109 research outputs found
Categorical Invariants of Graphs and Matroids
Graphs and matroids are two of the most important objects in combinatorics.We study invariants of graphs and matroids that behave well with respect to
certain morphisms by realizing these invariants as functors from a category of
graphs (resp. matroids).
For graphs, we study invariants that respect deletions and contractions ofedges. For an integer , we define a category of of graphs of genus at most
g where morphisms correspond to deletions and contractions. We prove that this
category is locally Noetherian and show that many graph invariants form finitely
generated modules over the category . This fact allows us to exihibit many
stabilization properties of these invariants. In particular we show that the torsion
that can occur in the homologies of the unordered configuration space of n points
in a graph and the matching complex of a graph are uniform over the entire family
of graphs with genus .
For matroids, we study valuative invariants of matroids. Given a matroid,one can define a corresponding polytope called the base polytope. Often, the base
polytope of a matroid can be decomposed into a cell complex made up of base
polytopes of other matroids. A valuative invariant of matroids is an invariant that
respects these polytope decompositions. We define a category of matroids
whose morphisms correspond to containment of base polytopes. We then define the
notion of a categorical matroid invariant which categorifies the notion of a valuative
invariant. Finally, we prove that the functor sending a matroid to its Orlik-Solomon
algebra is a categorical valuative invariant. This allows us to derive relations among
the Orlik-Solomon algebras of a matroid and matroids that decompose its base
polytope viewed as representations of any group whose action is compatible with
the polytope decomposition.
This dissertation includes previously unpublished co-authored material
LIPIcs, Volume 251, ITCS 2023, Complete Volume
LIPIcs, Volume 251, ITCS 2023, Complete Volum
IQP Sampling and Verifiable Quantum Advantage: Stabilizer Scheme and Classical Security
Sampling problems demonstrating beyond classical computing power with noisy
intermediate-scale quantum (NISQ) devices have been experimentally realized. In
those realizations, however, our trust that the quantum devices faithfully
solve the claimed sampling problems is usually limited to simulations of
smaller-scale instances and is, therefore, indirect. The problem of verifiable
quantum advantage aims to resolve this critical issue and provides us with
greater confidence in a claimed advantage. Instantaneous quantum
polynomial-time (IQP) sampling has been proposed to achieve beyond classical
capabilities with a verifiable scheme based on quadratic-residue codes (QRC).
Unfortunately, this verification scheme was recently broken by an attack
proposed by Kahanamoku-Meyer. In this work, we revive IQP-based verifiable
quantum advantage by making two major contributions. Firstly, we introduce a
family of IQP sampling protocols called the \emph{stabilizer scheme}, which
builds on results linking IQP circuits, the stabilizer formalism, coding
theory, and an efficient characterization of IQP circuit correlation functions.
This construction extends the scope of existing IQP-based schemes while
maintaining their simplicity and verifiability. Secondly, we introduce the
\emph{Hidden Structured Code} (HSC) problem as a well-defined mathematical
challenge that underlies the stabilizer scheme. To assess classical security,
we explore a class of attacks based on secret extraction, including the
Kahanamoku-Meyer's attack as a special case. We provide evidence of the
security of the stabilizer scheme, assuming the hardness of the HSC problem. We
also point out that the vulnerability observed in the original QRC scheme is
primarily attributed to inappropriate parameter choices, which can be naturally
rectified with proper parameter settings.Comment: 22 pages, 3 figure
Induced log-concavity of equivariant matroid invariants
Inspired by the notion of equivariant log-concavity, we introduce the concept
of induced log-concavity for a sequence of representations of a finite group.
For an equivariant matroid equipped with a symmetric group action or a finite
general linear group action, we transform the problem of proving the induced
log-concavity of matroid invariants to that of proving the Schur positivity of
symmetric functions. We prove the induced log-concavity of the equivariant
Kazhdan-Lusztig polynomials of -niform matroids equipped with the action of
a finite general linear group, as well as that of the equivariant
Kazhdan-Lusztig polynomials of uniform matroids equipped with the action of a
symmetric group.
As a consequence of the former, we obtain the log-concavity of
Kazhdan-Lusztig polynomials of -niform matroids, thus providing further
positive evidence for Elias, Proudfoot and Wakefield's log-concavity conjecture
on the matroid Kazhdan-Lusztig polynomials. From the latter we obtain the
log-concavity of Kazhdan-Lusztig polynomials of uniform matroids, which was
recently proved by Xie and Zhang by using a computer algebra approach. We also
establish the induced log-concavity of the equivariant characteristic
polynomials and the equivariant inverse Kazhdan-Lusztig polynomials for
-niform matroids and uniform matroids.Comment: 36 page
Generating Polynomials of Exponential Random Graphs
The theory of random graphs describes the interplay between probability and graph theory: it is the study of the stochastic process by which graphs form and evolve. In 1959, ErdoÌs and ReÌnyi defined the foundational model of random graphs on n vertices, denoted G(n, p) ([ER84]). Subsequently, Frank and Strauss (1986) added a Markov twist to this story by describing a topological structure on random graphs that encodes dependencies between local pairs of vertices ([FS86]). The general model that describes this framework is called the exponential random graph model (ERGM).
In the past, determining when a probability distribution has strong negative dependence has proven to be difficult ([Pem00, BBL09]). The negative dependence of a probability distribution is characterized by properties of its corresponding generating polynomial ([BBL09]). This thesis bridges the theory of exponential random graphs with the geometry of their generating polynomials, namely, when and how they satisfy the stable or Lorentzian properties ([Wag09, BBL09, BH20, AGV21]). We provide necessary and sufficient conditions as well as full characterizations of the parameter space for when this model has a stable or Lorentzian generating polynomial. This is done using a well-developed dictionary between probability distributions and their corresponding multiaffine generating polynomials.
In particular, we characterize when the generating polynomial of a random graph model with a large symmetry group is irreducible. We assert that the edge parameter of the exponential random graph model does not affect stability and that the triangle and k-star parameters are necessarily related if the model is stable or Lorentzian. We also provide full Lorentzian and stable characterizations for the model on K3 and a Lorentzian characterization for specializations of the model on K4
Computation and Physics in Algebraic Geometry
Physics provides new, tantalizing problems that we solve by developing and implementing innovative and effective geometric tools in nonlinear algebra. The techniques we employ also rely on numerical and symbolic computations performed with computer algebra.
First, we study solutions to the Kadomtsev-Petviashvili equation that arise from singular curves. The Kadomtsev-Petviashvili equation is a partial differential equation describing nonlinear wave motion whose solutions can be built from an algebraic curve. Such a surprising connection established by Krichever and Shiota also led to an entirely new point of view on a classical problem in algebraic geometry known as the Schottky problem. To explore the connection with curves with at worst nodal singularities, we define the Hirota variety, which parameterizes KP solutions arising from such curves. Studying the geometry of the Hirota variety provides a new approach to the Schottky problem. We investigate it for irreducible rational nodal curves, giving a partial solution to the weak Schottky problem in this case.
Second, we formulate questions from scattering amplitudes in a broader context using very affine varieties and D-module theory. The interplay between geometry and combinatorics in particle physics indeed suggests an underlying, coherent mathematical structure behind the study of particle interactions. In this thesis, we gain a better understanding of mathematical objects, such as moduli spaces of point configurations and generalized Euler integrals, for which particle physics provides concrete, non-trivial examples, and we prove some conjectures stated in the physics literature.
Finally, we study linear spaces of symmetric matrices, addressing questions motivated by algebraic statistics, optimization, and enumerative geometry. This includes giving explicit formulas for the maximum likelihood degree and studying tangency problems for quadric surfaces in projective space from the point of view of real algebraic geometry
LIPIcs, Volume 261, ICALP 2023, Complete Volume
LIPIcs, Volume 261, ICALP 2023, Complete Volum
LIPIcs, Volume 274, ESA 2023, Complete Volume
LIPIcs, Volume 274, ESA 2023, Complete Volum
Graph Coverings with Few Eigenvalues or No Short Cycles
This thesis addresses the extent of the covering graph construction. How much must a cover X resemble the graph Y that it covers? How much can X deviate from Y? The main statistics of X and Y which we will measure are their regularity, the spectra of their adjacency matrices, and the length of their shortest cycles. These statistics are highly interdependent and the main contribution of this thesis is to advance our understanding of this interdependence. We will see theorems that characterize the regularity of certain covering graphs in terms of the number of distinct eigenvalues of their adjacency matrices. We will see old examples of covers whose lack of short cycles is equivalent to the concentration of their spectra on few points, and new examples that indicate certain limits to this equivalence in a more general setting. We will see connections to many combinatorial objects such as regular maps, symmetric and divisible designs, equiangular lines, distance-regular graphs, perfect codes, and more. Our main tools will come from algebraic graph theory and representation theory. Additional motivation will come from topological graph theory, finite geometry, and algebraic topology
What is in# P and what is not?
For several classical nonnegative integer functions, we investigate if they
are members of the counting complexity class #P or not. We prove #P membership
in surprising cases, and in other cases we prove non-membership, relying on
standard complexity assumptions or on oracle separations.
We initiate the study of the polynomial closure properties of #P on affine
varieties, i.e., if all problem instances satisfy algebraic constraints. This
is directly linked to classical combinatorial proofs of algebraic identities
and inequalities. We investigate #TFNP and obtain oracle separations that prove
the strict inclusion of #P in all standard syntactic subclasses of #TFNP-1
- âŠ