Joel Hakavuori (Aalto) - August 23, at 12:00, hall M203

Title: Subadditivity of shifts, Eilenberg-Zilber shuffle products and cohomology of lattices

Abstract: We show that the maximal shifts in the minimal free resolution of a monomial ideal form a subadditive sequence, answering a conjecture of Avramov, Conca and Iyengar. To do so, we study different models for the cohomology rings of posets and lattices, introduce the Eilenberg-Zilber shuffle product for lattices and use it to establish vanishing theorems and models for homology for lattices. The talk is based on joint work with Karim Adiprasito, Minas Margaritis and Eran Nevo.

Gunnar Fløystad (University of Bergen) - May 29, at 15:15, on Zoom

Title: Polarizations of artin monomial ideals define triangulated balls

Abstract: We show that any polarization of an artin monomial ideal defines a triangulated ball, via the Stanley-Reisner correspondence.  This proves a conjecture of A.Almousa, H.Lohne and the speaker.

Geometrically, polarizations of ideals containing the ideal (x_1^{a_1}, ...., x_n^{a_n})  define full-dimensional triangulated balls on the sphere which is the join of boundaries of simplices of dimensions a_1-1, ... , a_n-1.

Combinatorially, these triangulated balls derive from subsets T of products of finite sets A_1 x A_2 x ... x A_n which we call bitight.  The subset T and its complement fulfill exchange conditions similar to that of matroids.

Petteri Kaski (Aalto) - May 22, at 15:15, hall M2

Title: Constructive and nonconstructive enumeration

Abstract: This talk will be a primer to combinatorial constructive and nonconstructive enumeration up to isomorphism and will consist of two parts (a) a recall/primer of finite groups and group actions to capture pertinent isomorphism relations and (b) an introduction to algorithmic isomorph-free exhaustive generation techniques, in particular to Brendan McKay's [J Algorithms 1998] influential canonical construction path technique.

Irene Heinrich (TU Darmstadt) - May 15, at 15:15, hall M2

Title: Colored highly regular graphs

Abstract: A coloured graph is k-ultrahomogeneous if every isomorphism between two induced subgraphs of order at most k extends to an automorphism. A coloured graph is t-tuple regular if the number of vertices adjacent to every vertex in a set S of order at most k depends only on the isomorphism type of the subgraph induced by S. We classify the finite vertex-coloured k-ultrahomogeneous graphs and the finite vertex-coloured l-tuple regular graphs for k at least 4 and l at least 5, respectively. Our theorem in particular classifies finite vertex-coloured ultrahomogeneous graphs, where ultrahomogeneous means the graph is simultaneously k-ultrahomogeneous for all k.

Miika Hannula (University of Helsinki) - April 24, at 15:15, hall M2

Title: On database dependencies and information inequalities

Abstract: In databases, notions of dependence and independence play a crucial role. For instance, every database relation typically has a key, which is a set of attributes that functionally determines the remaining attributes in the relation. By taking a uniform distribution over the database relation, these dependency notions can be recast using Shannon’s information measures. Consequently, logical implication between database dependencies can (sometimes) be reconceptualized as an information inequality, that is, a linear inequality over entropies. In this talk we review these connections and also consider information inequalities from a computational perspective. Unlike logical implication in database theory, not much seems to be known about the exact computational complexity of decision problems associated with information inequalities.

Olga Kuznetsova (Aalto) - April 17, at 15:15, hall M2

Title: Weak maximum likelihood threshold of coloured Gaussian graphical models

Abstract: Colored Gaussian graphical models are linear concentration models arising from undirected graphs with a coloring in its vertices and edges. Given a coloured Gaussian graphical models, one may be interested to know how many observations are necessary for a maximum likelihood estimate to exist with positive probability. This is called the weak maximum likelihood threshold of a graph. We discuss computational and algebraic methods for studying the weak maximum likelihood threshold of a graph.

Tobias Boege (Aalto) - March 27, at 15:15, hall M2

Title: Matroids in information theory

Abstract: Matroids capture combinatorial properties of independence in linear algebra and graph theory. It is less well-known that (poly)matroidal structures also appear as entropy functions of discrete random variables. We give an introduction to this point of view and survey relevant results and examples

Fausto Barbero (University of Helsinki) - February 17, at 13:00, hall Y229c

Title: Expressivity of languages for probabilistic causal reasoning

Abstract: Causal reasoning, as developed e.g. in the works of J. Pearl and of Spirtes, Glymour & Scheines, is a collection of effective but rather disorganized mathematical techniques and ad hoc formalisms. With Gabriel Sandu, we proposed a semantic framework (causal multiteam semantics) by means of which many of the statements typical of causal reasoning (concerning e.g. conditional probabilities, "do expressions", Pearl's "counterfactuals") can be decomposed into three simpler elements: (marginal) probability statements, and two distinct conditionals, which describe the effects, respectively, of an action and of an increase in information.

In this talk I will present part of a joint work with Jonni Virtema. We gave abstract characterizations of the expressive power of a number of candidate languages for probabilistic causal reasoning; these characterizations involve identifying three special classes of linear inequalities.

By analyzing the geometry of these inequalities as interpreted in standard n-simplexes, we used the characterization results to prove that the languages form a proper hierarchy (i.e., that distinct levels of generality of the syntax correspond to distinct levels of expressivity) and to prove some further undefinability result.

Sándor Kisfaludi-Bak (Aalto) - February 7, at 14:00, hall M237

Title: On geometric variants of the traveling salesman problem

Abstract: In the classic Euclidean traveling salesman problem, we are given n points in the Euclidean plane, and the goal is to find the shortest round trip that visits all the points. We will briefly discuss how modern algorithmic and lower bound tools allowed us to find (conditionally) optimal exact and approximation algorithms for this problem, while the closely related Steiner tree problem seems to resist many similar attempts. We will then turn to the traveling salesman or Steiner tree with "neighborhoods". Here instead of points, we are given a set of affine subspaces, and the goal is to find the shortest round trip or tree that intersects each subspace. It turns out that these problems have a different computational complexity than the classic problems with points: they require a completely novel approach for the hyperplane case, while the other cases remain largely mysterious.

Tuomas Kelomäki (Aalto) - January 31, at 14:00, hall M237

Title: Using discrete Morse theory algebraically, part 2

Tuomas Kelomäki (Aalto) - January 24 (2023), at 14:00, hall M237

Title: Using discrete Morse theory algebraically

Abstract: This is an introductory talk on discrete Morse theory and especially on Sköldberg's algebraic formulation of it. The goal is to learn the method through several examples. If the time permits, we will also derive finite versions of several classical results in homological algebra from the theory.

Oscar Kivinen (EPFL, Lausanne) - December 16 (2022), at 15:30, hall M237

Title: Plane curves: a bridge between number theory, knots, and physics

Abstract: Plane curves are some of the simplest classical algebro-geometric objects, that most of us encounter in middle school (over the real numbers). Especially their singularity theory remains a source of many interesting discoveries. For example, it turns out many moduli spaces of sheaves on singular plane curves are related to 1) arithmetic problems arising in the Langlands program 2) physics of some 3d/4d supersymmetric gauge theories.

On the other hand, in most cases a lot of data about the singularities is controlled by the knots arising as their links, and one may ask how the relevant knot theory is reflected in the themes 1) and 2) above. I will give an introduction to plane curves and discuss the relation to 1) and possibly 2), highlighting a recent result about "orbital integrals” on p-adic groups (obtained by myself and Tsai) where the topology of the singularities informs the arithmetic. Little background knowledge will be required, apart from knowing what knots, finite fields, and the general linear group are.

Toni Annala (Institute for Advanced Study) - December 16, at 14:15, hall M237

Title: Topologically protected tricolorings

Abstract: Topological vortices are codimension-one topological defects that arise in various physical systems, such as liquid crystals, Bose--Einstein condensates, and vacuum structures of Yang--Mills theories. I will explain how, under certain homotopical assumptions that are satisfied in many realistic systems, topological vortex configurations admit faithful presentations in terms of colored link diagrams. The most well-known coloring scheme of links is given by tricolorings: each arc of the link diagram is colored by one of three possible colors (red, green, or blue) in such a way that, in each crossing, either all arcs have the same color, or all arcs have a different color. A tricolored link is topologically protected if it cannot be transformed into a disjoint union of unlinked simple loops by a sequence of color-respecting isotopies and color-respecting local cut-and-paste operations. The latter operations are referred to as allowed local surgeries. We use equivariant bordism groups of three-manifolds to construct invariants of colored links that are conserved in allowed local surgeries, and employ the invariant to classify all tricolored links up to local surgeries. The talk is based on joint work with Hermanni Rajamäki, Roberto Zamora Zamora, and Mikko Möttönen.

Teemu Lundström (Aalto) - November 18, at 2:15, hall M237. 

Title: Yamada Polynomials from Transfer-Matrix Methods 

Abstract: Spatial graphs are graphs that are embedded in three-dimensional space. They are in some sense a generalization of knots and the theory of spatial graphs is closely related to knot theory. In knot theory, one can distinguish between two inequivalent knots by computing some algebraic invartiant of them, for example, the Jones polynomial. Similar invariants have been invented for spatial graphs and one important such invariant is the Yamada Polynomial, first introduced by Shuji Yamada in 1989. In this talk I will introduce spatial graphs and the Yamada polynomial defined for them. I will focus on computing the polynomial for certain classes of graphs that have a layer-like structure. Computing the Yamada polynomial for a spatial graph can be computationally demanding, but by focusing on such graphs we are able to apply the so called transfer-matrix method with which we are able to find a closed form formula for two infinite families of spatial graphs. 

Muhammad Ardiyansyah (Aalto) - November 9, at 3pm, hall M2. 

Title: Dimensions of the factor analysis model and its higher-order generalizations

Abstract: The factor analysis model is a statistical model where a certain number of hidden random variables, called factors, affect linearly the behaviour of another set of observed random variables, with additional random noise. The main assumption of the model is that the factors and the noise are Gaussian random variables. In this talk, we do not assume that the factors and the noise are Gaussian, hence the higher order moment and cumulant tensors of the observed variables are generally nonzero. This motivates the generalized notion of kth-order factor analysis model, that is the family of all random vectors in a factor analysis model where the factors and the noise have finite and possibly nonzero moment and cumulant tensors up to order k. This subset may be described as the image of a polynomial map onto a Cartesian product of symmetric tensor spaces. We provide its dimension and conditions under which the image has positive codimension. This talk is based on joint work with Luca Sodomaco.

Lisa Nicklasson (Università di Genova) - October 20, at 11:15, on Zoom

Title: Ideals arising from Bayesian networks

Abstract: A Bayesian network is a statistical model which can be presented graphically by a directed acyclic graph. The nodes in the graph are discrete random variables, and the edges encode dependencies between the variables. Bayesian nets can also be described algebraically as varieties of homogeneous prime ideals. In this talk we will discuss connections between algebraic properties of such ideals and combinatorial properties of the graphs. In particular, we would like to understand when the variety is toric and when the ideal is quadratic.

Gregory Arone (Stockholm University) - October 13, at 11:15, hall M2

Title: The S_n-equivariant topology of partition complexes

Abstract: Let n be a positive integer. Consider the poset of partitions of the set {1, ... , n}, ordered by refinement. Its geometric realization is a topological space that encodes information about the combinatorial properties of the partition poset. We obtain a sequence of spaces T_1, T_2, ..., T_n, ...,. In fact it is a symmetric sequence of spaces, by which we mean that the n-th space T_n has a natural action of the symmetric group S_n.  These spaces have many interesting properties, and they arise in a number of places in mathematics, from the study of Lie algebras to algebraic topology to mathematical biology. We will survey some of the properties and applications of the spaces T_1, ..., T_n,..., focusing on the properties of the action of the symmetric groups. In particular, we will give a "branching rule" that describes the restriction of T_n to a Young subgroup of S_n (this is joint work with Lukas Brantner). The proof uses discrete Morse theory, and it generalizes many previous results. I will show some applications and some open questions.

Milo Orlich (Aalto) - October 6, at 11:15, hall M2

Title: Asymptotic results on Betti numbers of edge ideals of graphs via critical graphs

Abstract: To any graph G one can associate its edge ideal. One of the most famous results in combinatorial commutative algebra, Hochster's formula, describes the Betti numbers of the edge ideal in terms of combinatorial information on the graph G. More explicitly, each specific Betti number is given in terms of the presence of certain induced subgraphs in G. The machinery of critical graphs, relatively recently introduced by Balogh and Butterfield, deals with characterizing asymptotically the structure of graphs based on their induced subgraphs. In a joint work with Alexander Engström, we apply these techniques to Betti numbers and regularity of edge ideals. We introduce parabolic Betti numbers, which constitute a non-trivial portion of the Betti table. Usually, the vanishing of a Betti number has little impact on the rest of the Betti table. I this talk I will describe our main results, which show that on the other hand the vanishing of a parabolic Betti number determines asymptotically the structure and regularity of the graphs with that Betti number equal to zero.

Tobias Boege (Aalto) - September 28, 2pm, hall Y307

Title: Ingleton's inequality for entropies

Abstract: The Ingleton inequality is a necessary condition for a matroid to be linearly representable and it comes in the form of a linear inequality in its rank function. In a probability-theoretic reinterpretation of the inequality, linear subspaces are replaced by discrete random variables and ranks by Shannon entropies. In this setting, the Ingleton inequality no longer holds universally for representable rank functions but only if additional linear constraints are assumed.

In this talk, I give an overview of these so-called conditional Ingleton inequalities, their historical roots and my own contribution to finishing their classification for four discrete random variables.