type complexes with orthogonality relations''
Abstract: In matroid theory the lattice of flats and the independence complex ofa
matroid are well studied posets and simplicial complexes.
Classical results in geometric combinatorics show that the order complex
of the lattice of flats and the independence complex are well behaved
combinatorially adn geometrically. In this talk we compare these results
with old and new results on analogous constructions which use the
orthogonality relation and non-degeneracy condition when starting with
vectors spaces over a finite field
equipped with a non-degenerate form. In particular we are interested in
the unitary case where our results allow to weaken conditions
restricting the know cases of a conjecture by Quillen on the poset of
p-subgroups of a finite group.
This is joint work with Kevin Piterman (Buenos Aires University).
Organizing committee:
Patricia Hersh (UO)
John Machacek (UO)
Ben Young (UO)