The AGC seminar at Aalto aims to provide talks relating broadly to algebra, geometry, combinatorics and their interplay. The seminar is usually held live at the department in Aalto Undergraduate Centre on Mondays starting at 14:15.
The seminar is organised by the research groups of Oscar Kivinen and Kaie Kubjas. If you're interested in giving a talk send an email to the current managers
emilia[dot]takanen[at]aalto[dot]fi
nataliia[dot]kushnerchuk[at]aalto[dot]fi
You can find our old webpage on Google Sites.
TBA
This is an expository talk about polytopes. I will introduce what polytopes are and how to construct them. I will talk about faces, face lattices, f-vectors and duality of polytopes and what is meant by the "combinatorics of polytopes". I will talk about some classical results and conjectures on polytopes and how to turn combinatorial objects into polytopes. Everyone is welcome.
The first part of the talk is a high level "What is?" talk introducing homological algebra with a view towards algebraic geometry, a setting where there is a contravariant algebra of functions accompanying the topology. There will however also be examples from more familiar topologies to introduce simplicial sets. We will finish with an explanation of basic symplectic geometry from this viewpoint. The second part of the talk is an introduction to my line of research for the purposes of a midterm review. The previous part will be used to explain the intuition behind derived algebraic geometry and how symplectic geometry naturally lifts to this setting as shifted symplectic geometry. The (-1)-shifted symplectic setting has a nice local model and a natural connection to Milnor fibres and vanishing cycles. The motivic incarnations of these objects give well behaving invariants that have in some cases been shown to be functorial w.r.t. correspondences. Finally we show how character varieties of 3-manifolds have a natural (-1)-shifted symplectic structure and how this could be leveraged to obtain motivic invariants of 3-manifolds.
The central problem in scene analysis is finding the space of d-dimensional polyhedral caps with a fixed projection to (d-1)-dimensional space. In particular, we are interested in which sets of points in (d-1)-dimensional space are the projections of non-trivial polyhedral caps, in the sense that the hyperplanes of the polyhedral cap are distinct. Given the combinatorial structure of the polyhedral cap, studying the space of d-dimensional polyhedral caps with a fixed generic projection becomes a combinatorial problem. Specifically, liftings of generic projections can be studied via a lifting matrix. Whiteley characterised independence in the row-matroid of the lifting matrix for generic projections. The dual problem to studying scenes over generic projections is the problem of studying the space of parallel redrawings of hyperplane arrangements. In this talk, we will focus on liftability of non-generic projections, or, dually, parallel redrawings of non-generic hyperplane arrangements. For a class of polyhedral caps, we will see that the set of projections that lift to non-generic polyhedral caps with the correct combinatorial structure is given as the zero-set of a single polynomial, called the pure condition. We will see some basic properties of the pure condition, and how to easily compute it. The talk will be based on joint work with Daniel Bernstein.
We show that sufficiently low tensor rank for the balanced tripartitioning tensor $P_d(x,y,z)=\sum_{A,B,C\in\binom{[3d]}{d}:A\cup B\cup C=[3d]}x_Ay_Bz_C$ for a large enough constant $d$ implies uniform arithmetic circuits for the matrix permanent that are exponentially smaller than circuits obtainable from Ryser's formula. We show that the same low-rank assumption implies exponential time improvements over the state of the art for a wide variety of other related counting and decision problems. As our main methodological contribution, we show that the tensors $P_n$ have a desirable Kronecker scaling property: They can be decomposed efficiently into a small sum of restrictions of Kronecker powers of $P_d$ for constant $d$. We prove this with a new technique relying on Steinitz's lemma, which we hence call Steinitz balancing. As a consequence of our methods, we show that the mentioned low rank assumption (and hence the improved algorithms) is implied by Strassen's asymptotic rank conjecture [Progr. Math. 120 (1994)], a bold conjecture that has recently seen intriguing progress. Joint work with Andreas Björklund, Tomohiro Koana, and Jesper Nederlof; cf. https://arxiv.org/abs/2504.05772.
We study stucture theory of complete discrete valued fields to understand the absolute Galois group of \mathbb{C}((x)) and describe some of its structure for the local field \mathbb{F}_p((x)). In the first part, we cover topics such as Hensel's Lemma, unramified, totally ramified, and tamely ramified extensions, ramification groups, and Artin-Schreier extensions. In the second part we look at specific extensions of characteristic p>0 fields and their Galois groups as well as describe the algebraic closure of \overline{\mathbb{F}_p}((x)) through the introduction of generalized power series.
We classify arrangements of three ellipsoids in space up to rigid isotopy classes, focusing on nondegenerate configurations that avoid singular intersections. Our approach begins with a combinatorial description of differentiable closed curves on the projective plane that intersect a given arrangement of lines transversally. This framework allows us to label classes of spectral curves associated with ellipsoid configurations, which are real plane quartic curves. We determine necessary and sufficient conditions for these classes to be inhabited through arguments coming from linear algebra, algebraic geometry, combinatorics, and by computations in Mathematica and Macaulay2.
This Zoom seminar is also watchable in M2! Tensors are higher-dimensional generalisations of matrices, and likewise the main notion of complexity on matrices - their rank - may be extended to tensors. Unlike in the matrix case however, there is no single canonical notion of rank for tensors, and the most suitable notion often depends on the application that one has in mind. The most frequently used notion so far has been the tensor rank (hence its name), but several other notions and their applications have blossomed in recent years, such as the slice rank, partition rank, analytic rank, subrank, and geometric rank. Unlike their counterparts for the rank of matrices, many of the basic properties of the ranks of tensors are still not well understood. After reviewing the definitions of several of these rank notions, I will present a number of results of a type that arises in many cases when one attempts to generalise a basic property of the rank of matrices to these ranks of tensors: the naive extension of the original property fails, but it admits a rectification which is simultaneously not too complicated to state and in a spirit that is very close to that of the original property from the matrix case. Link to Zoom: https://aalto.zoom.us/j/66860024175
I will report on two recent papers establishing some geometric and categorical properties of character sheaves. In one, joint with Gonin and Ionov, we give a new proof and an extension to non-unipotent setting of the t-exactness of the composition of Radon and Harish-Chandra transforms for character sheaves. In another, joint with Bezrukavnikov, Ionov and Varshavsky, we show that this can be used to compute the Drinfeld center of the abelian Hecke category attached to the same reductive group.
Order polytopes and chain polytopes, associated with partially ordered sets, were introduced by Stanley in 1986. In 2016, Hibi and Li proposed the following conjecture concerning the number of faces of these polytopes in each dimension: (a) For any dimension i (≥1), the number of i-dimensional faces of an order polytope does not exceed that of the corresponding chain polytope. (b) If the numbers of i-dimensional faces of both polytopes coincide for some i (≥1), then the two polytopes are unimodularly equivalent. In this talk, we will provide an overview of the current progress on this conjecture and discuss results obtained specifically for faces of simplices.
The classical tool for representing cause-effect relationships in statistical modeling is a Bayesian network. This statistical model postulates noisy functional dependencies among random variables according to a directed graph. Despite their widespread use, Bayesian networks have two shortcomings: (1) different causal mechanisms may define the same statistical model, so cause-effect relationships cannot be deduced reliably from observational data alone, and (2) if the directed graph contains cycles (which may be interpreted as "feedback loops"), the model is no longer globally identifiable. A different paradigm in causal modeling has recently been proposed, which takes the stationary distributions of a family of diffusion processes as a statistical model. This temporal perspective easily accommodates feedback loops. For Ornstein--Uhlenbeck processes whose drift matrix has a specified sparsity pattern, this results in semialgebraic statistical models of Gaussian random variables now known as graphical continuous Lyapunov models. In this talk I want to introduce these models and survey recent results on identifiability, model equivalence and conditional independence. I will also discuss the algebraic challenges that lie ahead. This is based on joint works with Carlos Améndola, Mathias Drton, Benjamin Hollering, Sarah Lumpp, Pratik Misra and Daniela Schkoda.
Geometric Langlands seems very abstract, does it have any concrete applications? Ill explain how to use it to construct (new) vertex algebras.
The original idea of formality applies to a dg algebra, such as cochains on a space, and characterizes those dg algebras whose structure can be recovered from their cohomology, up to quasi-isomorphism. There is an obstruction-theoretic perspective on formality: a dg algebra is formal if and only if a certain characteristic class, called the Kaledin class, vanishes. From studying this class one deduces the formality of the algebra of cochains on spheres and other highly connected spaces. In this talk I will describe how an extension of this definition for properadic algebras allows us to address formality questions not only of space in its own, but of (possibly noncommutative) spaces endowed with a certain type of Poincaré duality structure. I will then describe a simple calculation of these obstructions for spheres. This is joint work with Coline Emprin.
The exponent $\sigma(T)$ of a tensor $T\in\mathbb{F}^d\otimes\mathbb{F}^d\otimes\mathbb{F}^d$ over a field $\mathbb{F}$ captures the base of the exponential growth rate of the tensor rank of $T$ under Kronecker powers. Tensor exponents are fundamental from the standpoint of algorithms and computational complexity theory; for example, the exponent $\omega$ of matrix multiplication can be characterized as $\omega=2\sigma(\mathrm{MM}_2)$, where $\mathrm{MM}_2\in\mathbb{F}^4\otimes\mathbb{F}^4\otimes\mathbb{F}^4$ is the tensor that represents $2\times 2$ matrix multiplication. Our main result is an explicit construction of a sequence $\mathcal{U}_d$ of zero-one-valued tensors that is universal for the worst-case tensor exponent; more precisely, we show that $\sigma(\mathcal{U}_d)=\sigma(d)$ where $\sigma(d)=\sup_{T\in\mathbb{F}^d\otimes\mathbb{F}^d\otimes\mathbb{F}^d}\sigma(T)$. We also supply an explicit universal sequence $\mathcal{U}_\Delta$ localised to capture the worst-case exponent $\sigma(\Delta)$ of tensors with support contained in $\Delta\subseteq [d]\times[d]\times [d]$; by combining such sequences, we obtain a universal sequence $\mathcal{T}_d$ such that $\sigma(\mathcal{T}_d)=1$ holds if and only if Strassen's asymptotic rank conjecture [Progr. Math. 120 (1994)] holds for $d$. Finally, we show that the limit $\lim_{d\rightarrow\infty}\sigma(d)$ exists and can be captured as $\lim_{d\rightarrow\infty} \sigma(D_d)$ for an explicit sequence $(D_d)_{d=1}^\infty$ of tensors obtained by diagonalisation of the sequences $\mathcal{U}_d$. As our second result we relate the absence of polynomials of fixed degree vanishing on tensors of low rank, or more generally asymptotic rank, with upper bounds on the exponent $\sigma(d)$. Using this technique, one may bound asymptotic rank for all tensors of a given format, knowing enough specific tensors of low asymptotic rank. Joint work with Mateusz Michałek (U. Konstanz). arXiv: https://arxiv.org/abs/2404.06427
Counting the number of higher dimensional partitions is a hard classical problem. Computing the motive of the Hilbert schemes of points is even harder, and should be seen as the geometric counterpart of the classical combinatorial problem. I will discuss some structural formulas for the generating series of both problems, their stabilisation properties when the dimension grows very large and how to apply all of this to obtain (infinite) new examples of motives of singular Hilbert schemes. This is joint work with M. Graffeo, R. Moschetti and A. Ricolfi.
Characters play a key role in representation theory. Lusztigs character sheaves and Springer theory provide one way to work with characters geometrically. In this talk I will explain how to develop the theory of character sheaves in the context of graded Lie algebras. Graded Lie algebras naturally arise from the Moy-Prasad filtration of p-adic groups. In the graded case interesting Hessenberg varieties arise. Affine bundles over these varieties provide a paving of certain affine Springer fibers. At the end of the talk I will explain how one obtains a complete classification of the cuspidal character sheaves on graded Lie algebras via a near by cycle construction. The new results presented are joint work with Grinberg, Liu, Tsai, and Xue.
A generalized polygon is a point-line incidence structure that includes projective planes (generalized 3-gons). Incidence graphs of generalized m-gons are connected bipartite graphs of diameter m and girth 2m. Associated to any group presentation is a graph called the star graph, which encodes structural information about the group defined by the presentation. Transitional behaviour can occur for groups defined by presentations whose star graph components are incidence graphs of generalized polygons; such presentations are called special. A cyclic presentation of a group is a type of group presentation that admits a cyclic symmetry. In this talk I will discuss joint work with Ihechukwu Chinyere in which we classify the special cyclic presentations.
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