Who we are



Current members


External collaborators


Current Postdoctoral Researchers


Current Graduate Students


  • Dario Antolini

    Supervisors: Alessandro Oneto, Elisa Postinghel
    started in November 2023

  • Enrica Barrilli

    Supervisors: Bernard Mourrain (INRIA), Alessandra Bernardi
    started in November 2024
    (joint with INRIA, U. Côte d'Azur)

  • Oriol Reig Fité

    Supervisors: Alessandra Bernardi, Bernard Mourrain (INRIA)
    started in November 2024
    (joint with Inria U. Côte d'Azur)

  • Otto Schmidt

    Supervisors: Iacopo Carusotto, Simon Telen (MPI Lipzig)
    started in November 2024
    (joint with MPI Leipzig)

Current Master Students

  • Martino Lepri
    Supervisor: Alessandro Oneto
  • Massimiliano Osti
    Supervisor: Elisa Postinghel
  • Ilaria Pelegrinelli
    Supervisor: Alessandro Oneto



Presentation of the Lab @ TensorDay 2025

Former PostDocs

  • Tomasz Mańdziuk

    (2023-2024)

  • Alex Casarotti

    (2021-2023)

  • Maciej Gałązka

    (2024-2025)

  • Pablo Mazòn

    (2024-2025)

Former PhD students


  • Valentina Amitrano

    Digital Quantum Computing for Many-Body Simulations
    Supervisors: Francesco Pederiva, Alessandra Bernardi
    PhD defense (in Physics): 13 December 2023


  • Vincenzo Galgano

    Secant varieties of Spinor varieties and of other generalized Grassmannians
    Supervisors: Alessandra Bernardi, Giorgio Ottaviani
    PhD defense: 18 December 2023


  • Claudia De Lazzari

    Algebraic, geometric and numerical methods for Tensor Network Varieties
    Supervisors: Alessandra Bernardi, Iacopo Carusotto
    PhD defense: 7 December 2022


  • Pierpaola Santarsiero

    Identifiability of small rank tensors and related problems
    Supervisors: Alessandra Bernardi, Edoardo Ballico
    PhD defense: 1 April 2022


  • Reynaldo Staffolani

    Schur apolarity and how to use it
    Supervisor: Alessandra Bernardi
    PhD defense: 14 February 2022

Former Master students

  • Vincenzo Antonio Isoldi
    Schur apolarity and the dual Terracini lemma for flag varieties: a geometric and representation-theoretic approach
    Supervisors: Alessandra Bernardi, Stefano Canino
    A.Y.:2025-2026


  • Jose Enrique Manrique Santillana
    Perineural invasion in pancreatic cancer: clinical significance using multple and SVD-based imputation
    Supervisors: Alessandra Bernardi
    A.Y.:2024-2025


  • Simone Patscheider
    Geometric Measure of Entanglement via Tensor Decomposition
    Supervisor: Alessandra Bernardi
    A.Y.: 2022-2023


  • Surbhi Malhotra
    at CIBIO - Dept. Cellular, Computational and Integrative Biology (U. Trento)
    Multi-omics integrative analysis via tensor decompositions
    Supervisors: Mario Lauria, Alessandro Oneto
    A.Y.: 2022-2023


  • Giulia Bazzucco
    Multivariate function approximation via Chebyshev interpolation and Tucker decomposition
    Supervisors: Alessandro Oneto, Nick Vannieuwenhoven (KU Leuven, Belgium)
    A.Y.: 2021-2022


  • Davide Tezza
    Design automatization of Quantum Machine Learning algorithms
    Supervisors: Massimiliano Incudini (U. Verona & Data Reply), Alessandro Oneto
    A.Y.: 2021-2022


  • Martina Iannacito
    HOSVD for multispectral images. A numerical approach to plant biodiversity estimation
    Supervisors Alessandra Bernardi, Duccio Rocchini
    A.Y.: 2018-2019


  • Daniele Taufer
    at Galileian School of Higher Education
    Symmetric tensor decompositions
    Supervisor: Alessandra Bernardi
    A.Y.: 2016-2017


Masterclass: Tensor decompositions and their applications

by Nick Vannieuwenhoven (Assistant Professor, KU Leuven, Belgium)

Trento, 8-17 November 2022.
nick

Multidimensional datasets, in which data can vary in more than two directions, became popular over the past two decades as computational and storage resources increased along with algorithmic innovations for the processing of such data. Multidimensional data poses several challenges, ranging from their interpretation and the extraction of meaningful insights from them, their processing and visualisation, and their storage and archiving. In this Masterclass, we study tensor decompositions, which are algorithmic techniques designed to tackle the foregoing challenges. Tensor decompositions extend the idea of matrix decompositions (like singular value decomposition, principal component analysis, and nonnegative matrix factorization) as instruments for the analysis of data that varies in only two directions to more directions.

All of the main tensor decompositions will be covered, namely:
- tensor rank decomposition or canonical polyadic decomposition,
- Tucker decomposition,
- tensor train decomposition, and
- hierarchical Tucker decomposition.
For each of these decompositions, we will investigate their definition, their main theoretical properties, the main algorithms for their computation, and a worked-out example of how they can be employed to analyse or process multidimensional datasets.
Detailed contents.
1. Introduction
  - Recap of matrix decompositions
  - Basic tensor notation
  - The tensor product and rank-1 tensors
  - Tensor product basis
  - Tensor product subspace

2. Tucker decomposition
  - Definition
  - HOSVD algorithm
  - Truncation strategies: parallel, sequential
  - Approximation quality
  - Application: hyperspectral image compression

3. Tensor trains and hierarchical Tucker decomposition
  - Definition
  - Nested tensor subspaces
  - SVD-based leaves-to-root and root-to-leaves algorithms
  - Approximation quality
  - Application: accelerating scientific computations

4-5. Tensor rank decomposition
  - Definition
  - Theoretical properties: subspace constraint, ill-posedness, conditioning, identifiability
  - Jennrich's pencil-based algorithm including numerical properties
  - Optimization algorithms: steepest descent, Gauss-Newton, Riemannian Gauss-Newton
  - Extra constraints (smoothness, nonnegativity), regularization
  - Application: full data analysis example for solar power prediction


Lecture 1: Introduction

Lecture 2: Tucker Decomposition

Lecture 3: Tensor Trains Decomposition

Lecture 4: Tensor Rank Decomposition

Lecture 5: Approximation by a Tensor Rank Decomposition

Masterclass: Algebraic statistics and related topics

by Kaie Kubjas (Assistant Professor, AAlto University, Finland)

Trento, 17-21 October 2022.

kaie   intro_algstat_poster

In this course, we will cover selected topics from algebraic statistics including:
  - conditional independence,
  - likelihood inference,
  - graphical models,
  - nonnegative matrix factorizations.
Time permitting further topics will be covered.

At the end of this course, the student can:
  - list topics in algebraic statistics;
  - recognize problems in statistics that are answerable by algebraic methods;
  - assess which algebraic methods are suitable for solving a problem;
  - apply basic algebraic tools to solve a problem.

Masterclass: Advances in solving polynomial equations

by Daniele Taufer (FWO PostDoctoral Researcher, KU Leuven, Belgium)


Povo 1 (Polo Ferrari), Trento
20 April-26 May 2023, four lectures

daniele

Schedule and rooms: (the indicated time is the local one in Trento: CET (Central European Time))
April 20, 8:30-11:30 | Room A217
April 21, 9:30-10:30 | Room A224
May 25, 8:30-11:30 | Room A217
May 26, 9:30-10:30 | Room A224

A huge amount of problems arising from algebra, geometry and number theory are linked to the solution of distinct polynomial systems over prescribed fields. The study of their solutions and the increasing need for root explicit detection for real-world applications have been leading to vibrant research in computational algebra, which has become prominent in both theoretical and applied uses. This masterclass aims at providing the attendees with the basic notion of this discipline, together with several modern results pointing to open research lines. We aim at be covering the advances in both normal form and homotopy continuation methods, which are two of the leading approaches for polynomial root detection. Such different approaches will be explained, compared and tested against explicit examples of practical interest, highlighting their strong points and limitations.

References:

Masterclass: Introduction to Algebraic Vision and Multifocal Tensors

by Kathlén Kohn (KTH Stockholm, Sweden)


Povo 1 (Polo Ferrari), Trento
9-13 October 2023, five lectures, two hours each day

Schedule and rooms: (the indicated time is the local one in Trento: CET (Central European Time))
Monday 9, 11:30 - 13:30 | Room A220
Tuesday 10, 13:30 - 15:30 | Room A110
Wednesday 11, 12:30 - 14:30 | Room A219
Thursday 12, 13:30 - 15:30 | Room A108
Friday 13, 09:30 - 11:30 | Room A218

Registration Form. link
The registration is free, but highly recommended in order to facilitate the communication of information and ZOOM links before and during the masterclass.

Remote participation via ZOOM platform will be available. The link will be shared with the registered participants.

Algebraic Vision is the amalgamation of Algebraic Geometry and Computer Vision. The reconstruction of 3D scenes from 2D images is a classical task in computer vision. We explore the inherent geometric structures of this task and the algebraic methods that are employed in state-of-the-art solvers. At the heart of this are finite rational maps, multifocal tensors, and Euclidean distance minimization over multiview varieties.

kathlen kathlen

Masterclass: Tensor Decompositions and Applications in Multi-Omics Data Analysis

by Neriman Tokcan (U. Massachussets, Boston)


neriman
Povo 1 (Polo Ferrari), Trento
25-29 November 2024, five lectures, two hours each day

REGISTRATION FORM Deadline: 25 October 2024.
The registration is free, but mandatory in order to facilitate communications of information and ZOOM links before and during the masterclass.

Remote participation via ZOOM platform will be available. The link will be shared to the registered participants.
Schedule
  • Monday, Nov 25: 11:30 - 13:30 (Room A108)
  • Tuesday, Nov 26: 15:30 - 17:30 (Room A102)
  • Wednesday, Nov 27: 12:30 - 14:30 (Room A203)
  • Thursday, Nov 28: 10:30 - 12:30 (Room A209)
  • Friday, Nov 29: 11:30 - 13:30 (Room A209)
Description
Omics technologies, including genomics, transcriptomics, proteomics, and metabolomics, have revolutionized biological research by enabling comprehensive, high-throughput analysis of molecular components within cells and organisms. The resulting high-dimensional datasets pose significant analytical challenges, particularly in integrating diverse data types and uncovering complex biological relationships. Tensor-based approaches have emerged as powerful tools for analyzing these high-dimensional omics data, offering advantages over traditional matrix-based methods in capturing complex, multi-way relationships.

This masterclass delves into tensor decomposition techniques and their applications in omics data analysis. Participants will explore the mathematical foundations of tensors, key methods such as CP (CANDECOMP/PARAFAC), Tucker, and tensor train decompositions, as well as probabilistic decompositions. We discuss how tensors can naturally represent multi-dimensional omics datasets and how tensor factorization methods enable dimensionality reduction while preserving important structural information. A comprehensive biological background and an overview of relevant public databases and resources are provided to contextualize the computational methods. Case studies will illustrate the application of tensor methods for tasks such as identifying biological patterns, integrating multiple types of omics data, imputing missing values, and uncovering cell-cell interaction patterns, while demonstrating their impact in systems biology, cancer research, personalized medicine, and drug discovery. Challenges and future directions for tensor-based omics data analysis will also be discussed.

By the end of this masterclass, participants will grasp tensor decomposition techniques, gain practical experience with omics data, and stay updated on the latest trends in tensor-based multi-omics research.
Topics
  • Introduction to Tensors and Tensor Rank
  • Tensor Decomposition Techniques
  • Applications in Multi-Omics and Case Studies
  • Emerging Trends in Tensor Methods and Applications
  • Hands-on Tutorial
  • Tensor operations and decompositions
  • Applying tensor decompositions to real omics datasets
  • Visualization, interpretation, and performance evaluation techniques


No biological background is required.

Computational Requirements
The computational parts of this masterclass will involve hands-on practice with tensor decompositions using the following software and languages:
Python: We will be using popular Python libraries such as Tensorly, NumPy, and SciPy for tensor computations. Participants should come prepared with Python installed (along with the necessary libraries). We will use Jupyter Notebook or Jupyterlab. Prior to the class, participants will be provided with a detailed guide on how to install the necessary Python libraries and how to use Jupyter Notebook/Jupyterlab.
Data sets and code templates will be provided in advance.

Masterclass and PhD course
Tensors and their applications: foundations and recent advances

Lecturer: Jeroen Zuiddam (U. Amsterdam / QuSoft)
Exercise sessions: Maxim van den Berg (Ruhr-U. Bochum / U. Amsterdam)



The course is organized within the Trento-Verona PhD School in Mathematics and the Transdisciplinary Doctoral Program, and is sponsored by the TensorDec Laboratory and Q@TN.
All materials and information for the Masterclass (including ZOOM link) will be available on a private channel on Zulip. All registered participants have received an email invitation: please accept it, as no further updates will be sent by email. If you did not receive the invitation, your registration email address may be incorrect, please check your registration email and contact us if needed (alessandra.bernardi@unitn.it; alessandro.oneto@unitn.it).

Location
Povo 1 (Polo Ferrari), Trento
The lectures will be also streamed online and recorded. PhD students who cannot attend in person can include the course in their study plan.

Schedule
13-24 October 2025, five lectures per week, two hours each day
  • Monday: 11:30 - 13:30 (Room A108)
  • Tuesday: 15:30 - 17:30 (Room A107)
  • Wednesday: 12:30 - 14:30 (15 October: Room A221 | 22 October: Room A210)
  • Thursday: 10:30 - 12:30 (Room A209)
  • Friday: 13:30 - 15:30 (Room A212)
Description
Tensors play a central role in various areas of mathematics, physics and computer science: from constructing fast matrix multiplication algorithms in algebraic complexity theory to the study of entanglement in quantum information. While matrices (order-two tensors) are very well understood, tensors have a much more intricate structure, and much about them is still unknown, despite much interest.

In this lecture series we will discuss foundations and recent advances in the theory of tensors and their applications, building on Strassen’s pioneering perspective developed in his quest to understand the complexity of matrix multiplication. Along the way we will see various applications, important tensor parameters (ranks), and old and new methods and results.

In the first week (October 13-17, 2025), we will focus on foundations, important applications in various areas, the notion of asymptotic rank and Strassen's bold conjecture about it, other notions of rank and their relations, lower bound methods and barriers.

In the second week (October 20-24, 2025), we will dive into more advanced directions including some recent ones: the study of the asymptotic behaviour of tensors via asymptotic spectrum duality and via techniques from representation theory (Schur-Weyl duality and moment polytopes), algebraic geometry and quantum information.

Secure Communication in the Digital Age: Exploring the Synergy between Cryptography, Algebra, and Industrial Mathematics


by Blended Intensive Programmes (BIP) Erasmus+





Dates:
- In-person meeting at Adam Mickiewicz University in Poznań, Poland in the week 25.09 - 29.09.2023.
- In the period of 30.09 - 24.10.2023 students will be engaged with preparation of the projects.
- The final 3-day virtual workshop will take place during the period 25.10-27.10.2023.

Topics:
Cryptography, Elliptic Curves, Lattices, Quantum Computing, Applications of algebra in other fields, Cryptanalysis, Cybersecurity.

Target students:
Master level students with broad interest in modern development in algebra, algebraic geometry, computer cryptography, quantum computing and applications to applied and industrial mathematics.

Visit the website for more details: https://bipcrypt.wmi.amu.edu.pl/#
The interested students are invited to send by June 15 a manifestation of interest to stage.dipmat@unitn.it with a motivation letter and the list of successfully passed exams.
Students may refer to this web page https://international.unitn.it/outgoing/eciu-university-challenges-and-micro-modules-0
The sections students can refer to are documents, economic support and recognition. They will receive the grant after the mobility.

Initiative supported by Math Department with the TensordDec Laboratory.
Reference Professor for the initiative: Alessandra Bernardi.

Industrial & Public AI Challenge: apply Artificial Intelligence techniques to real data provided by enterprises and public administrations


by Fondazione Hub Innovazione Trentino (HIT)





Applications are now open for the Industrial & Public AI Challenge, the new innovation contest by Fondazione Hub Innovazione Trentino (HIT) and the University of Trento. The initiative targets university students, recent graduates and PhD students interested in working on improving decision-making and operational processes through the application of Artificial Intelligence solutions.

During the Challenge, starting in September and running until December 2023, teams of students from different departments of the University of Trento will work with the support of researchers and startups on solving optimization or problem-solving challenges in one or more production processes (Industrial AI Challenge), or on optimizing processes and event forecasting that affect the organization and delivery of public services, for those who choose to work on challenges brought by the Public Administration (Public AI Challenge).

Solvers will apply Data Science and Machine Learning techniques to datasets supplied by enterprises and PA to create predictive models and data-driven solutions to public and industrial Challenges related to, for example, automated product quality control, predictive maintenance, production planning optimization, and supply chain optimization.

Interested students should apply by filling out this form by June 30, 2023.

All info about the challenge and how to apply, included a video of presentation of the challenge by Nicola Doppio (HIT): link.
For more information write to us at: ai-challenge@trentinoinnovation.eu

The Industrial AI Challenge is an initiative organized by Fondazione Hub Innovazione Trentino (HIT) , in collaboration with the University of Trento , FBK - Fondazione Bruno Kessler, Trentino Sviluppo , Confindustria Trento , DIH - Digital Innovation Hub Trentino Alto-Adige / Südtirol and PAT - Autonomous Province of Trento.

Training

Courses

Tensor Decomposition for Big Data Analysis
by A. Bernardi

Master level course in Data Science, Mathematics and Statistics for Life and Social Sciences, Mathematics for Life and Data Sciences.
An introduction to big data science from tensor decompositons perspective.


Geometry and Topology for Data Analysis
by F. Fallucca (previously, by A. Oneto)

Master level course in Data Science, Mathematics and Statistics for Life and Social Sciences, Mathematics for Life and Data Sciences.
A first course in algebraic topology, numerical algebraic geometry, with a view towards applications in data analysis.


Masterclasses

Every year we offer a masterclass to our students: a short and intense introduction to state-of-the-art topics within the area of applications of algebra and geometry. The lectures are given by an international expert invited by the lab.

2025, October 13-24. Tensors and their applications: foundations and recent advances , by Jeroen Zuiddam (U. Amsterdam / QuSoft)
2024, November 25-29. Tensor Decompositions and Applications in Multi-Omics Data Analysis , by Neriman Tokcan (U. Massachussets, Boston, USA)
2023, October 9-13. Introduction to Algebraic Vision and Multifocal Tensors, by Kathlén Kohn (KTH Stockholm, Sweden)
2023, April 20-21, May 25-26. Advances in Solving Polynomial Equations, by Daniele Taufer (KU Leuven, Belgium)
2022, October 17-21. Algebraic Statistics and related topics, by Kaie Kubjas (Aalto U.)
2021, November 8-17. Tensor Decomposition and their applications, by Nick Vannieuwenhoven (KU Leuven)

Check here for more details.

Lab's Activities

Masterclasses

We organize every year at least one masterclass to offer our students a short and intense introduction on state-of-the-art topics within the area of applications of algebra and geometry. The lectures are given by an international expert invited by the lab.
Check here past and future schedule.


TensorDays

Every year, we organize a day to present the laboratory to Master students. During these meetings we introduce our activities and our areas of research. We also invite guests such as former students and members of the laboratory as well as collaborators.
Check here past and future schedule.


TensorDec Lab Seminars

We periodically organize seminars that are aimed not only to our colleagues researchers but also to master and PhD students. Here the page dedicated to past and future seminars.
Check here past and future schedule.



Related activities and events

This is a list of activities within the lab's scope that involve the active participation of some of its members.


Running activities

Applied Algebraic Geometry Seminar

Joint cycle of Seminars between Bologna, Ferrara, Firenze, Siena, Trento, Torino on Applied Algebraic Geometry in particular on arguments related to tensor decompositions. The seminars are held between Bologna e Firenze, every three weeks.
Organisers: A. Bernardi, E. Carlini, L. Chiantini, C. Flavi, A. Gimigliano, M. Mella, G. Ottaviani


Information, Algebra and Geometry Workgroup

Joint cycle of interdisciplinary meetings at the University of Trento aimed at laying the foundations for a common language between Geometry, Algebra and Physics starting from Quantum Information.
Organizers: A. Bernardi, I. Carusotto, F. Pederiva, F. Hauke



Past activities

Trento, 30 June - 2 July 2025
INABAG Conference 2025: with a day in honour of Sandro Gimigliano

The conference was the second event of the Italian Network for Applied and Birational Algebraic Geometry (INABAG), a group of geometers and algebraists based in Italy, whose common research interests lie within the theory of tensors from the commutative algebraic, the birational algebro-geometric, and the applied viewpoints. The INABAG Conference 2025 gathered international experts to present recent achievements as well as discuss future lines of research. The conference has also been an occasion to celebrate the remarkable career and contributions of Sandro Gimigliano, a crucial member and guiding figure of the Network, honouring his 70th birthday. The special session highlighted Sandro Gimigliano’s work, innovative ideas, and lasting impact on the field of tensor decomposition and secant varieties through some of his collaborators and former students.
Organizers: E. Guardo, A. Oneto, E. Postinghel, P. Santarsiero


MPI MiS Leipzig, 28-30 November 2022
Pfaffians, Tensors, and Applications

The intention of this meeting is to encourage and facilitate the interaction between different communities engaging with Pfaffians or other tensor-related structures. Particular emphasis is put on explicit methods and applications to different areas of mathematics or other sciences. The goals of the workshop are to connect different communities, exchange knowledge, and inspire one another. The workshop features invited and contributed talks, as well as "open mic" and free discussion sessions in order to promote more informal discussions.
Organizers: D. Faenzi, J. Maglione, M. Stanojkovski


Levico Terme, 26-28 July 2021
Tensor Networks: Quantum Physics, Geometry and Applications

The goal of the workshop is to develop the connections between the mathematics and physics communities working on tensor network. It is aimed at researchers from both areas that are active on the topic, from experienced PhD students over postdocs to permanent faculty. As a key novelty, it will be a hands-on event with a limited number of talks and several sessions of active work. During these sessions, participants will distribute themselves in small groups and will actively work on a specific cutting-edge research problem. Each problem will be proposed by a world expert who will be in charge of directing the activities of the group.Thus, we hope to foster strong scientific collaborations that last well beyond the workshop.
Organizers: A. Bernardi, I. Carusotto, P. Hauke


TensorDays


We regularly organize a day to present the laboratory to Master students. During these meetings we introduce our activities and our areas of research. We also invite guests such as former students and members of the laboratory as well as collaborators.

TensorDay 2025

29 October 2025, Povo2, Room B108, h.09.00 - h.11.30

09.00 - 09.30
Welcome and Opening
Introduction to the TensorDec Laboratory and its mission.


09.40 - 10.20
Dino Zardi (Università di Trento)
Tensors meet... Fluid Mechanics


Bio. Dino Zardi is a full professor of Atmospheric Physics at the Department of Civil, Environmental and Mechanical Engineering (DICAM) of the University of Trento. He holds a Master's degree in Physics from the University of Bologna and a PhD degree in Hydrodynamics from the University of Genoa.

10.30 - 11.10
Claudia De Lazzari (QTI srl)
Tensors meet... Quantum Cryptography


Bio. Claudia De Lazzari is a Quantum Cryptographer at QTI SRL. She holds a Master's degree in Mathematics from the University of Padova and a PhD degree from the University of Trento.

11.15 - 11.30
Q&A with the speakers and the Lab members

TensorDay 2024

9 December 2024, Polo Ferrari - Povo1, Room A108, h.09.30 - h.12.30

09.30 - 09.40
Welcome and Opening

09.40 - 10.05
Research Group Presentation
Lab members have showcased research opportunities, spotlighting diverse cutting-edge work.



10.10 - 10.40
Daniele Taufer (KU Leuven)
Why do we like to shatter tensors?


Abstract. I will give a gentle introduction to tensors, and discuss why breaking them down has been attracting considerable research interest. Surprisingly enough, many modern scientifc challenges arise from, or can be rephrased as, the problem of decomposing tensors into simpler pieces. I will precisely describe what this problem consists of when the considered tensors are symmetric, highlighting its beautiful algebraic nature. Finally, I will present some instances of compelling applications of tensor decomposition, involving computational root fnding and complexity theory.

Bio. Daniele is an FWO postdoctoral fellow (project number: 12ZZC23N) in the NUMA research unit at KU Leuven. Since his master's years, he has actively collaborated with the members of the TensorDec lab.


10.40 - 11.10
Iacopo Carusotto (CNR-INO and U. Trento)
Tensors: the physicist's all-rounder


Abstract. In this talk I will give an overview of a few areas of the physical sciences where tensors and their decomposition appear as a crucial tool for important problems of contemporary research. Special attention will be paid to tensor networks which are heavily used by researchers at the BEC Center as a powerful tool to describe the many-body physics of quantum fuids of atoms or photons. I will sketch the key ideas behind these applications and the mathematical structure of some main problems. I will fnally outline a few promising avenues for new math-physics collaborations.

Bio. Iacopo works as Research Director (Dirigente di Ricerca) at INO-CNR BEC Center (Trento). He is also a partner in TENORS: A Marie Skłodowska-Curie Doctoral Network for 2024-2027 joint with Trento's Math Dept.


11.30 - 12.00
Pierpaola Santarsiero (U. Bologna)
Decoding tensors through a geometric lens


Abstract. Tensors are fundamental objects in various areas of mathematics and applied sciences, including physics and data analysis. In this talk, we will explore the world of tensors and their decompositions from a geometric perspective. We will focus on the identifability problem — determining whether a tensor can be uniquely decomposed — a challenge that is not only crucial for its practical applications but also represents a signifcant theoretical problem on its own. Finally, I will provide some prime examples to illustrate the power of geometric tools in addressing tensor-related questions and challenges arising from other mathematical areas.

Bio. Pierpaola is a postdoctoral researcher within the Italian Network for Applied and Birational Algebraic Geometry INABAG. She received her PhD degree from the University of Trento and has been one of the first PhD students from the TensorDec Lab to graduate.


12.00 - 12.30
Discussion Panel
Meet our younger members.

TensorDay 2023

21 November 2023, Polo Ferrari - Povo1

09.00 - 09.15
Welcome and Opening

09.15 - 10.00
Research Group Presentation
The members will present the research lines pursued within the Lab.

10.00 - 11.00
Simon Telen (MPI MiS Leipzig, DE)
Tensors, polynomials and applications

Abstract. Tensors are multidimensional arrays. They are useful in many applications. I will discuss tensors and their associated geometric objects from the point of view of computational algebraic geometry. I will show how they appear in algebraic statistics, and discuss how to decompose tensors using algebraic techniques. I will also show how tensor decomposition is used to solve polynomial equations.
Short Bio. Group Leader of the Numerical Nonlinear Algebra research group at the Max-Planck Institute for Mathematics in the Sciences in Leipzig; European Young Academy Fellow; Partner in TENORS: A Marie Skłodowska-Curie Doctoral Network for 2024-2027 joint with Trento's Math and Physics Dept.

11.15 - 11.45
Martina Iannacito (KU Leuven, BE)
Discovering tensors: their challenges and applications

Abstract. Tensors, regarded as mathematical tools, have gained widespread utilization across diverse fields due to their practical advantages. In various disciplines, researchers employ tensor factorization techniques to tackle computationally challengin problems, examine large datasets, and formulate better descriptions of intricate phenomena. In this presentation, following the evolution of my tensor studies, I will briefly introduce some classical tensor methods, focusing on an applicative problem where they find use, ranging from remote sensing to multilinear algebra and numerical simulation. Particular attention will be on understanding how and why a tensor technique is chosen, depending on the field and the problem to address. The talk aims to illustrate the benefits and drawbacks of these choices, which each discipline faces differently. By elucidating these hurdles, the presentation seeks to provide insights into current research directions arising from real-world, computational or applied problems.
Short Bio. Postdoctoral researcher from the Electrical Engineering Department at KU Leuven and former master student of the TensorDec laboratory, with a thesis on biodiversity estimation using tensor decomposition.

11.45 - 12.15
Industrial Partners
Learn about potential traineeships and master thesis projects at:
BlueTensor (Trento, IT)
Equal1 (Dublin, IR)

TensorDec Lab Seminars

A.Y. 2025-2026


Working Group Meetings
A series of periodic meetings where some of us present some topic that they are studying to the rest of the group

Thursday 12 March 2026
Vincenzo Antonio Isoldi
Flag Varieties: From Minimal Orbits to a Dual Terracini Lemma

Wednesday 25 February 2026
Stefano Canino
A new bound on the rank of tensor product of W-states

Monday 9 February 2026
Dario Antolini
Cayley-Menger varieties, Euclidean measurement sets and flattenable graphs

Wednesday 17 December 2025
Mima Stanojkovski
Tensor invariants for the group isomorphism problem

Thursday 30 October 2025
Enrica Barrilli
Computing Generalized Additive Decompositions of symmetric tensors

Thursday 23 October 2025
Sara Marziali (U. Siena)
Product of tensors and description of the network for matrix multiplication via Strassen's algorithm

Wednesday 15 October 2025
Oriol Reig Fité
Local cactus rank and structure tensors of Artinian Gorenstein algebras


within the series Mathematics for Machine Learning
Tuesday 9 September 2025
Guido Francisco Montúfar Cuartas (BUCLA Mathematics and Statistics & Data Science)
Geometric View of Restricted Boltzmann Machines

A.Y. 2024-2025


Working Group Meetings
A series of periodic meetings where some of us present some topic that they are studying to the rest of the group

Thursday 3 April 2025
Oriol Reig Fité
An extension theorem on truncated Henkel operators

Thursday 3 April 2025
Oriol Reig Fité
An extension theorem on truncated Henkel operators

Tuesday 18 March 2025
Alberto Tabarelli De Fatis
Tensor Networks in strongly interacting one-dimensional quantum systems

Thursday 13 February 2025
Otto T.P. Schmidt
Quantum many-body theory and tensors

Thursday 30 January 2025
Jose Enrique Manrique Santillana
Perineural Invasion in Pancreatic Cancer: Clinical significance using Multiple and SVD-Based Imputation

Thursday 19 December 2024
Marco Calderini
Perfect Nonlinear Functions: Combinatorial and Geometric Aspects

Thursday 5 December 2024
Daniele Taufer (KU Leuven)
Polynomials interpolating generalizing additive decomposition

Thursday 21 November 2024
Pablo González-Mazòn
On the generic identifiability of binary tensors

Thursday 17 October 2024
Oriol Reig Fité
Introduction to Berkovich spaces

Thursday 3 October 2024
Dario Antolini
Hadamard ranks and tropical geometry


within the joint series ALgebra+GeomSem
Thursday 29 May 2025
Marzio Mula (UniBw Munich, Germany), Federico Pintore, and Daniele Taufer (KU Leuven, Belgium)
The Hessian of elliptic curves

Friday 16 May 2025
Nick Vannieuwenhoven (KU Leuven, Belgium)
Hadamard-Hitchcock decompositions


within the series GeomSem
Thursday 10 October 2024
Khazhgali Kozhasov (U. Nice, France)
On the minimal ED degree of Segre-Veronese varieties

A.Y. 2023-2024


Working Group Meetings
A series of periodic meetings where some of us present some topic that they are studying to the rest of the group

Wednesday 29 May 2024
Sara Marziali (U. Siena)
Product of Tensors and Description of Networks

Wednesday 15 May 2024
Dario Antolini
Hadamard products of linear spaces and tropical geometry

Monday 6 May 2024
Pablo González-Mazòn
Center point theorem for polynomial superlevel sets

Thursday 2 May 2024
Maciej Gałązka
Border rank of monomials: an overview

Wednesday 10 April 2024
Gilberto Bini (U. Palermo, Italy)
Grassmann Tensors

Wednesday 3 April 2024
Asma Momal
Introduction to computer vision

Wednesday 20 March 2024
Alessandro Oneto
Tensors in Rigidity Theory and Cayley-Menger varieties

Wednesday 13 March 2024
Dario Antolini
Rigidity theory and low-rank matrix completion


Series: Geometry and Topology for Data Analysis
This series of seminars is part of the homonym master level course.

Thursday 24 May 2024 (Povo 1 / online)
Andrea Guidolin (KTH)
Methods for topological analysis of biological data

Thursday 23 May 2024 (Povo 1 / online)
Chiara Romanengo (CNR-IMATI)
Recognition and representation of curves and surfaces in digital models via the Hough transform

Thursday 16 May 2024 (Povo 1 / online)
Angelica Torres (MPI Leipzig)
Algebraic techniques in the Structure-from-Motion problem

Friday 3 May 2024 (Povo 1 / online)
Massimiliano Datres (UniTn/FBK)
Dimensionality Reduction: UMAP and applications, II

Thursday 2 May 2024 (Povo 1 / online)
Massimiliano Datres (UniTn/FBK)
Dimensionality Reduction: UMAP and applications, I



within the series GeomSem
Wednesday 29 November 2023
Alexander Taveira Blomenhofer (CWI Amsterdam, The Netherlands)
Nondefectivity of GL-invariant secant varieties

A.Y. 2022-2023


Working Group Meetings
A series of periodic meetings where some of us present some topic that they are studying to the rest of the group

Wednesday 3 May 2023
Mima Stanojkovski
Studying p-groups via Pfaffians

Wednesday 12 April 2023
Vincenzo Galgano
Equivariant Euler characteristic on Permutohedral Varieties

Wednesday 29 March 2023
Alessandro Oneto
Waring decompositions of special ternary forms with different Hilbert functions

Wednesday 15 March 2023
Tomasz Mańdziuk
Limits of saturated ideals

Wednesday 1 March 2023
Valentina Amitrano
Explicit tensor formalism for Quantum Computing

Tuesday 14 February 2023
Alex Casarotti
Identifiability of sums of powers

Tuesday 7 February 2023
Cosimo Flavi (U. Firenze, Italy)
Ranks of powers of quadratic forms

Friday 20 January 2023
Vincenzo Galgano
Identifiability and singular locus of secant varieties to Grassmannians



Series: Geometry and Topology for Data Analysis
This series of seminars is part of the homonym master level course.

Thursday 11 May 2023
Sara Scaramuccia (U. Roma Tor Vergata)
Data Classification via Persistent Homology

Wednesday 10 May 2023
Monica Moroni (FBK)
Dimensionality Reduction: UMAP and applications, II

Thursday 4 May 2023
Timhoty Duff (U. Washington)
Solving camera relative pose problems with homotopy continuation

Wednesday 3 May 2023
Orlando Marigliano (KTH Stockholm)
Algebraic Statistical Models

Thursday 27 April 2023 (Povo1)
Monica Moroni (FBK)
Dimensionality Reduction: UMAP and applications, I


within the series Mathematics for Data Science, Artificial Intelligence and Machine Learning
Wednesday 30 November 2022
Neriman Tokcan (Broad Institute of MIT & Harvard)
Non-negative consensus tensor factorization and applications in genomics


within the series MathBites Trento
Tuesday 22 November 2022
Marta Casanellas (UPC Barcelona)
Phylogenetics from an algebraic perspective

A.Y. 2021-2022


Thursday 19 May 2022
Daniele Taufer (CISPA Helmotz Center, Saarbrücken)
An invitation to computational and symbolic algebra


Series: Geometry and Topology for Data Analysis
This series of seminars is part of the homonym master level course.

Thursday 26 May 2022
Martina Scolamiero (KTH Stockholm)
Topological data analysis methods for neuroscience

Wednesday 25 May 2022
Paul Braiding (U. Osnabrück)
Numerical Algebraic Geometry

Thursday 19 May 2022
Nicole Bussola (OROBIX Life, Bergamo)
Topological applications for pattern discovery in precision medicine

Thursday 12 May 2022
Ulderico Fugacci (CNR-IMATI Genova)
Persistent homology for shape comparison

Wednesday 11 May 2022
Marina Garrote-López (U. Alaska Fairbanks)
Algebraic degrees of phylogenetic varieties


within the series Mathematics for Data Science, Artificial Intelligence and Machine Learning
Wednesday 20 April 2022
Daniele Castellana (U. Pisa)
A Tensor Framework for Learning in Structured Domains
video

A.Y. 2020-2021


Online PhD seminar
Our graduate students collected all Italian graduate students on topics related to tensor decompositions in a weekly cycle of online seminars where graduate students have the possibility to share their latest discoveries or the problems they are currently studying in front of the senior members of the Italian community.
Organizers: C. Delazzari, V. Galgano, P. Santarsiero, R. Staffolani

Masterclasses

Masterclass: Tensors and their applications: foundations and recent advances

Trento, 13-24 October 2025.
Held by J. Zuiddam (U. Amsterdam, The Netherlands).


Tensors play a central role in various areas of mathematics, physics and computer science: from constructing fast matrix multiplication algorithms in algebraic complexity theory to the study of entanglement in quantum information. While matrices (order-two tensors) are very well understood, tensors have a much more intricate structure, and much about them is still unknown, despite much interest. In this lecture series we will discuss foundations and recent advances in the theory of tensors and their applications, building on Strassen's pioneering perspective developed in his quest to understand the complexity of matrix multiplication. Along the way we will see various applications, important tensor parameters (ranks), and old and new methods and results.

Masterclass: Tensor Decompositions and Applications in Multi-Omics Data Analysis

Trento, 25-29 November 2024.
Held by N. Tokcan (U. Massachussets, Boston, USA).


Omics technologies, including genomics, transcriptomics, proteomics, and metabolomics, have revolutionized biological research by enabling comprehensive, high-throughput analysis of molecular components within cells and organisms. The resulting high-dimensional datasets pose significant analytical challenges, particularly in integrating diverse data types and uncovering complex biological relationships. Tensor-based approaches have emerged as powerful tools for analyzing these high-dimensional omics data, offering advantages over traditional matrix-based methods in capturing complex, multi-way relationships.

Masterclass: Introduction to Algebraic Vision and Multifocal Tensors

Trento, 9-13 October 2023.
Held by K. Kohn (KTH Stockholm, Sweden).


Algebraic Vision is the amalgamation of Algebraic Geometry and Computer Vision. The reconstruction of 3D scenes from 2D images is a classical task in computer vision. We explore the inherent geometric structures of this task and the algebraic methods that are employed in state-of-the-art solvers. At the heart of this are finite rational maps, multifocal tensors, and Euclidean distance minimization over multiview varieties.

Masterclass: Advances in solving polynomial equations

Trento, 20-21 April, 25-26 May 2023.
Held by D. Taufer (KU Leuven, Belgium).


A huge amount of problems arising from algebra, geometry and number theory are linked to the solution of distinct polynomial systems over prescribed fields. The study of their solutions and the increasing need for root explicit detection for real-world applications have been leading to vibrant research in computational algebra, which has become prominent in both theoretical and applied uses. This masterclass aims at providing the attendees with the basic notion of this discipline, together with several modern results pointing to open research lines. We aim at be covering the advances in both normal form and homotopy continuation methods, which are two of the leading approaches for polynomial root detection. Such different approaches will be explained, compared and tested against explicit examples of practical interest, highlighting their strong points and limitations.

Masterclass: Algebraic Statistics and related topics

Trento, 17-21 October 2022.
Held by K. Kubjas (Aalto U., Finland).


In this course, we will cover selected topics from algebraic statistics including conditional independence likelihood inference, graphical models, nonnegative matrix factorizations. Time permitting further topics will be covered.
At the end of this course, the student can:
  • list topics in algebraic statistics
  • recognize problems in statistics that are answerable by algebraic methods
  • assess which algebraic methods are suitable for solving a problem
  • apply basic algebraic tools to solve a problem.

Masterclass: Tensor decompositions and their applications

Trento, 8-17 November 2021.
Held by N. Vannieuwenhoven (KU Leuven, Belgium).


Multidimensional datasets, in which data can vary in more than two directions, became popular over the past two decades as computational and storage resources increased along with algorithmic innovations for the processing of such data. Multidimensional data poses several challenges, ranging from their interpretation and the extraction of meaningful insights from them, their processing and visualisation, and their storage and archiving. In this Masterclass, we will study tensor decompositions, which are algorithmic techniques designed to tackle the foregoing challenges. Tensor decompositions extend the idea of matrix decompositions (like singular value decomposition, principal component analysis, and nonnegative matrix factorization) as instruments for the analysis of data that varies in only two directions to more directions.

The slides of the course can be found here.

Research Interests

We collect here some of the main research lines of the group with some recent literature. Other topics that we have been recently discussing can be found in the series of Working Group Meetings whose titles are listed within the Lab Seminars.

This page has to be considered as work in progress, if you want to know more let us know!

Additive decompositions of polynomials and identifiability of Gaussian mixture models
The moments of Gaussian mixture models are higher degree polynomials that can be represented as polynomials in the linear forms associated to the mean vectors and the quadrics defined by the covariance matrices. The study of such additive decompositions can be used to find theoretical and algorithmic results about such algebraic statistical models. In the particular case of centered Gaussian mixture models, such decompositions are sums of powers of quadrics.

Recent literature:
- Blomenhofer, Alexander Taveira, et al. "Identifiability for mixtures of centered Gaussians and sums of powers of quadratics." Bull. of LMS, to appear (2023).
- Casarotti, Alex, and Elisa Postinghel. "Waring identifiability for powers of forms via degenerations." arXiv preprint arXiv:2301.05276 (2023).


Defectivity of Segre-Veronese varieties
A classical question in algebraic geometry regards the classification of defective varieties, i.e., varieties whose secant varieties have dimension smaller than the expected. In the last decades, this question regained a lot of attention thanks to the Alexander-Hirschowitz Theorem (1995) which completely classified Veronese varieties and the relation with tensor decompositions, indeed, secant varieties of Veronese, Segre and Segre-Veronese varieties are parametrized by tensors of bounded rank. A challenging question is to extend the classification of defective varieties to the latter families of varieties.

Recent literature:
- Ballico, Edoardo, Alessandra Bernardi, and Maria Virginia Catalisano. "Higher Secant Varieties of Pn x P1 Embedded in bi-degree (a,b)." Communications in algebra 40.10 (2012): 3822-3840.
- Bernardi, Alessandra, et al. "The Hitchhiker guide to: Secant varieties and tensor decomposition." Mathematics 6.12 (2018): 314.
- Galuppi, Francesco, and Alessandro Oneto. "Secant non-defectivity via collisions of fat points." Advances in Mathematics 409 (2022): 108657.
- Laface, Antonio, and Elisa Postinghel. "Secant varieties of Segre-Veronese embeddings of." Mathematische Annalen 356.4 (2013): 1455-1470.


Hadamard products of Tensors and Restricted Boltzmann Machines
Restricted Boltzmann Machines are graphical statistical models whose joint probability distributions correspond to Hadamard products, i.e., coefficient-wise products, of sums of rank-one tensors. Even if they are a natural generalization of the classical tensor rank decomposition, from the point of view of algebraic statistics these models started to be studied only recently.

Recent literature:
- Montúfar, G. (2018). Restricted boltzmann machines: Introduction and review. In Information Geometry and Its Applications: On the Occasion of Shun-ichi Amari's 80th Birthday, IGAIA IV Liblice, Czech Republic, June 2016 (pp. 75-115). Springer International Publishing.
- Cueto, María Angélica, Jason Morton, and Bernd Sturmfels. "Geometry of the restricted Boltzmann machine." Algebraic methods in statistics and probability II 516 (2010): 135-153.
- Oneto, A. and Vannieuwenhoven, N. "Hadamard-Hitchcock Decompositions of Tensors", in preparation.


Strassen's additivity of the tensor rank
It is an interesting question to determine whether the rank of the sum of two tensors in independent vector spaces equals the sum of their individual ranks. A positive answer was previously known as Strassen's conjecture until Shitov showed that counterexamples exist, asymptotically, for very large tensor spaces. On the other hands additivity holds for small three-factor tensors and, under certain conditions in the symmetric case and for monomials.

Recent literature:
- Buczynski, J.; Postinghel, E.; Rupniewski, F., "On Strassen's rank additivity for small three-way tensors" On Strassen's rank additivity for small three-way tensors". SIAM J. Matrix Anal. Appl. 41 (2020), no. 1, 106-133
- Carlini, E.; Catalisano, M. V.; Geramita, A. V., "The solution to the Waring problem for monomials and the sum of coprime monomials". J. Algebra 370 (2012), 5-14.
- Carlini, E.; Catalisano, M. V.; Oneto, A., "Waring loci and the Strassen conjecture". Adv. Math. 314 (2017), 630-662.
- Shitov, Y., "Counterexamples to Strassen's direct sum conjecture", Acta Math. 222 (2019), no. 2, 363-379.


Studying p-groups via their intrinsic geometry
Finite groups are understood externally via their simple composition factors (Jordan-Hoelder) and internally via their Sylow p-subgroup (thanks to Sylow theory). Though the classification of finite simple groups is now complete, the classification of finite p-groups is far out of reach. In the study of general questions about finite p-groups it is frequently beneficial to focus on groups in natural families. Often, this yields additional geometric information, as for instance algebraic varieties (such as Pfaffians) attached to the natural tensor structure. This is helpful in the determination of group invariants like the number of conjugacy classes, faithful dimensions, automorphism group sizes, and the number of immediate descendants.

Recent literature:
- Marcus du Sautoy and Michael Vaughan-Lee. "Non-PORC behaviour of a class of descendant p-groups." Journal of Algebra 361:287-312, 2012.
- Eamonn O'Brien and Christoper Voll. "Enumerating classes and characters of p-groups." Transactions of the American Mathematical Society 367(11):7775-7796, 2015.
- Mohammad Bardestani, Keivan Mallahi-Karai, and Hadi Salmasian. "Kirillov's orbit method and polynomiality of the faithful dimension of p-groups." Compositio Mathematica, 155(8):1618-1654, 2019.
- Mima Stanojkovski and Christopher Voll. "Hessian matrices, automorphisms of p-groups. and torsion points of elliptic curves." Mathematische Annalen 381, 593-629 (2021).
- Tobias Rossmann and Christopher Voll. "Groups, graphs, and hypergraphs: average sizes of kernels of generic matrices with support constraints." arXiv preprint arXiv:1908.09589 (2019), to appear in Memoirs of the American Mathematical Society.
- Tobias Rossmann. "On the enumeration of orbits of unipotent groups over finite fields." arXiv preprint arXiv:2208.04646 (2022)
- Joshua Maglione and Mima Stanojkovski. "Smooth cuboids in group theory." arXiv preprint arXiv:2212.03941 (2022)