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BEGIN:VEVENT
SUMMARY:Matthew Baker (Georgia Tech School of Mathematics)
DTSTART;VALUE=DATE-TIME:20221004T200000Z
DTEND;VALUE=DATE-TIME:20221004T210000Z
DTSTAMP;VALUE=DATE-TIME:20240423T110231Z
UID:Matroids/1
DESCRIPTION:Title: Foundations of Matroids\nby Matthew Baker (Georgia Tech School of Mat
hematics) as part of Matroids - Combinatorics\, Algebra and Geometry Semin
ar\n\nLecture held in Room 210\, The Fields Institute.\n\nAbstract\nMatroi
d theorists are interested in questions concerning representability of mat
roids over fields. More generally\, one can ask about representability ove
r partial fields in the sense of Semple and Whittle. Pendavingh and van Zw
am introduced the universal partial field of a matroid\, which governs the
representations of over all partial fields. Unfortunately\, most matroids
are not representable over any partial field\, and in this case\, the uni
versal partial field is not defined. Oliver Lorscheid and I have introduce
d a generalization of the universal partial field which we call the founda
tion of a matroid\; it is always well-defined. The foundation is a type of
algebraic object which we call a pasture\; pastures include both hyperfie
lds and partial fields. As a particular application of this point of view\
, I will explain the classification of all possible foundations for matroi
ds having no minor isomorphic to U(2\,5) or U(3\,5). Among other things\,
this provides a short and conceptual proof of the 1997 theorem of Lee and
Scobee which says that a matroid is both ternary and orientable if and onl
y if it is dyadic.\n
LOCATION:https://researchseminars.org/talk/Matroids/1/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Bennet Goeckner (University of San Diego)
DTSTART;VALUE=DATE-TIME:20221006T190000Z
DTEND;VALUE=DATE-TIME:20221006T200000Z
DTSTAMP;VALUE=DATE-TIME:20240423T110231Z
UID:Matroids/2
DESCRIPTION:Title: Type cones and products of simplices\nby Bennet Goeckner (University
of San Diego) as part of Matroids - Combinatorics\, Algebra and Geometry S
eminar\n\nLecture held in Room 210\, The Fields Institute.\n\nAbstract\nA
polytope $P$ is the convex hull of finitely many points in Euclidean space
. For polytopes $P$ and $Q$\, we say that $Q$ is a Minkowski summand of $P
$ if there exists some polytope $R$ such that $Q+R=P$. The type cone of $P
$ encodes all of the (weak) Minkowski summands of P. In general\, combinat
orially isomorphic polytopes can have different type cones. We will first
consider type cones of polygons\, and then show that if $P$ is combinatori
ally isomorphic to a product of simplices\, then the type cone is simplici
al. This is joint work with Federico Castillo\, Joseph Doolittle\, Michael
Ross\, and Li Ying.\n
LOCATION:https://researchseminars.org/talk/Matroids/2/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Jessica Sidman (Mount Holyoke College)
DTSTART;VALUE=DATE-TIME:20221011T190000Z
DTEND;VALUE=DATE-TIME:20221011T200000Z
DTSTAMP;VALUE=DATE-TIME:20240423T110231Z
UID:Matroids/3
DESCRIPTION:Title: Matroid varieties\nby Jessica Sidman (Mount Holyoke College) as part
of Matroids - Combinatorics\, Algebra and Geometry Seminar\n\nLecture held
in Room 210\, The Fields Institute.\n\nAbstract\nLet $x$ denote a $k$-dim
ensional subspace of $\\mathbb{C}^n$ and let $A_x$ be a $k\\times n$ matri
x whose rows are a basis for $x$. The matroid $M_x$ on the columns of $A_x
$ is invariant under a change of basis for $x$. What can we say about the
set $\\Gamma_x$ of all $k$-dimensional subspaces $y$ such that $M_y = M_x?
$. We will explore this question algebraically\, showing that for some mat
roids that arise geometrically many non-trivial equations vanishing on $\\
Gamma_x$ can be derived geometrically. This is joint work with Will Traves
and Ashley Wheeler.\n
LOCATION:https://researchseminars.org/talk/Matroids/3/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Criel Merino (Instituto de Matematicas UNAM)
DTSTART;VALUE=DATE-TIME:20221013T190000Z
DTEND;VALUE=DATE-TIME:20221013T200000Z
DTSTAMP;VALUE=DATE-TIME:20240423T110231Z
UID:Matroids/4
DESCRIPTION:Title: The h-vector of a matroid complex\, paving matroids and the chip firing g
ame.\nby Criel Merino (Instituto de Matematicas UNAM) as part of Matro
ids - Combinatorics\, Algebra and Geometry Seminar\n\nLecture held in Room
210 The Fields Institute.\n\nAbstract\nA non-empty set of monomials $\\Si
gma$ is a multicomplex if for any monomial $z$ in $\\Sigma$ and a monomial
$z'$ such that $z'|z$\, we have that $z'$ also belongs to $\\Sigma$. A mu
lticomplex $\\Sigma$ is called pure if all its maximal elements have the s
ame degree. This notion is a generalization of the simplicial complex\, an
d several invariants extend directly\, as the $f$-vector of a multicomplex
\, which is the vector that lists the monomials grouped by degrees. A non-
empty set of monomials $\\Sigma$ is a multicomplex if for any monomial $z$
in $\\Sigma$ and a monomial $z'$ such that $z'|z$\, we have that $z'$ als
o belongs to $\\Sigma$. A multicomplex $\\Sigma$ is called pure if all its
maximal elements have the same degree. This notion is a generalization of
the simplicial complex\, and several invariants extend directly\, as the
$f$-vector of a multicomplex\, which is the vector that lists the monomial
s grouped by degrees. The relevance of multicomplexes in matroid theory is
partly due to a 1977 Richard Stanley conjecture that says that the $h$-ve
ctor of a matroid complex is the $f$-vector of a pure multicomplex. This h
as been proved for several families of matroids. In this talk\, we review
some results of Stanley’s conjecture\, mainly for paving and cographic m
atroids. A paving matroid is one in which all its circuits have a size of
at least the rank of the matroid. While\, the chip firing game is a solita
ire game played on a connected graph $G$ that surprisingly is related to t
he $h$-vector of the bond matroid of $G$.\n
LOCATION:https://researchseminars.org/talk/Matroids/4/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Olivier Bernardi (Brandeis University)
DTSTART;VALUE=DATE-TIME:20221018T190000Z
DTEND;VALUE=DATE-TIME:20221018T200000Z
DTSTAMP;VALUE=DATE-TIME:20240423T110231Z
UID:Matroids/5
DESCRIPTION:Title: Universal Tutte polynomial\nby Olivier Bernardi (Brandeis University)
as part of Matroids - Combinatorics\, Algebra and Geometry Seminar\n\nLec
ture held in Room 210 The Fields Institute.\n\nAbstract\nThe Tutte polynom
ial is an important matroid invariant. We will explain a natural way to e
xtend the Tutte polynomial from matroids to polymatroids. The Tutte polyno
mial can then be expressed as a sum over the points of the polymatroid (th
is is an extension of the basis extension of the classical definition of
the Tutte polynomial in terms of activities). Our definition is related to
previous works of Cameron and Fink and of Kálmán and Postnikov. \n\nOne
of the great properties of our Tutte polynomial is that it is polynomial
in the values of the rank function of the polymatroid. In other words\, we
can define a "universal Tutte polynomial" $T_n$ in $2+(2^n−1)$ variable
s that specialize to the Tutte polynomials of all polymatroids on n elemen
ts (the $2^n-1$ extra variables correspond to the non-trivial values of th
e rank function). \n\nThis is joint work with Tamás Kálmán and Alex Pos
tnikov.\n
LOCATION:https://researchseminars.org/talk/Matroids/5/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Graham Denham (Western University)
DTSTART;VALUE=DATE-TIME:20221020T190000Z
DTEND;VALUE=DATE-TIME:20221020T200000Z
DTSTAMP;VALUE=DATE-TIME:20240423T110231Z
UID:Matroids/6
DESCRIPTION:Title: Lagrangian Geometry\nby Graham Denham (Western University) as part of
Matroids - Combinatorics\, Algebra and Geometry Seminar\n\nLecture held i
n Room 210 The Fields Institute.\n\nAbstract\nIn joint work with Federico
Ardila and June Huh\, we introduce the conormal fan of a matroid\, which i
s an analogue of the Bergman fan. We use it to give a Lagrangian interpre
tation of the Chern-Schwartz-MacPherson cycle of a matroid. We also develo
p tools for tropical Hodge theory to show that the conormal fan satisfies
Poincaré duality\, the Hard Lefschetz property\, and the Hodge--Riemann r
elations. Together\, these imply conjectures of Brylawski and Dawson abou
t the log-concavity of the h-vectors of the broken circuit complex and ind
ependence complex of a matroid.\n
LOCATION:https://researchseminars.org/talk/Matroids/6/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Swee Hong Chan (Rutgers University)
DTSTART;VALUE=DATE-TIME:20221101T190000Z
DTEND;VALUE=DATE-TIME:20221101T200000Z
DTSTAMP;VALUE=DATE-TIME:20240423T110231Z
UID:Matroids/9
DESCRIPTION:Title: Combinatorial atlas for log-concave inequalities\nby Swee Hong Chan (
Rutgers University) as part of Matroids - Combinatorics\, Algebra and Geom
etry Seminar\n\nLecture held in Room 210 The Fields Institute.\n\nAbstract
\nThe study of log-concave inequalities for combinatorial objects have see
n much progress in recent years. One such progress is the solution to the
strongest form of Mason’s conjecture (independently by Anari et. al. and
Brándën-Huh). In the case of graphs\, this says that the sequence $f_k$
of the number of forests of the graph with $k$ edges\, form an ultra log-
concave sequence. In this talk\, we discuss an improved version of all the
se results\, proved by using a new tool called the combinatorial atlas met
hod. This is a joint work with Igor Pak. This talk is aimed at a general a
udience.\n
LOCATION:https://researchseminars.org/talk/Matroids/9/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Mariel Supina (KTH Royal Institute of Technology)
DTSTART;VALUE=DATE-TIME:20221103T190000Z
DTEND;VALUE=DATE-TIME:20221103T200000Z
DTSTAMP;VALUE=DATE-TIME:20240423T110231Z
UID:Matroids/10
DESCRIPTION:Title: The universal valuation of Coxeter matroids\nby Mariel Supina (KTH R
oyal Institute of Technology) as part of Matroids - Combinatorics\, Algebr
a and Geometry Seminar\n\nLecture held in Room 210 The Fields Institute.\n
\nAbstract\nMatroids subdivisions have rich connections to geometry\, and
thus we are often interested in functions on matroids that behave nicely w
ith respect to subdivisions\, or "valuations". Matroids are naturally link
ed to the symmetric group\; generalizing to other finite reflection groups
gives rise to Coxeter matroids. I will give an overview of these ideas an
d then present some work with Chris Eur and Mario Sanchez on constructing
the universal valuative invariant of Coxeter matroids. Since matroids and
their Coxeter analogues can be understood as families of polytopes with sp
ecial combinatorial properties\, I will present these results from a polyt
opal perspective.\n
LOCATION:https://researchseminars.org/talk/Matroids/10/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Federico Castillo (Universidad Catolica de Chile)
DTSTART;VALUE=DATE-TIME:20221108T200000Z
DTEND;VALUE=DATE-TIME:20221108T210000Z
DTSTAMP;VALUE=DATE-TIME:20240423T110231Z
UID:Matroids/11
DESCRIPTION:Title: Trying to generalize Pick's theorem.\nby Federico Castillo (Universi
dad Catolica de Chile) as part of Matroids - Combinatorics\, Algebra and G
eometry Seminar\n\nLecture held in Room 210 The Fields Institute.\n\nAbstr
act\nWe will review Pick's theorem in dimension 2 and see some ideas for a
n extension. Based on understanding the Todd class of toric varieties\, a
variety of local formulas have been proposed. Each of these local formulas
can be seen as a higher Pick's theorem. I will present a conjecture and e
vidence for one particular formula to become "the" natural extension.\n
LOCATION:https://researchseminars.org/talk/Matroids/11/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Mario Sanchez (Cornell University)
DTSTART;VALUE=DATE-TIME:20221117T200000Z
DTEND;VALUE=DATE-TIME:20221117T210000Z
DTSTAMP;VALUE=DATE-TIME:20240423T110231Z
UID:Matroids/14
DESCRIPTION:Title: Valuations and Hopf Monoid of Generalized Permutahedra\nby Mario San
chez (Cornell University) as part of Matroids - Combinatorics\, Algebra an
d Geometry Seminar\n\nLecture held in Room 210 The Fields Institute.\n\nAb
stract\nMany combinatorial objects\, such as matroids\, graphs\, and poset
s\, can be realized as generalized permutahedra - a beautiful family of po
lytopes. These realizations respect the natural multiplication of these ob
jects as well as natural "breaking" operations. Surprisingly many of the i
mportant invariants of these objects\, when viewed as functions on polytop
es are valuations\, that is\, they satisfy an inclusion-exclusion formula
with respect to subdivisions. In this talk\, I will discuss work with Fede
rico Ardila that describes the relationship between the algebraic structur
e on generalized permutahedra and valuations. Our main contribution is a n
ew easy-to-apply method that converts simple valuations into more complica
ted ones. New examples of valuations coming from this method include the K
azhdan-Lustzig polynomials of matroids and the motivic zeta functions of m
atroids.\n
LOCATION:https://researchseminars.org/talk/Matroids/14/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Jorge Olarte Parra (Technische Universitat Berlin)
DTSTART;VALUE=DATE-TIME:20221122T200000Z
DTEND;VALUE=DATE-TIME:20221122T210000Z
DTSTAMP;VALUE=DATE-TIME:20240423T110231Z
UID:Matroids/15
DESCRIPTION:Title: Transversal Valuated Matroids\nby Jorge Olarte Parra (Technische Uni
versitat Berlin) as part of Matroids - Combinatorics\, Algebra and Geometr
y Seminar\n\nLecture held in Room 210 The Fields Institute.\n\nAbstract\nT
ransversal matroids are a class of matroids that arises from matchings in
bipartite graphs. This has a generalization to valuated matroids which ari
se from tropical minors of a matrix. This is the so-called tropical Stiefe
l map. Brualdi and Dinolt gave a construction that characterizes all bipar
tite graphs that represent a given transversal matroid. We show a valuated
analogue of this result. In other words\, we characterize all matrices wi
th fixed maximal tropical minors. This has several geometric interpretatio
ns\, such as characterizing tropical hyperplanes with fixed stable interse
ctions.This is joint work with Alex Fink.\n
LOCATION:https://researchseminars.org/talk/Matroids/15/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Anastasia Nathanson (University of Minnesota)
DTSTART;VALUE=DATE-TIME:20221124T200000Z
DTEND;VALUE=DATE-TIME:20221124T210000Z
DTSTAMP;VALUE=DATE-TIME:20240423T110231Z
UID:Matroids/16
DESCRIPTION:Title: Volume polynomials of matroids\nby Anastasia Nathanson (University o
f Minnesota) as part of Matroids - Combinatorics\, Algebra and Geometry Se
minar\n\nLecture held in Room 210 The Fields Institute.\n\nAbstract\nAssoc
iated to any divisor in the Chow ring of a simplicial tropical fan\, we di
scuss the construction a family of polytopal complexes\, called normal com
plexes\, which we propose as an analogue of the well-studied notion of nor
mal polytopes from the setting of complete fans. The talk will describe ce
rtain closed convex polyhedral cones of divisors for which the “volume
” of each divisor in the cone—that is\, the degree of its top power—
is equal to the volume of the associated normal complexes. We will discuss
the theory of normal complexes developed in talk as a polytopal model und
erlying the combinatorial Hodge theory pioneered by Adiprasito\, Huh\, and
Katz.\n
LOCATION:https://researchseminars.org/talk/Matroids/16/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Lauren Novak (University of Washington)
DTSTART;VALUE=DATE-TIME:20221129T200000Z
DTEND;VALUE=DATE-TIME:20221129T210000Z
DTSTAMP;VALUE=DATE-TIME:20240423T110231Z
UID:Matroids/17
DESCRIPTION:Title: Mixed Volumes of normal complexes\nby Lauren Novak (University of Wa
shington) as part of Matroids - Combinatorics\, Algebra and Geometry Semin
ar\n\nLecture held in Room 210 The Fields Institute.\n\nAbstract\nIn 2021\
, Nathanson and Ross demonstrated that a geometric object called a normal
complex is the correct object to unite the algebraic concept of a tropical
fan's volume polynomial with an actual geometric volume computation. That
same year\, expanding on the work of Adiprasito\, Huh\, and Katz in provi
ng log-concavity in characteristic polynomials of matroids\, Amini and Piq
uerez established that mixed degrees of divisors of certain classes of tro
pical fans are log-concave. Given that mixed volumes generate log-concave
sequences\, we develop a definition of mixed volumes of normal complexes a
nd use our theory to establish new techniques for determining log-concavit
y for mixed degrees of divisors of a broad class of tropical fans.\n
LOCATION:https://researchseminars.org/talk/Matroids/17/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Dustin Ross (San Francisco State University)
DTSTART;VALUE=DATE-TIME:20221201T193000Z
DTEND;VALUE=DATE-TIME:20221201T203000Z
DTSTAMP;VALUE=DATE-TIME:20240423T110231Z
UID:Matroids/18
DESCRIPTION:Title: Matroid Psi Classes\nby Dustin Ross (San Francisco State University)
as part of Matroids - Combinatorics\, Algebra and Geometry Seminar\n\nLec
ture held in Room 210 The Fields Institute.\n\nAbstract\nMatroid Chow ring
s have played a central role in recent developments in matroid theory. In
this talk\, I’ll discuss parallels between matroid Chow rings and Chow r
ings of moduli spaces of curves\, leading to a new and simplified understa
nding of many important properties of matroid Chow rings.\n
LOCATION:https://researchseminars.org/talk/Matroids/18/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Colin Crowley (University of Wisconsin Madison)
DTSTART;VALUE=DATE-TIME:20221206T200000Z
DTEND;VALUE=DATE-TIME:20221206T210000Z
DTSTAMP;VALUE=DATE-TIME:20240423T110231Z
UID:Matroids/19
DESCRIPTION:Title: Hyperplane arrangements and compactifications of vector groups\nby C
olin Crowley (University of Wisconsin Madison) as part of Matroids - Combi
natorics\, Algebra and Geometry Seminar\n\nLecture held in Room 210 The Fi
elds Institute.\n\nAbstract\nSchubert varieties of hyperplane arrangements
\, also known as matroid Schubert varieties\, play an essential role in th
e proof of the Dowling-Wilson conjecture and in Kazhdan-Lusztig theory for
matroids. We study these varieties as equivariant compactifications of af
fine spaces\, and give necessary and sufficient conditions to characterize
them. We also generalize the theory to include partial compactifications
and morphisms between them. Our results resemble the correspondence betwee
n toric varieties and polyhedral fans.\n
LOCATION:https://researchseminars.org/talk/Matroids/19/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Shiyue Li (Brown University)
DTSTART;VALUE=DATE-TIME:20221208T200000Z
DTEND;VALUE=DATE-TIME:20221208T210000Z
DTSTAMP;VALUE=DATE-TIME:20240423T110231Z
UID:Matroids/20
DESCRIPTION:Title: K-rings of wonderful varieties and matroids\nby Shiyue Li (Brown Uni
versity) as part of Matroids - Combinatorics\, Algebra and Geometry Semina
r\n\nLecture held in Room 210 The Fields Institute.\n\nAbstract\nAs we hav
e seen in many talks in this wonderful seminar\, the wonderful variety of
a realizable matroid and its Chow ring have played key roles in solving ma
ny long-standing open questions in combinatorics and algebraic geometry. Y
et\, its $K$-rings are underexplored until recently. I will be sharing wit
h you some discoveries on the $K$-rings of the wonderful variety associate
d with a realizable matroid: an exceptional isomorphism between the $K$-ri
ng and the Chow ring\, with integral coefficients\, and a Hirzebruch–Rie
mann–Roch-type formula. These generalize a recent construction of Berget
–Eur–Spink–Tseng on the permutohedral variety. We also compute the E
uler characteristic of every line bundle on wonderful varieties\, and give
a purely combinatorial formula. This in turn gives a new valuative invari
ant of an arbitrary matroid. As an application\, we present the $K$-rings
and compute the Euler characteristic of arbitrary line bundles of the Deli
gne–Mumford–Knudsen moduli spaces of rational stable curves with disti
nct marked points. Joint with Matt Larson\, Sam Payne and Nick Proudfoot.\
n
LOCATION:https://researchseminars.org/talk/Matroids/20/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Laura Escobar Vega (Washington University in St. Louis)
DTSTART;VALUE=DATE-TIME:20221025T190000Z
DTEND;VALUE=DATE-TIME:20221025T200000Z
DTSTAMP;VALUE=DATE-TIME:20240423T110231Z
UID:Matroids/21
DESCRIPTION:Title: Partial Permutohedra\nby Laura Escobar Vega (Washington University i
n St. Louis) as part of Matroids - Combinatorics\, Algebra and Geometry Se
minar\n\nLecture held in Room 210 The Fields Institute.\n\nAbstract\nParti
al permutohedra are lattice polytopes which were recently introduced\nand
studied by Heuer and Striker. For positive integers $m$ and $n$\, the part
ial permutohedron~$\\cP(m\,n)$ is the convex hull of all vectors in $\\{0\
,1\,\\ldots\,n\\}^m$ with distinct nonzero entries. In this talk I will pr
esent results on the face lattice\, volume and Ehrhart polynomial of $\\cP
(m\,n)$. This is joint work with Behrend\, Castillo\, Chavez\, Diaz-Lopez\
, Harris and Insko.\n
LOCATION:https://researchseminars.org/talk/Matroids/21/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Vasu Tewari (University of Hawaii)
DTSTART;VALUE=DATE-TIME:20221027T190000Z
DTEND;VALUE=DATE-TIME:20221027T200000Z
DTSTAMP;VALUE=DATE-TIME:20240423T110231Z
UID:Matroids/22
DESCRIPTION:Title: Quotienting by quasisymmetric polynomials\nby Vasu Tewari (Universit
y of Hawaii) as part of Matroids - Combinatorics\, Algebra and Geometry Se
minar\n\nLecture held in Room 210 The Fields Institute.\n\nAbstract\nWe in
troduce a new basis for the polynomial ring which has its genesis in the c
omputation of the cohomology class of the permutahedral variety. We will s
ee that this basis is very well-behaved in regards to reduction modulo the
ideal of quasisymmetric polynomials. This has interesting ramifications o
f a discrete-geometric flavour that we will discuss -- for instance\, a mu
ltivariate analogue of mixed Eulerian numbers comes up naturally\, amongst
other things. \nJoint with Philippe Nadeau (CNRS and Univ. Lyon).\n
LOCATION:https://researchseminars.org/talk/Matroids/22/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Connor Simpson (University of Wisconsin Madison)
DTSTART;VALUE=DATE-TIME:20221110T200000Z
DTEND;VALUE=DATE-TIME:20221110T210000Z
DTSTAMP;VALUE=DATE-TIME:20240423T110231Z
UID:Matroids/26
DESCRIPTION:Title: Simplicial generators for Chow rings of matroids\nby Connor Simpson
(University of Wisconsin Madison) as part of Matroids - Combinatorics\, Al
gebra and Geometry Seminar\n\nLecture held in Room 210 The Fields Institut
e.\n\nAbstract\nThe simplicial generators of a Chow ring of a matroid are
divisors pulled back from a product of projective spaces. Multiplying them
can be interpreted combinatorially\, yielding a simple formula for the vo
lume polynomial of the Chow ring of a matroid. If time permits\, we will d
iscuss further developments these generators have inspired since their int
roduction. Joint work with Chris Eur & Spencer Backman.\n
LOCATION:https://researchseminars.org/talk/Matroids/26/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Jose Bastidas (Université du Québec à Montréal)
DTSTART;VALUE=DATE-TIME:20221115T200000Z
DTEND;VALUE=DATE-TIME:20221115T210000Z
DTSTAMP;VALUE=DATE-TIME:20240423T110231Z
UID:Matroids/27
DESCRIPTION:Title: Polytope algebra of generalized permutohedra\nby Jose Bastidas (Univ
ersité du Québec à Montréal) as part of Matroids - Combinatorics\, Alg
ebra and Geometry Seminar\n\nLecture held in Room 210 The Fields Institute
.\n\nAbstract\nDanilov-Koshevoy\, Postnikov\, and Ardila-Benedetti-Doker t
aught us that any generalized permutahedron is a signed Minkowski sum of t
he faces of the standard simplex. In other words\, these faces correspond
to a maximal linearly independent collection of rays in the deformation co
ne of the permutahedron. In contrast\, Ardila-Castillo-Eur-Postnikov obser
ved that the faces of the cross-polytope only span a subspace of roughly h
alf the dimension of the deformation cone of the type B permutahedron. In
this talk\, we use McMullen's polytope algebra to help explain this phenom
enon.\n
LOCATION:https://researchseminars.org/talk/Matroids/27/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Max Kutler (Ohio State University)
DTSTART;VALUE=DATE-TIME:20230117T200000Z
DTEND;VALUE=DATE-TIME:20230117T210000Z
DTSTAMP;VALUE=DATE-TIME:20240423T110231Z
UID:Matroids/28
DESCRIPTION:Title: Motivic Zeta functions of hyperplane arrangements\nby Max Kutler (Oh
io State University) as part of Matroids - Combinatorics\, Algebra and Geo
metry Seminar\n\nLecture held in Room 210 The Fields Institute.\n\nAbstrac
t\nWe associate to any matroid a motivic zeta function. If the matroid is
representable by a complex hyperplane arrangement\, then thiscoincides wit
h the motivic Igusa zeta function of the arrangement.Although the motivic
zeta function is a valuative invariant which is finer than the characteris
tic polynomial\, it is not obvious how one should extract meaningful combi
natorial data from the motivic zeta function. One strategy is to specializ
e to the topological zeta function. I will survey what is known about thes
e functions and discuss some open questions.\n
LOCATION:https://researchseminars.org/talk/Matroids/28/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Hunter Novak Spink (Stanford University)
DTSTART;VALUE=DATE-TIME:20230119T200000Z
DTEND;VALUE=DATE-TIME:20230119T210000Z
DTSTAMP;VALUE=DATE-TIME:20240423T110231Z
UID:Matroids/29
DESCRIPTION:Title: Log-concavity of matroid h-vectors and mixed Eulerian numbers\nby Hu
nter Novak Spink (Stanford University) as part of Matroids - Combinatorics
\, Algebra and Geometry Seminar\n\nLecture held in Room 210 The Fields Ins
titute.\n\nAbstract\nThe combinatorial Chow ring of a matroid produces log
-concave sequences from a list of polytopes called "generalized permutahed
ra". If we take S_n-invariant polytopes\, then we obtain matroid invariant
s\, but what invariants do we obtain? We will discuss how many such invari
ants are linear combinations of the h-vector of the independence complex o
f a matroid by "mixed Eulerian numbers"\, and how this proves a strengthen
ing of a conjecture of Dawson on the log-concavity of the matroid h-vector
.\n
LOCATION:https://researchseminars.org/talk/Matroids/29/
END:VEVENT
BEGIN:VEVENT
SUMMARY:June Huh (Princeton University)
DTSTART;VALUE=DATE-TIME:20230124T200000Z
DTEND;VALUE=DATE-TIME:20230124T210000Z
DTSTAMP;VALUE=DATE-TIME:20240423T110231Z
UID:Matroids/30
DESCRIPTION:Title: Stellahedral geometry of matroids\nby June Huh (Princeton University
) as part of Matroids - Combinatorics\, Algebra and Geometry Seminar\n\nLe
cture held in Room 210 The Fields Institute.\n\nAbstract\nThe 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 stella
hedral toric variety. Based on joint work with Chris Eur and Matt Larson\,
https://arxiv.org/abs/2207.10605.\n
LOCATION:https://researchseminars.org/talk/Matroids/30/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Diane Maclagan (University of Warwick)
DTSTART;VALUE=DATE-TIME:20230126T200000Z
DTEND;VALUE=DATE-TIME:20230126T210000Z
DTSTAMP;VALUE=DATE-TIME:20240423T110231Z
UID:Matroids/31
DESCRIPTION:Title: Tropical ideals\nby Diane Maclagan (University of Warwick) as part o
f Matroids - Combinatorics\, Algebra and Geometry Seminar\n\nLecture held
in Room 210 The Fields Institute.\n\nAbstract\nTropical ideals are the bui
lding blocks for the theory of tropical schemes\, which are part of a prog
ram to build foundations for tropical geometry. From a matroid perspectiv
e\, they are "towers" of (valuated) matroids: for each degree d we give a
(valuated) matroid on the set of monomials of degree d in a polynomial rin
g. I will introduce this theory\, and explain the relevance of better und
erstanding these matroids for tropical and algebraic geometry.\n
LOCATION:https://researchseminars.org/talk/Matroids/31/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Alex Fink (Queen Mary University of London)
DTSTART;VALUE=DATE-TIME:20230131T200000Z
DTEND;VALUE=DATE-TIME:20230131T210000Z
DTSTAMP;VALUE=DATE-TIME:20240423T110231Z
UID:Matroids/32
DESCRIPTION:Title: Matrix orbit closures and their Hilbert functions\nby Alex Fink (Que
en Mary University of London) as part of Matroids - Combinatorics\, Algebr
a and Geometry Seminar\n\nLecture held in Room 210 The Fields Institute.\n
\nAbstract\nIf an ordered point configuration in projective space is repre
sented by a matrix of coordinates\, the resulting matrix is determined up
to the action of the general linear group on one side and the torus of dia
gonal matrices on the other. We study orbits of matrices under the action
of the product of these groups\, as well as their images in quotients of t
he space of matrices like the Grassmannian. The main question is what prop
erties of closures of these orbits are determined by the matroid of the po
int configuration\; the main result is that their equivariant K-classes ar
e so determined. I will also draw connections to positivity and the work o
f Berget\, Eur\, Spink and Tseng.\n
LOCATION:https://researchseminars.org/talk/Matroids/32/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Chris Eur (Harvard University)
DTSTART;VALUE=DATE-TIME:20230202T200000Z
DTEND;VALUE=DATE-TIME:20230202T210000Z
DTSTAMP;VALUE=DATE-TIME:20240423T110231Z
UID:Matroids/33
DESCRIPTION:Title: Signed permutohedra and delta matroids\nby Chris Eur (Harvard Univer
sity) as part of Matroids - Combinatorics\, Algebra and Geometry Seminar\n
\nLecture held in Room 210 The Fields Institute.\n\nAbstract\nDelta-matroi
ds are type B generalizations of matroids in the theory of Coxeter matroid
s. We generalize many recent developments in the algebraic geometry of mat
roids to that of delta-matroids. These include the theory of tautological
classes of delta-matroids\, and formulas for volumes of type B generalized
permutohedra. In particular\, for a class of delta-matroids that includes
all graphical delta-matroids\, we show a log-concavity for "Tutte polynom
ials" of delta-matroids. Time permitting\, we'll indicate some future dire
ctions in the pursuit of a "Hodge theory" for Coxeter matroids.\n
LOCATION:https://researchseminars.org/talk/Matroids/33/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Petter Brändén (KTH Royal Institute of Technology)
DTSTART;VALUE=DATE-TIME:20230207T200000Z
DTEND;VALUE=DATE-TIME:20230207T210000Z
DTSTAMP;VALUE=DATE-TIME:20240423T110231Z
UID:Matroids/34
DESCRIPTION:Title: Lorentzian polynomials on cones (Part I).\nby Petter Brändén (KTH
Royal Institute of Technology) as part of Matroids - Combinatorics\, Algeb
ra and Geometry Seminar\n\nLecture held in Room 210 The Fields Institute.\
n\nAbstract\nWe show how the theory of Lorentzian polynomials extends to c
ones other than the positive orthant\, and how this may be used to prove H
odge-Riemann relations of degree one for Chow rings. Joint work with Jonat
han Leake.\n
LOCATION:https://researchseminars.org/talk/Matroids/34/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Michael Joswig (MPI MiS Leipzig)
DTSTART;VALUE=DATE-TIME:20230209T200000Z
DTEND;VALUE=DATE-TIME:20230209T210000Z
DTSTAMP;VALUE=DATE-TIME:20240423T110231Z
UID:Matroids/35
DESCRIPTION:Title: Generalized Permutahedra and Positive Flag Dressians\nby Michael Jos
wig (MPI MiS Leipzig) as part of Matroids - Combinatorics\, Algebra and Ge
ometry Seminar\n\nLecture held in Room 210 The Fields Institute.\n\nAbstra
ct\nWe study valuated matroids\, their tropical incidence relations\, flag
matroids and total positivity. This leads to a characterization of subdiv
isions of regular permutahedra into generalized permutahedra. Further\, we
get a characterization of those subdivisions arising from positive valuat
ed flag matroids.\n
LOCATION:https://researchseminars.org/talk/Matroids/35/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Matthieu Piquerez (University of Nantes)
DTSTART;VALUE=DATE-TIME:20230214T200000Z
DTEND;VALUE=DATE-TIME:20230214T210000Z
DTSTAMP;VALUE=DATE-TIME:20240423T110231Z
UID:Matroids/36
DESCRIPTION:Title: Tropical Hodge theory\nby Matthieu Piquerez (University of Nantes) a
s part of Matroids - Combinatorics\, Algebra and Geometry Seminar\n\nLectu
re held in Room 210 The Fields Institute.\n\nAbstract\nDuring the last dec
ade\, several important long-standing conjectures about matroids\, as the
Heron-Rota-Welsh conjecture\, has been solved thanks to the development of
the combinatorial Hodge theory by Huh and his collaborators. Classical Ho
dge theory is about the cohomology of complex varieties. For matroids repr
esentable over the complex field\, this theory applied to some complex var
ieties associated to the matroids implies the Heron-Rota-Welsh conjecture.
For a general matroid\, Adiprasito\, Huh and Katz achieved to develop a c
ombinatorial Hodge theory for (Chow rings of) matroids which works as if o
ne can associate a complex variety to the matroid\, though this is not the
case. The proof is very clever but does not give much insight into why th
is combinatorial Hodge theory works in general.\n\nActually\, every matroi
d is in some sense representable over the tropical hyperfield. Moreover\,
in a joint work with Amini\, we developed a tropical Hodge theory. Hence\,
to every matroid one can associate a tropical variety (the canonically co
mpactified Bergman fan)\, and the Hodge properties of this variety imply t
he Heron-Rota-Welsh conjecture. We thus get a geometric proof of the conje
cture\, as well as an extension of the applicability of the combinatorial
Hodge theory. The heart of our proof relies on a very interesting inductio
n\, based on the deletion-contraction induction."\n
LOCATION:https://researchseminars.org/talk/Matroids/36/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Benjamin Schroter (Goethe-Universität Frankfurt)
DTSTART;VALUE=DATE-TIME:20230216T200000Z
DTEND;VALUE=DATE-TIME:20230216T210000Z
DTSTAMP;VALUE=DATE-TIME:20240423T110231Z
UID:Matroids/37
DESCRIPTION:Title: Valuative invariants for large classes of matroids\nby Benjamin Schr
oter (Goethe-Universität Frankfurt) as part of Matroids - Combinatorics\,
Algebra and Geometry Seminar\n\nLecture held in Room 210 The Fields Insti
tute.\n\nAbstract\nValuations on polytopes are maps that combine the geome
try of polytopes with relations in a group. It turns out that many importa
nt invariants of matroids are valuative on the collection of matroid base
polytopes\, e.g.\, the Tutte polynomial and its specializations or the Hil
bert–Poincaré series of the Chow ring of a matroid.\n\nIn this talk I w
ill present a framework that allows us to compute such invariants on large
classes of matroids\, e.g.\, sparse paving and elementary split matroids\
, explicitly. The concept of split matroids introduced by Joswig and mysel
f is relatively new. However\, this class appears naturally in this contex
t. Moreover\, (sparse) paving matroids are split. I will demonstrate the f
ramework by looking at Ehrhart polynomials and further examples. \n\nThis
talk is based on the preprint `Valuative invariants for large classes of m
atroids' which is joint work with Luis Ferroni.\n
LOCATION:https://researchseminars.org/talk/Matroids/37/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Johannes Rau (Universidad de los Andes)
DTSTART;VALUE=DATE-TIME:20230228T190000Z
DTEND;VALUE=DATE-TIME:20230228T200000Z
DTSTAMP;VALUE=DATE-TIME:20240423T110231Z
UID:Matroids/38
DESCRIPTION:Title: The Lefschetz-Hopf trace formula for matroidal automorphisms\nby Joh
annes Rau (Universidad de los Andes) as part of Matroids - Combinatorics\,
Algebra and Geometry Seminar\n\nLecture held in Room 210 The Fields Insti
tute.\n\nAbstract\nThe Lefschetz-Hopf trace formula is a beautiful topolog
ical statement that relates the fixed points of a map to an alternating su
m of traces on homology groups. The related Poincaré-Hopf index formula c
omputes the Euler characteristic of a space in terms of the zeros of a vec
tor field. In my talk\, I want to present analogous statements in tropical
geometry\, in particular\, for matroid fans. In doing so\, we use two ing
redients that have received much attention over the last years: tropical h
omology on the homology side and tropical intersection theory on the fixed
point/vector field side. The two sides will be connected using a certain
variant of the beta invariant.\n
LOCATION:https://researchseminars.org/talk/Matroids/38/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Edvard Aksnes (University of Oslo)
DTSTART;VALUE=DATE-TIME:20230302T200000Z
DTEND;VALUE=DATE-TIME:20230302T210000Z
DTSTAMP;VALUE=DATE-TIME:20240423T110231Z
UID:Matroids/39
DESCRIPTION:Title: Tropical Poincaré duality spaces\nby Edvard Aksnes (University of O
slo) as part of Matroids - Combinatorics\, Algebra and Geometry Seminar\n\
nLecture held in Room 210 The Fields Institute.\n\nAbstract\nBergman fans
of matroids are the tropical equivalent of linear spaces. In the context o
f tropical homology\, such fans can be shown to satisfy tropical Poincaré
duality\, i.e. a duality between tropical homology and cohomology. Thanks
to work of Amini and Piquerez\, satisfying tropical Poincaré duality is
related to the combinatorial Hodge theory of the Matroid Chow ring of Adip
rasito-Huh-Katz. In this talk\, we will give some results on the problem o
f classifying which fans satisfy tropical Poincaré duality\, and give som
e perspectives on recent work relating tropical Poincaré duality and the
cohomology of subvarieties of tori.\n
LOCATION:https://researchseminars.org/talk/Matroids/39/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Kris Shaw (University of Oslo)
DTSTART;VALUE=DATE-TIME:20230307T200000Z
DTEND;VALUE=DATE-TIME:20230307T210000Z
DTSTAMP;VALUE=DATE-TIME:20240423T110231Z
UID:Matroids/40
DESCRIPTION:Title: The birational geometry of matroids\nby Kris Shaw (University of Osl
o) as part of Matroids - Combinatorics\, Algebra and Geometry Seminar\n\nL
ecture held in Room 210 The Fields Institute.\n\nAbstract\nIn this talk\,
I will consider isomorphisms of Bergman fans of matroids. Motivated by al
gebraic geometry\, such isomorphisms can be considered as matroid analogs
of birational maps. If the matroids in question are not totally disconnect
ed\, I will explain that an isomorphism respecting their fine fan structur
es must be induced by a matroid isomorphism. However\, if we switch to the
coarse fan structure\, this is no longer the case. I will introduce Cremo
na automorphisms of the coarse structure of certain Bergman fans. These pr
oduce a class of Bergman fan isomorphisms which are not induced by matroid
automorphisms. I will then explain that the automorphism group of the coa
rse fan structure is generated by matroid automorphisms and Cremona maps i
n the case of rank 3 matroids which are not parallel connections and for m
odularly complemented matroids. This is based on joint work with Annette W
erner.\n
LOCATION:https://researchseminars.org/talk/Matroids/40/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Jonathan Leake (University of Waterloo)
DTSTART;VALUE=DATE-TIME:20230309T200000Z
DTEND;VALUE=DATE-TIME:20230309T210000Z
DTSTAMP;VALUE=DATE-TIME:20240423T110231Z
UID:Matroids/41
DESCRIPTION:Title: Lorentzian polynomial on cones Part II\nby Jonathan Leake (Universit
y of Waterloo) as part of Matroids - Combinatorics\, Algebra and Geometry
Seminar\n\nLecture held in Room 210 The Fields Institute.\n\nAbstract\nIn
Part I\, Petter Brändén extended the theory of Lorentzian polynomials to
cones beyond the positive orthant\, and showed how this could be used to
prove Hodge-Riemann relations of degree one for Chow rings. In this talk\,
we will recap the main points of his talk\, and then we will apply the th
eory of Lorentzian polynomials on cones to some specific important example
s. Part I will not be required to understand this talk. Joint work with Pe
tter Brändén.\n
LOCATION:https://researchseminars.org/talk/Matroids/41/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Joseph Bonin (The George Washington University)
DTSTART;VALUE=DATE-TIME:20230321T190000Z
DTEND;VALUE=DATE-TIME:20230321T200000Z
DTSTAMP;VALUE=DATE-TIME:20240423T110231Z
UID:Matroids/42
DESCRIPTION:Title: An overview of recent developments on non-isomorphic matroids that have
the same G-invariant\nby Joseph Bonin (The George Washington Universit
y) as part of Matroids - Combinatorics\, Algebra and Geometry Seminar\n\nL
ecture held in Room 210 The Fields Institute.\n\nAbstract\nThe G-invariant
\, which was introduced by Derksen\, is a matroid invariant that contains
all of the data in the Tutte polynomial\, and far more. The theory of chr
omatic uniqueness and chromatic equivalence for graphs\, based on the chro
matic polynomial\, has been developed extensively\, and there are many res
ults for the analogous notions using the Tutte polynomial\, for both graph
s and matroids. The corresponding questions for the G-invariant are ripe
for exploration. This talk will survey recent results that yield non-isomo
rphic matroids that have the same G-invariant\, all aimed at shedding ligh
t on what minimal structure determines the G-invariant.\n
LOCATION:https://researchseminars.org/talk/Matroids/42/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Suho Oh (Texas State University)
DTSTART;VALUE=DATE-TIME:20230323T190000Z
DTEND;VALUE=DATE-TIME:20230323T200000Z
DTSTAMP;VALUE=DATE-TIME:20240423T110231Z
UID:Matroids/43
DESCRIPTION:Title: Extending Shellings\nby Suho Oh (Texas State University) as part of
Matroids - Combinatorics\, Algebra and Geometry Seminar\n\nLecture held in
Room 210 The Fields Institute.\n\nAbstract\nThe independent complex of a
matroid is a shellable simplicial complex: their facets can be ordered nic
ely\, which translates to interesting properties in algebra and combinator
ics. Simon in 1994 conjectured that any shellable complex can be extended
to the k-skeleton of a simplex while maintaining the shelling property. We
go over various tools and results related to this problem. In particular\
, we will be going over a recent joint work with M. Coleman\, A. Dochterma
nn and N. Geist on proving this conjecture for a smaller class\, which con
tains the entire class of matroids.\n
LOCATION:https://researchseminars.org/talk/Matroids/43/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Luis Ferroni (KTH Royal Institute of Technology)
DTSTART;VALUE=DATE-TIME:20230328T190000Z
DTEND;VALUE=DATE-TIME:20230328T200000Z
DTSTAMP;VALUE=DATE-TIME:20240423T110231Z
UID:Matroids/44
DESCRIPTION:Title: Ehrhart polynomials of matroid polytopes\nby Luis Ferroni (KTH Royal
Institute of Technology) as part of Matroids - Combinatorics\, Algebra an
d Geometry Seminar\n\nLecture held in Room 210 The Fields Institute.\n\nAb
stract\nA fundamental invariant associated to a lattice polytope is its Eh
rhart polynomial\, which encodes the number of lattice points inside all t
he integer dilations of the polytope and much more arithmetic\, algebraic
and combinatorial information. One may associate to any matroid two polyto
pes called respectively the base polytope and the independence polytope\;
both of these polytopes can be seen as part of the larger family of genera
lized permutohedra. A conjecture of De Loera\, Haws and Köppe asserted th
at the Ehrhart polynomials of base polytopes of matroids had positive coef
ficients only\; more generally\, Castillo and Liu conjectured this was tru
e for all generalized permutohedra (in particular\, also for independence
polytopes). We will show how to construct counterexamples to these conject
ures\; we will exhibit examples of matroids whose base and independence po
lytopes attain negative Ehrhart coefficients. On the positive side\, we wi
ll discuss about some families of matroids that satisfy Ehrhart positivity
. Several open problems regarding Ehrhart polynomials of matroids will be
stated.\n
LOCATION:https://researchseminars.org/talk/Matroids/44/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Spencer Backman (University of Vermont)
DTSTART;VALUE=DATE-TIME:20230330T190000Z
DTEND;VALUE=DATE-TIME:20230330T200000Z
DTSTAMP;VALUE=DATE-TIME:20240423T110231Z
UID:Matroids/45
DESCRIPTION:Title: Higher Categorical Associahedra\nby Spencer Backman (University of V
ermont) as part of Matroids - Combinatorics\, Algebra and Geometry Seminar
\n\nLecture held in Room 210 The Fields Institute.\n\nAbstract\nThe associ
ahedron is a well-known poset with connections to many different areas of
combinatorics\, algebra\, geometry\, topology\, and physics. The associahe
dron has several different realizations as the face poset of a convex poly
tope and one realization\, due to Loday\, is a generalized permutahedron\,
i.e. a polymatroid. From the perspective of symplectic geometry\, the ass
ociahedron encodes the combinatorics of morphisms in the Fukaya category o
f a symplectic manifold. In 2017\, Bottman introduced a family of posets c
alled 2-associahedra which encode the combinatorics of functors between Fu
kaya categories\, and he conjectured that they can be realized as the face
posets of convex polytopes. In this talk we will introduce categorical n-
associahedra as a natural extension of associahedra and 2-associahedra\, a
nd we will produce a family of complete polyhedral fans called velocity fa
ns whose face posets are the categorical n-associahedra. Categorical n-ass
ociahedra cannot be realized by generalized permutahedra or any of their k
nown extensions. On the other hand\, our velocity fan specializes to the n
ormal fan of Loday’s associahedron suggesting a new extension of general
ized permutahedra. This is joint work with Nathaniel Bottman and Daria Pol
iakova.\n
LOCATION:https://researchseminars.org/talk/Matroids/45/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Matt Larson (Stanford University)
DTSTART;VALUE=DATE-TIME:20230404T190000Z
DTEND;VALUE=DATE-TIME:20230404T200000Z
DTSTAMP;VALUE=DATE-TIME:20240423T110231Z
UID:Matroids/46
DESCRIPTION:Title: Geometry of polymatroids\nby Matt Larson (Stanford University) as pa
rt of Matroids - Combinatorics\, Algebra and Geometry Seminar\n\nLecture h
eld in Room 210 The Fields Institute.\n\nAbstract\nJust as matroids are co
mbinatorial abstractions of hyperplane arrangements\, polymatroids are com
binatorial abstractions of subspace arrangements. In recent years\, algebr
aic geometry has inspired many theorems about matroids. I will describe wo
rk establishing polymatroidal analogues of some of these results\, often b
y reducing to the case of matroids. Joint work with Colin Crowley\, Christ
opher Eur\, June Huh\, Connor Simpson\, and Botong Wang.\n
LOCATION:https://researchseminars.org/talk/Matroids/46/
END:VEVENT
BEGIN:VEVENT
SUMMARY:James Oxley (Louisiana State University)
DTSTART;VALUE=DATE-TIME:20230406T190000Z
DTEND;VALUE=DATE-TIME:20230406T200000Z
DTSTAMP;VALUE=DATE-TIME:20240423T110231Z
UID:Matroids/47
DESCRIPTION:Title: Graphs\, Matroids\, Cographs and Comatroids\nby James Oxley (Louisia
na State University) as part of Matroids - Combinatorics\, Algebra and Geo
metry Seminar\n\nLecture held in Room 210 The Fields Institute.\n\nAbstrac
t\n“If a theorem about graphs can be expressed in terms of edges and cir
cuits only it probably exemplifies a more general theorem about matroids.
” These words of Bill Tutte from 1979 have had a profound influence on r
esearch in matroid theory. This talk will discuss an example of recent wor
k that was motivated by Tutte’s guiding principle. The class of cographs
or complement- reducible graphs is the class of graphs that can be genera
ted from K1 using the operations of disjoint union and complementation. We
define 2-cographs to be the graphs we get by also allowing the operation
of 1-sum. By analogy\, we introduce the class of binary comatroids as the
class of matroids that can be generated from the empty matroid using the o
perations of direct sum and taking complements inside of binary projective
space. This talk will explore the properties of 2-cographs and binary com
atroids. The main results characterize these classes in terms of forbidden
induced minors. This is joint work with Jagdeep Singh.\n
LOCATION:https://researchseminars.org/talk/Matroids/47/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Federico Ardila (San Francisco State University)
DTSTART;VALUE=DATE-TIME:20230411T190000Z
DTEND;VALUE=DATE-TIME:20230411T200000Z
DTSTAMP;VALUE=DATE-TIME:20240423T110231Z
UID:Matroids/48
DESCRIPTION:Title: Combinatorial Intersection Theory: A Few Examples\nby Federico Ardil
a (San Francisco State University) as part of Matroids - Combinatorics\, A
lgebra and Geometry Seminar\n\nLecture held in Room 210 The Fields Institu
te.\n\nAbstract\nIntersection theory studies how subvarieties of an algebr
aic variety X intersect. Algebraically\, this information is encoded in th
e Chow ring A(X). When X is the toric variety of a simplicial fan\, Brion
gave a presentation of A(X) in terms of generators and relations\, and Ful
ton and Sturmfels gave a ""fan displacement rule” to intersect classes i
n A(X)\, which holds more generally in tropical intersection theory. In th
ese settings\, intersection theoretic questions translate to algebraic com
binatorial computations in one point of view\, or to polyhedral combinator
ial questions in the other. Both of these paths lead to interesting combin
atorial problems\, and in some cases\, they are important ingredients in t
he proofs of long-standing conjectural inequalities.\n\nThis talk will sur
vey a few problems on matroids and root systems that arise in combinatoria
l intersection theory. It will feature joint work with Montse Cordero\, Gr
aham Denham\, Chris Eur\, June Huh\, Carly Klivans\, and Raúl Penaguião.
The talk will not assume previous knowledge of the words in the abstract.
\n
LOCATION:https://researchseminars.org/talk/Matroids/48/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Cynthia Vinzant (University of Washington)
DTSTART;VALUE=DATE-TIME:20230413T190000Z
DTEND;VALUE=DATE-TIME:20230413T200000Z
DTSTAMP;VALUE=DATE-TIME:20240423T110231Z
UID:Matroids/49
DESCRIPTION:Title: Positively Hyperbolic Varieties\, Tropicalization\, and Positroids\n
by Cynthia Vinzant (University of Washington) as part of Matroids - Combin
atorics\, Algebra and Geometry Seminar\n\nLecture held in Room 210 The Fie
lds Institute.\n\nAbstract\nHyperbolic varieties are a generalization of r
eal-rooted polynomials for varieties of codimension more than one. One pro
minent example is the image of linear space under coordinate-wise inversio
n. I will discuss the combinatorial structure of varieties that are hyperb
olic with respect to the nonnegative Grassmannian\, which is intimately re
lated with the theory of positroids. These varieties generalize multivaria
te stable polynomials and their tropicalizations are locally subfans of th
e type-A hyperplane arrangement\, in which the maximal cones satisfy a non
-crossing condition. This is based on joint work with Felipe Rincón and J
osephine Yu.\n
LOCATION:https://researchseminars.org/talk/Matroids/49/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Joseph Kung (University of North Texas)
DTSTART;VALUE=DATE-TIME:20230418T190000Z
DTEND;VALUE=DATE-TIME:20230418T200000Z
DTSTAMP;VALUE=DATE-TIME:20240423T110231Z
UID:Matroids/50
DESCRIPTION:Title: Tutte polynomial evaluations which are exponential sums.\nby Joseph
Kung (University of North Texas) as part of Matroids - Combinatorics\, Alg
ebra and Geometry Seminar\n\nLecture held in Room 210 The Fields Institute
.\n\nAbstract\nAn exponential sum is a sum $\\sum_{I=0}^{m-1} a_i \\omega^
I$\, where $\\omega$ is a primitive $m$th root of unity. We will show se
veral examples of Tutte polynomial evaluations which are exponential sums.
In particular\, for a matroid $G$ representable over a finite field of o
rder $q$\, then the evaluation $q^{r(M)} \\chi (G\;q)$\, where $\\chi$ is
the characteristic polynomial\, can be written as an exponential sum in
which the coefficients $a_i$ have an enumerative interpretation.\n
LOCATION:https://researchseminars.org/talk/Matroids/50/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Christin Bibby (Louisiana State University)
DTSTART;VALUE=DATE-TIME:20230420T190000Z
DTEND;VALUE=DATE-TIME:20230420T200000Z
DTSTAMP;VALUE=DATE-TIME:20240423T110231Z
UID:Matroids/51
DESCRIPTION:Title: Matroid schemes and geometric posets\nby Christin Bibby (Louisiana S
tate University) as part of Matroids - Combinatorics\, Algebra and Geometr
y Seminar\n\nLecture held in Room 210 The Fields Institute.\n\nAbstract\nT
he intersection data of an arrangement of hyperplanes is described by a ge
ometric lattice\, or equivalently a simple matroid. There is a rich interp
lay between this combinatorial structure and the topology of the arrangeme
nt complement. In this talk\, we will similarly characterize the combinato
rial structure underlying certain arrangements of subvarieties by defining
a class of geometric posets and a generalization of matroids called matro
id schemes. We will discuss some notions from matroid theory which extend
to this setting and touch on the topological implications of this framewor
k.\n
LOCATION:https://researchseminars.org/talk/Matroids/51/
END:VEVENT
END:VCALENDAR