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Veranstaltungen
MEGA 2024

MPI für Mathematik in den Naturwissenschaften Leipzig (Leipzig) E1 05 (Leibniz-Saal)
Universität Leipzig (Leipzig) Felix-Klein-Hörsaal

conference

29.07.24 02.08.24

MEGA 2024

MEGA is the acronym for Effective Methods in Algebraic Geometry (and its equivalent in Italian, French, Spanish, German, Russian, etc.). This series of biennial international conferences, with the tradition dating back to 1990, is devoted to computational and application aspects of Algebraic Geometry and related topics, over any characteristics.

MEGA conferences have emerged from the leading research activity in Effective Algebraic Geometry developed in Europe over the last decades. They have established themselves as an important vehicle in the exploration of the broad interactions between Algebraic Geometry and Computational and Applied domains of Mathematics. The conferences have also increasingly been attracting young talented researchers, showing the strength and vitality of the community. The MEGA conferences aim at promoting the development of the subject through academic exchange between International researchers; at presenting the recent progresses of the domain through the presentation of selected talks; at updating the community knowledge through the invitation of talented researchers from Europe and outside; and at attracting and supporting young students and researchers to join the current research area.

Furthermore, please note that a satellite event will take place on Saturday, August 3rd, 2024, to celebrate Bernard Mourrain's 60th Birthday.

For all submission guidelines, click here. Note: The general registration is already closed. For late registrations only for speakers, poster presenters, and committee members, please send an email to the administrative contact.

See a list of accepted talks.

See a list of accepted posters.

Guidelines for poster presenters: Poster presenters are kindly asked to print their own poster and bring it to the session on Tuesday, July 30. The posters should be in A0 or A1 format.

Accommodation: We have blocked a limited number of hotel rooms for participants. You will receive further information about the hotel and the booking procedure in your registration confirmation.

See important practical information.

The registration in person is already closed, but you can still register for following the talks via Zoom.

Speakers

Francis Brown

Oxford University

Türkü Özlüm Çelik

Max Planck Institute of Molecular Cell Biology and Genetics

Carlos D'Andrea

Universitat de Barcelona

Daniel Erman

University of Wisconsin-Madison

Fulvio Gesmundo

Institut de Mathématiques de Toulouse

Victor Magron

CNRS LAAS

Fatemeh Mohammadi

KU Leuven

Anna Wienhard

Max Planck Institute for Mathematics in the Sciences

Josephine Yu

Georgia Institute of Technology

Program

08:45 - 09:00	Announcements
09:00 - 10:00	Carlos D'Andrea Toric Euler-Jacobi vanishing theorem and zeros at infinity Residues in complex manifolds are one of those kind of objects that can be approached both geometrically and algebraically. Their use in computational algebraic geometry is ubiquitous, they appear naturally in interpolation, duality, and several types of representation problems. In this talk, we will review them and their use, and then focus on some vanishing theorems for residues that are well-known classically, and have been extended recently to more general varieties. I will report on joint work with Alicia Dickenstein on the use of residues as witnesses of zeroes at the infinity of toric varieties. Slides
10:00 - 10:30	Coffee Break
	Parallel Sessions: Parallel Sessions at MPI MiS and Leipzig University (Felix-Klein-Hörsaal)
10:30 - 11:00	MPI MiS, Leibniz-Saal: Elias Tsigaridas Condition-based Low-Degree Approximation of Real Polynomial Systems. I: The Zero-Dimensional Case
	Leipzig University, Felix-Klein-Hörsaal: Julian Vill Efficiently deciding if an ideal is toric after a linear coordinate change We propose an effective algorithm that decides if a prime ideal can be made toric by a linear automorphism of the ambient space. If this is the case, the algorithm computes such a transformation explicitly. The algorithm can compute that all Gaussian graphical models on five vertices that are not toric from the start cannot be made toric by any linear change of coordinates. The same holds for all conditional independence ideals of undirected Gaussian graphical models on six vertices. Slides
11:00 - 11:30	MPI MiS, Leibniz-Saal: Sebastian Debus Stable and Hurwitz slices, a degree principle and a generalized Grace-Walsh-Szegő theorem We consider affine varieties defined by multivariate complex symmetric polynomials and address the question whether such a variety contains a point with only few distinct real and imaginary coordinates. We present a degree principle for half-planes and a weak generalization of Grace-Walsh-Szegő’s coincidence theorem to symmetric polynomials that are not necessarily multiaffine but which can be expressed in few elementary symmetric polynomials. The results are derived from studying linear slices of the set of univariate stable polynomials with respect to a closed half-plane using Vieta’s formulas. Similarly, we study affine slices of the set of weakly Hurwitz polynomials. This is joint work with Cordian Riener and Robin Schabert Slides
	Leipzig University, Felix-Klein-Hörsaal: Angel Rios A toric degeneration of hyper-Kähler varieties
11:30 - 12:00	MPI MiS, Leibniz-Saal: Momonari Kudo The solving degree for computing Groebner bases of affine semi-regular polynomial sequences Slides
	Leipzig University, Felix-Klein-Hörsaal: Vincenzo Galgano Equivariant Euler characteristics on permutohedral varieties In the literature cohomology has been shown to be a useful tool for detecting invariants in combinatorics. Moreover, equivariant cohomology provides even finer invariants, and it becomes particularly practical for toric varieties. Binomial coefficients can be interpreted as a solution to an intersection problem on a permutohedral variety X. Via Hirzebruch-Riemann-Roch, this is equivalent to computing Euler characteristic of a specific element of K-theory of X which admits a natural lifting to equivariant K-theory. Thus the Euler characteristic (and hence binomial coefficients) may be upgraded to a Laurent polynomial. We provide and implement three different approaches, in particular a recursive one, to computing these polynomials. This is a joint work with Hanieh Keneshlou and Mateusz Michałek. Slides
12:00 - 14:00	Lunch
14:00 - 15:00	Josephine Yu Tropical Positivity and Symmetric Low Rank Matrices In 2003, Speyer and Williams introduced two definitions for positive tropicalizations of algebraic varieties over complex Puiseux series and proved their equivalence for Grassmannians. The two corresponding notions of positive tropical generators were studied by Brandenburg, Loho, and Sinn in 2022. In this talk, I will provide a brief introduction to tropical geometry and then present a combinatorial description of the tropicalization of the space of symmetric rank two matrices, along with their real and positive parts. We will compare the two notions of tropical positivity for rank 2 or corank 1 symmetric matrices. The results are based on joint works with Abeer Al Ahmadieh, May Cai, and Kisun Lee. Slides
15:00 - 15:30	Coffee Break
	Software Session: Software Session parallel at MPI MiS and Leipzig University (Felix-Klein-Hörsaal)
15:30 - 16:00	MPI MiS, Leibniz-Saal: Gianni Petrella Decomposing quiver moduli - a QuiverTools showcase We consider the moduli space of stable representations of a quiver and find a partial analogue of the semiorthogonal decomposition of the moduli space of vector bundles on curves. More precisely, we embed several copies of the derived category of a quiver and certain line bundles in the derived category of an associated moduli space of representations, and show by explicit computations their semiorthogonality. This gives the start of a semiorthogonal decomposition. Our results are obtained with QuiverTools, an open-source package of tools for quiver representations, their moduli spaces and their geometrical properties.
	Leipzig University, Felix-Klein-Hörsaal: Elima Shehu SAGBI and Gröbner Bases Detection In this talk, we will introduce the detection algorithm for SAGBI bases in polynomial rings, which is analogous to the Gröbner basis detection algorithm previously proposed by Gritzmann and Sturmfels. We will explore the software package SagbiGbDetection, and demonstrate its functionality with concrete examples. This package enables users to identify term orders for which a given set of input polynomials forms either a Gröbner basis for the ideal they generate or a SAGBI basis for the subalgebra.
16:00 - 16:30	MPI MiS, Leibniz-Saal: Leonie Kayser Mukai lifting of self-dual points in P^6 A set of 2n points in ℙ^{n-1} is self-dual if it is invariant under the Gale transform. Motivated by Mukai's work on canonical curves, Petrakiev showed that a general self-dual set of 14 points in ℙ^6 arises as the intersection of the Grassmannian Gr(2,6) in its Plücker embedding in ℙ^14 with a linear space of dimension 6. In this work we focus on the inverse problem of recovering such a linear space associated to a general self-dual set of points. We use numerical homotopy continuation to approach the problem and implement an algorithm in Julia to solve it. Along the way we also implement the forward problem of slicing Grassmannians and use it to experimentally study the real solutions to this problem. The presentation features a brief introduction to the topic as well as a demonstration of the implementation. This is joint work with Barbara Betti.
	Leipzig University, Felix-Klein-Hörsaal: Tyler Jarvis Chebyshev Subdivision and Reduction Methods for Solving Multivariable Systems of Equations I present a new algorithm and Python implementation, YRoots, for finding isolated zeros of a system of real-valued smooth (not necessarily polynomial) functions in a bounded interval in R^n. It uses the Chebyshev proxy method combined with a mixture of subdivision, reduction methods, and elimination checks that leverage special properties of Chebyshev polynomials. The method has quadratic convergence locally near simple zeros of the system. It also finds all nonsimple zeros, but convergence to those zeros is not guaranteed to be quadratic. Numerical evidence demonstrates that the method is both fast and accurate on a wide range of problems, significantly outperforming other standard methods on the problem of finding all real zeros in a bounded domain. See paper at arxiv.org. See the code on Github. Slides arxiv.org Github
16:30 - 17:00	MPI MiS, Leibniz-Saal: Francesco Nowell The GroebnerWalk package in OSCAR Abstract Slides
	Leipzig University, Felix-Klein-Hörsaal: Gabriel Ong A1-Brouwer Degrees in Macaulay2 The A^1-Brouwer degree can be thought of as an analogue of the topological Brouwer degree valued in the Grothendieck-Witt ring of symmetric bilinear forms. These A^1-degrees allow for the generalization of algebro-geometric invariants such as intersection numbers and Milnor numbers over arbitrary fields. We present the Macaulay2 package 'A1BrouwerDegrees' for computing local and global A^1-Brouwer degrees as well as for studying symmetric bilinear forms over fields. see paper at arxiv.org Slides
17:30 - 20:00	Board Meeting in G3 10, MPI MiS

09:00 - 10:00	Türkü Özlüm Çelik Algebraic Geometry and Complex Systems Algebraic geometry has revealed complex structures among fundamental natural phenomena, such as solitary waves. The Korteweg-de Vries (KdV) equation is a prominent nonlinear PDE modelling these waves, with solutions encoded by algebro-geometric objects such as -maximal real- algebraic curves and -totally positive- Grassmannians. The KdV equation transforms into the modified KdV (mKdV) equation, linking the dynamics of the underlying algebraic curve with its curvature. The mKdV equation also finds applications in DNA modelling. We will explore these relationships, along with the mathematical questions they raise, and showcase the computational machinery we have developed to advance this research. Our tools include computer algebra algorithms designed to study geometric data.
10:00 - 10:30	Coffee Break
	Parallel Sessions: Parallel Sessions at MPI MiS and Leipzig University (Felix-Klein-Hörsaal)
10:30 - 11:00	MPI MiS, Leibniz-Saal: Ettore Teixeira Turatti Generalized identifiability of sums of squares
	Leipzig University, Felix-Klein-Hörsaal: Benjamin Biaggi Border subrank via a generalised Hilbert-Mumford criterion The subrank of a bilinear map is the maximal number of independent scalar multiplications that can be linearly reduced to the bilinear map. Given a sufficiently general complex nxnxn tensor, we give an upper bound on the growth rate for the border subrank. Since this matches the growth rate for the generic (non-border) subrank recently established by Derksen-Makam-Zuiddam, we find that the generic border subrank has the same growth rate. In our proof, we use a generalisation of the Hilbert-Mumford criterion that we believe will be of independent interest. This talk is based on joint work with Chia-Yu Chang, Jan Draisma and Filip Rupniewski. Slides
11:00 - 11:30	MPI MiS, Leibniz-Saal: Thorsten Theobald Symmetric SAGE and SONC forms, exactness and quantitative gaps The classes of sums of arithmetic-geometric exponentials (SAGE) and of sums of nonnegative circuit polynomials (SONC) provide nonnegativity certificates which are based on the inequality of the arithmetic and geometric means. We study the cones of symmetric SAGE and SONC forms and their relations to the underlying symmetric nonnegative cone. As main results, we provide several symmetric cases where the SAGE or SONC property coincides with nonnegativity and we present quantitative results on the differences in various situations. The results rely on characterizations of the zeroes and the minimizers for symmetric SAGE and SONC forms, which we develop. Finally, we also study symmetric monomial mean inequalities and apply SONC certificates to establish a generalized version of Muirhead's inequality. Slides
	Leipzig University, Felix-Klein-Hörsaal: Francesco Galuppi Rank and symmetries of signature tensors Since decades, signature tensors are useful tools in stochastic analysis. Recently algebraic geometers began to study signatures as an interesting family of tensors in their own right. We study their algebraic properties, such as thir rank, their symmetry and their conciseness, and we outline how these results can be used in practise by data analysts.
11:30 - 12:00	MPI MiS, Leibniz-Saal: Lorenzo Baldi Effective Positivstellensatz over the Rational Numbers for Finite Semialgebraic Sets We study the problem of representing multivariate polynomials with rational coefficients, which are non-negative and strictly positive on semialgebraic sets, using rational sums of squares. Contrary to the classical Nullstellensatz, the worst case complexity analysis of these representation results, called \emph{effective Positivstellensatz}, is still for the most part an open area of research. We obtain existential results and degree bounds for such representations for finite semialgebraic sets, and apply them in the context of polynomial optimization. We retrieve results of Lasserre, Laurent and Parrilo on finite convergence of the moment-sums of squares hierarchy, showing furthermore the existence of sums of squares certificates of optimality over the rational numbers.
	Leipzig University, Felix-Klein-Hörsaal: Pablo Mazón Construction of birational trilinear volumes via tensor rank criteria Abstract Slides
12:00 - 14:00	Lunch
14:00 - 15:00	Victor Magron State polynomial optimization for nonlinear Bell inequalities This talk focuses on optimization over state polynomials, i.e., polynomials in noncommuting variables and formal states of their products. An archimedean Positivstellensatz in the spirit of Putinar and Helton-McCullough is presented leading to a hierarchy of semidefinite relaxations converging monotonically to the optimum of a state polynomial subject to state constraints. This hierarchy can be seen as a state analog of the Lasserre hierarchy for optimization of polynomials, and the Navascués-Pironio-Acín scheme for optimization of noncommutative polynomials. The motivation behind this theory arises from the study of correlations in quantum networks. Determining the maximal quantum violation of a polynomial Bell inequality for an arbitrary network is reformulated as a state polynomial optimization problem. Several examples of quadratic Bell inequalities in the bipartite scenario are analyzed. To reduce the size of the semidefinite programs, sparsity, sign symmetry and conditional expectation of the observables' group structure are exploited. This is based on the joint work with Igor Klep, Jurij Volčič, and Jie Wang. see paper at arxiv.org Slides
15:00 - 15:30	Coffee Break
	Parallel Sessions: Parallel Sessions at MPI MiS and Leipzig University (Felix-Klein-Hörsaal)
15:30 - 16:00	MPI MiS, Leibniz-Saal: Andreas Kretschmer Symmetric Ideals, Invariant Hilbert Schemes and Stabilization Symmetric ideals are ideals in a polynomial ring which are stable under all permutations of the variables. They appear at prominent places in algebraic combinatorics and asymptotic commutative algebra. A global study of zero-dimensional symmetric ideals is the study of the invariant Hilbert scheme for the action of the n-th symmetric group on affine n-space. In this talk I will introduce invariant Hilbert schemes in this setting and explain our irreducibility and smoothness results for certain cases. We have also classified all homogeneous symmetric ideals giving rise to a representation of dimension at most 2n and we can decide which of them define singular points of the invariant Hilbert scheme. I will end with several open questions. This is joint work with Sebastian Debus. Slides
	Leipzig University, Felix-Klein-Hörsaal: Elizabeth Pratt The Chow-Lam Form The classical Chow form encodes any projective variety by a single equation. In this talk, we will introduce the Chow-Lam form, which is a generalization of the Chow form to subvarieties of arbitrary Grassmannians. Like the Chow form, it has useful computational properties: for example, it gives us universal formulas for certain linear projections from one Grassmannian to another. Such formulas were pioneered by Thomas Lam for positroid varieties in the study of amplituhedra. This is joint work with Bernd Sturmfels.
16:00 - 16:30	MPI MiS, Leibniz-Saal: Mahrud Sayrafi Computing Direct Sum Decompositions The problems of finding isomorphism classes of indecomposable modules with certain properties, or determining the indecomposable summands of a module, are ubiquitous in commutative algebra, group theory, representation theory, and other fields. The purpose of this work is to describe and prove correctness of a practical algorithm for computing indecomposable summands of finitely generated modules over a finitely generated k-algebra, for k a field of positive characteristic. Our algorithm works over multigraded rings, which enables the computation of indecomposable summands of coherent sheaves on subvarieties of toric varieties (in particular, for varieties embedded in projective space). We also present multiple examples, including some which present previously unknown phenomena regarding the behavior of summands of Frobenius pushforwards and syzygies over Artinian rings. This is joint work with Devlin Mallory.
	Leipzig University, Felix-Klein-Hörsaal: Erin Connelly Geometric Interpretations of Compatibility for Fundamental Matrices Slides
16:30 - 17:00	MPI MiS, Leibniz-Saal: Mario González-Sánchez Projective Cohen-Macaulay monomial curves and their affine projections Abstract Slides
	Leipzig University, Felix-Klein-Hörsaal: Tommi Muller Algebraic Constraints on Common Lines in Cryo-EM We revisit the topic of common lines between projection images in single particle cryo-electron microscopy (cryo-EM). We derive a novel low-rank constraint on a certain 2𝑛 × 𝑛 matrix storing properly-scaled basis vectors for the common lines between 𝑛 projection images of one olecular conformation. Using this algebraic constraint and others, we give optimization algorithms to denoise common lines and recover the unknown 3D rotations associated to the images. As an application, we develop a clustering algorithm to partition a set of noisy images into homogeneous communities using common lines, in the case of discrete heterogeneity in cryo-EM.
17:30 - 20:00	Poster Session in University of Leipzig, Paulinum

09:00 - 10:00	Francis Brown 'Positive geometries’, canonical logarithmic forms and Hodge theory I will report on joint work with Clément Dupont, in which we define a notion of the genus of a pair (X,Y) of complex algebraic varieties, where Y is contained in X. Using this concept, we show that under some very general assumptions on X and Y, we can associate a canonical logarithmic differential form to singular chains on X whose boundary is contained in Y. This construction has many properties, including a recursive structure with respect to taking residues of forms and boundaries of chains. This talk will first review the concept of a positive geometry in physics, before covering elements of Hodge theory and the theory of logarithmic differential forms. Then I will explain the above construction which, strangely, maps homology to differential forms. The talk will be illustrated with a large number of examples, and end with some applications in the theory of periods.
10:00 - 10:30	Coffee Break
	Parallel Sessions: Parallel Sessions at MPI MiS and Leipzig University (Felix-Klein-Hörsaal)
10:30 - 11:00	MPI MiS, Leibniz-Saal: Jenya Soprunova The volume polynomial of lattice polygons Let P and Q be lattice polygons in the plane. Then their normalized volumes V(P) and V(Q) and their normalized mixed volume V(P,Q) satisfy the classical Minkowski inequality V(P)V(Q)
	Leipzig University, Felix-Klein-Hörsaal: Alex Dunbar A Topological Approach to Simple Descriptions of Convex Hulls of Sets Defined by Three Quadrics
11:00 - 11:30	MPI MiS, Leibniz-Saal: Gennadiy Averkov Mixed volumes of zonoids and the absolute value of the Grassmannian
	Leipzig University, Felix-Klein-Hörsaal: Niels Lubbes Self-intersections of surfaces that contain two circles through each point Surfaces containing two lines through each point are smooth quadrics as was discovered about 350 years ago by Sir Christopher Wren. The Möbius geometric analogues of such doubly ruled surfaces are "celestial surfaces", namely surfaces containing two circles through each point. Celestial surfaces are of interest in for example architecture (supporting frameworks of buildings), kinematics (configuration spaces of linkages) and computer vision (surface reconstruction from silhouettes and camera calibration). In this talk, we present a classification of the real singular loci of celestial surfaces. Although the simplest curve after a line is perhaps a circle, this classification is surprisingly diverse: there are 5 different possible configurations for the complex components and at least 7 different possible topological types for the real components. Our methods include intersection theory on real del Pezzo surfaces, Möbius geometry and factorization of quaternionic polynomials. Our classification plays a key role in the proof of a recent result which states that almost all celestial surfaces are homeomorphic to one of 4 topological normal forms. We do an effort to make our results and methods accessible to a wide audience. Slides
11:30 - 12:00	MPI MiS, Leibniz-Saal: Altan Berdan Kilic Polytope Structure of the Service Rate Region
	Leipzig University, Felix-Klein-Hörsaal: Tomoki Moriya Some explicit arithmetic on curves on genus three and their applications Slides
12:00 -	Free Afternoon

09:00 - 10:00	Daniel Erman Sheaf cohomology on weighted projective spaces My talk will focus on a new and efficient method for computing sheaf cohomology for subvarieties of a weighted projective space. I will explain how this algorithmic questions is closely related to novel generalizations of the notion of a minimal free resolution. This is joint work with Michael Brown.
10:00 - 10:30	Coffee Break
	Parallel Sessions: Parallel Sessions at MPI MiS and Leipzig University (Felix-Klein-Hörsaal)
10:30 - 11:00	MPI MiS, Leibniz-Saal: Ada Wang Identifiability of overcomplete independent component analysis ndependent component analysis (ICA) is a classical data analysis method to study mixtures of independent sources. An ICA model is said to be identifiable if the mixing can be recovered uniquely. Identifiability is known to hold if and only if at most one of the sources is Gaussian, provided the number of sources is at most the number of observations. In this talk, I will discuss our work to generalize the identifiability of ICA to the overcomplete setting, where the number of sources can exceed the number of observations. The underlying problem is algebraic and the proof studies linear spaces of rank one symmetric matrices. Based on joint work with Anna Seigal https://arxiv.org/abs/2401.14709. Slides
	Leipzig University, Felix-Klein-Hörsaal: Frédéric Bihan Sparse systems with high local multiplicity Consider a sparse system of n Laurent polynomials in n variables with complex coefficients and support in a finite lattice set A. The maximal number of isolated roots of the system in the complex n-torus is known to be the normalized volume of the convex hull of A (the BKK bound). We explore the following question: if the cardinality of A equals n+m+1, what is the maximum local intersection multiplicity at one point in the torus in terms of n and m? This study was initiated by Gabrielov in the multivariate case. We give an upper bound that is always sharp when m=1 and, under a generic technical hypothesis, it is considerably smaller for any dimension n and codimension m. We also present, for any value of n and m, a particular sparse system with high local multiplicity with exponents in the vertices of a cyclic polytope and we explain the rationale of our choice. Our work raises several interesting questions. Slides
11:00 - 11:30	MPI MiS, Leibniz-Saal: Benjamin Hollering Computing Implicitizations of Multi-Graded Polynomial Maps In this talk, we'll introduce a new method for computing the kernel of a polynomial map which is homogeneous with respect to a multigrading. We first demonstrate how to quickly compute a matrix of maximal rank for which the map has a positive multigrading. Then in each graded component we compute the minimal generators of the kernel in that multidegree with linear algebra. We have implemented our techniques in Macaulay2 and show that our implementation can compute many generators of low degree in examples where Gröbner basis techniques have failed. This includes several examples coming from phylogenetics where even a complete list of quadrics and cubics were unknown. When the multigrading refines total degree, our algorithm is embarrassingly parallel. This is joint work with Joseph Cummings.
	Leipzig University, Felix-Klein-Hörsaal: Josué Tonelli-Cueto On the Number of Real Zeros of Random Sparse Polynomial Systems Consider a polynomial system $f_1(x)=0,\ldots,f_n(x)=0$ of $n$ real polynomials in $n$ variables, where each $f_k$ has a prescribed set of exponent vectors in a set $A_k\subseteq \mathbb{Z}^n$ of size $t_k$. Kushnirenko's conjecture asks whether the number of positive zeros of such a polynomial system is of the form $\mathrm{poly}(t_1,\ldots,t_n)$. As of today, this is one of the most challenging problems in real algebraic geometry, being open even in the two-dimensional case!In this talk, we will present a probabilistic version of Kushnirenko's conjecture. We will show, among other variations of the result, that if the coefficients are independent Gaussian of any variance, then the expected number of positive zeros of the random system is bounded from above by $4^{-n} \prod_{k=1}^n t_k(t_k-1)$.This is joint work with Alperen A. Ergür and Maté T. Telek. Preprint available at https://arxiv.org/abs/2306.06784 Slides
11:30 - 12:00	MPI MiS, Leibniz-Saal: Shelby Cox ML Degrees of Brownian Motion Tree Models: Star Trees and Root Invariance
	Leipzig University, Felix-Klein-Hörsaal: Chiara Meroni Chebyshev Varieties Chebyshev varieties are algebraic varieties parametrized by Chebyshev polynomials or their multivariate generalizations. They play the role of toric varieties in sparse polynomial root finding, when monomials are replaced by Chebyshev polynomials. We introduce these objects and discuss their main properties, including dimension, degree, singular locus and defining equations, as well as some computational experiments. Slides
12:00 - 14:00	Lunch
14:00 - 15:00	Anna Wienhard Positivity in Lie groups and representation varieties In my talk I will discuss generalizations of total positivity in real semi simple Lie groups and applications of it to the study of representation varieties.
15:00 - 15:30	Coffee Break
	Parallel Sessions: Parallel Sessions at MPI MiS and Leipzig University (Felix-Klein-Hörsaal)
15:30 - 17:00	MPI MiS, Leibniz-Saal: Irina Kogan Equi-affine minimal-degree moving frames for polynomial curves We develop a theory and an algorithm for constructing minimal-degree polynomial moving frames for polynomial curves in an affine space. The algorithm is equivariant under volume-preserving affine transformations of the ambient space and the parameter shifts. We show that any matrix-completion algorithm can be turned into an equivariant moving frame algorithm via an equivariantization procedure that we develop. If a matrix-completion algorithm is of minimal degree then so is the resulting equivariant moving frame algorithm. We propose a novel minimal-degree matrix-completion algorithm, complementing the existing body of literature on this topic. This is a joint work with Hoon Hong, NC State University. Slides
	Leipzig University, Felix-Klein-Hörsaal: Pierpaola Santarsiero Degree of the subspace variety Subspace varieties are algebraic varieties whose elements are tensors with bounded multilinear rank. In this talk, we will understand how to compute their degrees by computing their volumes. Slides
16:00 - 16:30	MPI MiS, Leibniz-Saal: Matthew Faust Irreducibility of the Dispersion Polynomial for Periodic Graphs Given a ZZ^d-periodic graph G, a periodic potential together with a weighted graph Laplacian defines a discrete periodic operator, which acts on functions on the vertices of G. Floquet theory allows us to study the spectrum of this operator through a finite matrix with Laurent polynomial entries. The zero set of the corresponding characteristic polynomial is called the Bloch variety. We will focus on the reducibility of this variety, which provides insights into various spectral properties, such as quantum ergodicity. In particular, we will study how the reducibility of the Bloch variety is affected as one varies the period of the potential.
	Leipzig University, Felix-Klein-Hörsaal: Luca Sodomaco A vector bundle approach to Nash equilibria Using vector bundle techniques, we study the algebraic scheme of totally mixed Nash equilibria of an n-person game. This approach reveals intriguing similarities and differences among totally mixed Nash equilibria in Game Theory, eigenvectors, and singular vector tuples in Spectral Theory of higher-order tensors. First, a generic n-person game of format d=(d_1,...,d_n) does not admit totally mixed Nash equilibria if and only if the corresponding Segre variety of rank-one tensors of format d is dual defective. Consequently, we define the Nash resultant variety of games with dual defective format whose scheme of totally mixed Nash equilibria is nonempty. In particular, we show its irreducibility and compute its codimension and degree. Secondly, we study the algebraic variety of n-person games of format d whose scheme of totally mixed Nash equilibria is either non-reduced or has a positive dimensional component. We call it the Nash discriminant. In this talk, we describe entirely the Nash discriminant of a 3-person game of format (2,2,2), and we determine an irreducible decomposition of its singular locus. Finally, with the help of symbolic computations in Macaulay2, we show that the Nash discriminant of a 3-person game of format (2,2,3) is not irreducible.
16:30 - 17:00	MPI MiS, Leibniz-Saal: Jan Legerský Flexible pentagonal bipyramids and their Galois groups When a bipyramid flexes, the distance between the two opposite vertices of the two pyramids changes. Therefore, there is a map that associates each realization of the bipyramid to the distance between the two opposite vertices. From an algebraic point of view, this determines a field extension between the field of univariate rational functions and the field of rational functions on the configuration curve of the bipyramid. In this talk, we present a classification of flexible pentagonal bipyramids with respect to the Galois group of this field extension. As a consequence of the result, we construct an embedded flexible polyhedron with 8 vertices. This answers negatively a long-standing question whether the flexible embedded polyhedron by Steffen, which has 9 vertices, is an embedded flexible polyhedron with the least number of vertices.
	Leipzig University, Felix-Klein-Hörsaal: Matteo Gallet Eigenpoint collinearities of plane cubics
17:30 -	Beer garden

09:00 - 10:00	Fulvio Gesmundo Geometry and asymptotics in the study of tensors In the last decades, tensors have become central in several areas of pure and applied mathematics, leading to the rediscovery of many classical results and constructions across different languages and fields. In applied areas such as computational complexity and quantum information theory, there is significant interest in certain asymptotic properties of tensors, and often classical algebraic and geometric techniques have been successfully adapted to the asymptotic framework. In this talk, I will provide an overview of some of these research directions, focusing on different notions of tensor rank and tensor complexity and on the role of algebraic geometry in this area of study.
10:00 - 10:30	Coffee Break
	Parallel Sessions: Parallel Sessions at MPI MiS and Leipzig University (Felix-Klein-Hörsaal)
10:30 - 11:00	MPI MiS, Leibniz-Saal: Stefano Mereta The Fundamental theorem of tropical differential algebra over nontrivially valued fields and the radius of convergence of nonarchimedean differential equations We will discuss a fundamental theorem for tropical differential equations analogue of the fundamental theorem of tropical geometry in this context. We extend results from Aroca et al. and from Fink and Toghani, working only in the case of trivial valuation as introduced by Grigoriev, to (partial) differential equations with power series coefficients over any valued field. To do so, a crucial ingredient is the framework for tropical differential equations introduced by Giansiracusa and Mereta. As a corollary of the fundamental theorem, the radius of convergence of solutions of a differential equation over a nontrivially valued field can be computed tropically. This talk is based on results appearing in arXiv:2303.12124, joint with F. Gallinaro. Slides
	Leipzig University, Felix-Klein-Hörsaal: Shixuan Zhang Local minima in nonconvex sum-of-square optimization via syzygies We study spurious local minima in a nonconvex low-rank formulation of sum-of-squares optimization on a real variety $X$. We reformulate the problem of finding a spurious local minimum in terms of syzygies of the underlying linear series, and also bring in topological tools to study this problem. When the variety $X$ is of minimal degree, there exist spurious local minima if and only if both the dimension and the codimension of the variety are greater than one and $X$ is not the Veronese surface, answering a question by Legat, Yuan, and Parrilo. Moreover, for surfaces of minimal degree, we provide sufficient conditions to exclude points from being spurious local minima. In particular, all spurious local minima on the Veronese surface, corresponding to ternary quartics, lie on the boundary and can be written as a binary quartic, up to a linear change of coordinates, complementing work by Scheiderer on decompositions of ternary quartics as a sum of three squares. For general varieties of higher degree, we give examples and characterizations of spurious local minima in the interior, which can potentially lead to efficient low-rank sum-of-squares algorithms.
11:00 - 11:30	MPI MiS, Leibniz-Saal: Sarah Eggleston The amoeba dimension of a linear space Given a complex vector subspace V of C^n , the dimension of the amoeba of V∩(C^∗)^n depends only on the matroid that V defines on the ground set {1,...,n}. Here we prove that this dimension is given by the minimum of a certain function over all partitions of the ground set, as previously conjectured by Rau. We also prove that this formula can be evaluated in polynomial time.
	Leipzig University, Felix-Klein-Hörsaal: Emil Horobet The infinite Euclidean Distance discriminant In this talk, we study the infinite ED discriminant, the variety of data points with infinite critical points of the distance function from themselves to a given variety. We present many examples of varieties that have such a discriminant, and we show that this relates to instances of morphisms where we do not have the purity of the branch locus, we give sufficient conditions on the varieties to have a non-empty infinite discriminant and finally, we classify all such curves in arbitrary dimension and all surfaces in three-space which have a one-dimensional infinite discriminant, via the explicit construction of all such surfaces. Slides
11:30 - 12:00	MPI MiS, Leibniz-Saal: Máté László Telek Viro's patchworking and the signed reduced A- discriminant Computing the isotopy type of a hypersurface, defined as the positive real zero set of a polynomial, is a challenging problem in real algebraic geometry. For a fixed signed support, that is, for fixed exponent vectors and signs of the coefficients of the polynomial, the signed reduced -discriminant divides the coefficient space into chambers such that in each chamber the isotopy type of the hypersurfaces is constant. Within outer hambers the isotopy type of the hypersurface can be described by a olyhedral object. However, hypersurfaces associated with inner chambers might have isotopy types, which are more difficult to describe. In this talk, will discuss properties of the signed support that preclude the existence of such inner chambers of the signed reduced A-discriminant.
	Leipzig University, Felix-Klein-Hörsaal: Benjamin Lovitz A hierarchy of eigencomputations for polynomial optimization on the sphere
12:00 - 14:00	Lunch
14:00 - 15:00	Fatemeh Mohammadi Matroids in Applied Algebraic Geometry Matroids are combinatorial structures that generalize various notions of independence. Motivated by realizability and causality problems in statistics, we study the varieties associated with matroids, which can be reducible and exhibit arbitrary singularities. We focus on several families of matroids arising in statistics. Our main findings are threefold: (1) we describe an irreducible decomposition of these varieties combinatorially; (2) we explain the symmetric structures of these varieties, simplifying algebraic computations; (3) we compute the defining equations of these varieties. As an application, we explore the intersection axiom for conditional independence (CI) models in the presence of hidden variables. This talk is based on multiple joint projects: (a) with Emiliano Liwski, (b) with Kevin Grace and Oliver Clarke, (c) with Johannes Rauh, and (d) with Yulia Alexandr, Kristen Dawson, Hannah Friedman, Pardis Semnani, and Teresa Yu.

Participants

Firoozeh Aga (née Dastur)

RPTU Kaiserslautern and Fraunhofer ITWM Kaiserslautern

Dario Antolini

University of Trento

Gennadiy Averkov

Brandenburg University of Technology

Tabea Bacher

Max Planck Institute for Mathematics in the Sciences Leipzig

Lorenzo Baldi

University of Leipzig, MPI MiS

George Balla

Technische Universität Berlin

Alessandra Bernardi

University of Trento

Barbara Betti

Max Planck Institute for Mathematics in the Sciences

Benjamin Biaggi

University of Bern

Frédéric Bihan

Université Savoie Mont Blanc

Greg Blekherman

Georgia Tech

Tobias Boege

KTH Stockholm

Viktoriia Borovik

Osnabrück Universität

Alessio Borzi

MPI MiS Leipzig

Francis Brown

Oxford University

Clemens Brüser

TU Dresden

Stefano Canino

Politecnico di Torino

Jose Capco

University of Innsbruck

Laura Casabella

MPI MiS Leipzig

Türkü Özlüm Çelik

Max Planck Institute of Molecular Cell Biology and Genetics

Carles Checa

National and Kapodistrian University of Athens

Tracy Chin

University of Washington

Oliver Clarke

University of Edinburgh

Erin Connelly

Osnabrück University

Shelby Cox

MPI

Carlos D'Andrea

Universitat de Barcelona

Oliver Daisey

Durham University

Hana Dal Poz Kourimska

IST Austria

Alessandro Danelon

University of Michigan

Sebastian Debus

TU Chemnitz

Karel Devriendt

Max Planck Institute for Mathematics in the Sciences Leipzig

Sandra Di Rocco

KTH

Alicia Dickenstein

University of Buenos Aires

Ruiwen Dong

Saarland University

Jan Draisma

University of Bern

Alex Dunbar

Emory University

Rob Eggermont

Eindhoven University of Technology

Sarah Eggleston

University of Osnabrück

Boulos El Hilany

TU Braunschweig

Daniel Erman

University of Wisconsin-Madison

Matthew Faust

Texas A&M University

Kamillo Ferry

TU Berlin

Cosimo Flavi

Università degli Studi di Firenze

Hannah Friedman

UC Berkeley

Giulia Gaggero

Université de Neuchâtel

Vincenzo Galgano

MPI-CBG Dresden

Matteo Gallet

University of Trieste

Francesco Galuppi

University of Warsaw

Marina Garrote López

Max Planck Institute for Mathematics in the Sciences

Boštjan Gec

Jožef Stefan Institute

Fulvio Gesmundo

Institut de Mathématiques de Toulouse

Philippe Gimenez

University of Valladolid

Mario González-Sánchez

Universidad de Valladolid

Christiane Görgen

Leipzig University

Friedemann Groh

ISG Stuttgart

Samuel Gue

Swansea University

Sami Halaseh

Freie Universität

Jeroen Hanselman

RPTU Kaiserslautern-Landau

Sachi Hashimoto

Brown University

Paul Alexander Helminck

Tohoku University

Oskar Henriksson

University of Copenhagen

Milena Hering

University of Edinburgh

Benjamin Hollering

Technische Universität München

Emil Horobet

Sapientia Hungarian University of Transylvania

Benjamin Jany

Eindhoven University of Technology

Tyler Jarvis

Brigham Young University

Michael Joswig

TU Berlin

Jeong-Hoon Ju

Pusan National University

Jaewoo Jung

Institute for Basic Science

Yuta Kambe

Mitsubishi Electric

Leonie Kayser

Max Planck Institute for Mathematics in the Sciences Leipzig

Kim Lukas Kiehn

KTH Stockholm/University of Bonn

Altan Berdan Kilic

Eindhoven University of Technology

Joris Koefler

Max Planck Institute for Mathematics in the Sciences Leipzig

Irina Kogan

North Carolina State University

Andreas Kretschmer

MPI MiS Leipzig

Teresa Krick

University of Buenos Aires & Conicet

Kaie Kubjas

Aalto University

Momonari Kudo

Fukuoka Institute of Technology

Mikolaj Le Van

Pedagogical University of Cracow

Jan Legerský

Czech Technical University in Prague

Christian Lehn

Ruhr-Universität Bochum

Fabian Lenzen

TU Berlin

Janina Letz

University of California, Los Angeles / Universität Bielefeld

Benjamin Lovitz

Northeastern University

Niels Lubbes

Johannes Kepler University Linz

Dante Luber

TU Berlin

Natalia Machał

Pedagogical University of Cracow

Victor Magron

CNRS LAAS

Giacomo Maletto

KTH

Orlando Marigliano

University of Genoa

Pablo Mazón

Università di Trento

Amelie Menges

TU Dortmund

Stefano Mereta

KTH Royal Institute of Technology

Chiara Meroni

ETH Institute for Theoretical Studies

Łukasz Merta

University of the National Education Commission, Krakow

Pratik Misra

Technical University of Munich

Fatemeh Mohammadi

KU Leuven

Hyunsuk Moon

Korea Institute for Advanced Study

Tomoki Moriya

University of Birmingham

Chenqi Mou

Beihang University

Bernard Mourrain

Inria at Université Côte d'Azur

Yulia Mukhina

École Polytechnique, Institute Polytechnique de Paris

Tommi Muller

University of Oxford

Leonard Mushunje

columbia university

Elke Neuhaus

MPI MiS Leipzig

Thu Hien Nguyen

Julius Maximilians-Universität Würzburg

Francesco Nowell

TU Berlin

Umut Oktem

UC Davis

Janike Oldekop

TU Berlin

Alessandro Oneto

U. Trento

Gabriel Ong

University of Bonn

Birte Ostermann

Technische Universität Braunschweig

Giorgio Ottaviani

Universita' di Firenze

Soohyun Park

Hebrew University of Jerusalem

Euisung Park

Korea University

Niharika Paul

MPI MIS

Dmitrii Pavlov

MPI MiS

Gianni Petrella

University of Luxembourg

Anaëlle Pfister

Max Planck Institute for Mathematics in the Sciences Leipzig

Eric Pichon-Pharabod

Inria Saclay

Daniel Plaumann

TU Dortmund

Franziska Porkert

Universität Bonn

Irem Portakal

Max Planck Institute for Mathematics in the Sciences

Elizabeth Pratt

UC Berkeley

Muhammad Imran Qureshi

King Fahd University of Petroleum and Minerals

Claudiu Raicu

University of Notre Dame

Ali Raza

Higher Education Department

Bernhard Reinke

MPI MiS

Yue Ren

Durham University

Alvaro Ribot

Harvard University

Cordian Riener

UiT The Arctic University of Norway

Angel Rios

Université Paris-Saclay

Andrea Rosana

SISSA

Kemal Rose

KTH

Olivier Ruatta

Université de Limoges

Felix Rydell

KTH Royal Institute of Technology

Flavio Salizzoni

Université de Neuchatel

Pierpaola Santarsiero

University of Bologna

Anna-Laura Sattelberger

Max Planck Institute for Mathematics in the Sciences Leipzig

David Sawall

Universität Konstanz

Mahrud Sayrafi

Max Planck Institute for Mathematics in the Sciences

Robin Schabert

UiT - The Arctic University of Norway

Johannes Schmitt

Universität Siegen

Nicolà Schnieper

University of Bern

Lisa Seccia

University of Neuchâtel

Javier Sendra Arranz

Tübingen University

Tim Seynnaeve

KU Leuven

Elima Shehu

Otto-von-Guericke-Universität Magdeburg

Rainer Sinn

Universität Leipzig

Luca Sodomaco

KTH Royal Institute of Technology

Qiuye Song

Beihang University

Ivan Soprunov

Cleveland State University

Jenya Soprunova

Kent State University

Frank Sottile

Texas A&M University

Pedro Souza

Goethe-Universität Frankfurt am Main

Mima Stanojkovski

Università di Trento

Bernd Sturmfels

Max Planck Institute for Mathematics in the Sciences

Svala Sverrisdottir

University of California, Berkeley

Tomasz Szemberg

University of the National Education Comission

Daniele Taufer

KU Leuven

Ettore Teixeira Turatti

UiT - The Arctic University of Norway

Máté László Telek

University of Copenhagen

Simon Telen

Max Planck Institute for Mathematics in the Sciences

Thorsten Theobald

Goethe University Frankfurt

Josué Tonelli-Cueto

Johns Hopkins University

Elias Tsigaridas

Inria Paris and Sorbonne University

Julian Vill

OvGU Magdeburg

Cynthia Vinzant

University of Washington

Ada Wang

Harvard University

Lena Weis

Technische Universität Berlin

Jannik Wesner

TU Dortmund

Anna Wienhard

Max Planck Institute for Mathematics in the Sciences

Maximilian Wiesmann

MPI MiS

Benedikt Wolf

Julius-Maximilians-Universität Würzburg

Josephine Yu

Georgia Institute of Technology

Francesca Zaffalon

KU Leuvevn

Shixuan Zhang

Texas A&M University

Maciej Zięba

University of the National Education Commission

Marcin Zieliński

University of the National Education Commission, Krakow

Filip Zieliński

University of the National Education Commision

Local organizers

Christian Lehn

Ruhr-Universität Bochum

Irem Portakal

Max Planck Institute for Mathematics in the Sciences

Rainer Sinn

Universität Leipzig

Bernd Sturmfels

Max Planck Institute for Mathematics in the Sciences

Simon Telen

Max Planck Institute for Mathematics in the Sciences

Executive committee

Kaie Kubjas

Aalto University

Frank Sottile

Texas A&M University (TAMU)

Alessandra Bernardi

Università di Trento

Elisa Gorla

Université de Neuchâtel

Teresa Krick

Universidad de Buenos Aires

Pierre Lairez

École polytechnique

Claudiu Raicu

University of Notre Dame

Simon Telen

Max Planck Institute for Mathematics in the Sciences

Administrative Contact

Mirke Olschewski

Max Planck Institute for Mathematics in the Sciences Contact via Mail

Saskia Gutzschebauch

Max Planck Institute for Mathematics in the Sciences Contact via Mail

MEGA 2024

Links

Speakers

Francis Brown

Türkü Özlüm Çelik

Carlos D'Andrea

Daniel Erman

Fulvio Gesmundo

Victor Magron

Fatemeh Mohammadi

Anna Wienhard

Josephine Yu

Program

Participants

Firoozeh Aga (née Dastur)

Dario Antolini

Gennadiy Averkov

Tabea Bacher

Lorenzo Baldi

George Balla

Alessandra Bernardi

Barbara Betti

Benjamin Biaggi

Frédéric Bihan

Greg Blekherman

Tobias Boege

Viktoriia Borovik

Alessio Borzi

Francis Brown

Clemens Brüser

Stefano Canino

Jose Capco

Laura Casabella

Türkü Özlüm Çelik

Carles Checa

Tracy Chin

Oliver Clarke

Erin Connelly

Shelby Cox

Carlos D'Andrea

Oliver Daisey

Hana Dal Poz Kourimska

Alessandro Danelon

Sebastian Debus

Karel Devriendt

Sandra Di Rocco

Alicia Dickenstein

Ruiwen Dong

Jan Draisma

Alex Dunbar

Rob Eggermont

Sarah Eggleston

Boulos El Hilany

Daniel Erman

Matthew Faust

Kamillo Ferry

Cosimo Flavi

Hannah Friedman

Giulia Gaggero

Vincenzo Galgano

Matteo Gallet

Francesco Galuppi

Marina Garrote López

Boštjan Gec

Fulvio Gesmundo

Philippe Gimenez

Mario González-Sánchez

Christiane Görgen

Friedemann Groh

Samuel Gue

Sami Halaseh

Jeroen Hanselman

Sachi Hashimoto

Paul Alexander Helminck

Oskar Henriksson

Milena Hering

Benjamin Hollering

Emil Horobet

Benjamin Jany

Tyler Jarvis