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15.07.24 19.07.24

Summer School in Algebraic Combinatorics

Combinatorics is the study of finite and discrete structures. Starting from fundamental questions of ordering, decomposition and structuring of finitely many objects or states, combinatorics represents the nanotechnology of mathematics and its applications. Due to its interdisciplinarity, it is a central mathematical research area with influence across disciplinary boundaries, and hence plays a key role for Mathematics in the Sciences.
This summer school focuses on Algebraic Combinatorics. This branch employs methods of abstract algebra, notably group theory, representation theory and algebraic geometry, in various combinatorial contexts and, conversely, applies combinatorial techniques to problems in algebra. Key players include matroids, polytopes, hyperplane arrangements, root systems, generating functions, posets and lattices, symmetric functions, and Young tableux.
Three outstanding speakers (Chris Eur, Greta Panova and Vic Reiner) will present short courses on current topics in Algebraic Combinatorics. The lectures will take place from Monday afternoon to Friday afternoon.
This school is supported by the new DFG Priority program SPP "Combinatorial Synergies" (see the link below).

Registration is already closed.

Chris Eur: Log-concave rainbows and where to find them
Many combinatorial objects admit "signed" or more generally "colored" versions. We will study some general tools in combinatorial algebraic geometry that one can use to deduce numerical properties about these "colored" combinatorial objects, drawing upon joint works with Emily Clader, Chiara Damiolini, Alex Fink, Daoji Huang, Matt Larson, Shiyue Li, and Hunter Spink.

Greta Panova: Computational Complexity in Algebraic Combinatorics
Algebraic Combinatorics studies objects and quantities originating in Algebra, Representation Theory and Algebraic Geometry via combinatorial methods, finding formulas and neat interpretations. Some of its feats include the hook-length formula for the dimension of an irreducible symmetric group (\(S_n\)) module, or the Littlewood-Richardson rule to determine multiplicities of GL irreducibles in tensor products. Yet some natural multiplicities elude us, among them the fundamental Kronecker coefficients for the decomposition of tensor products of \(S_n\) irreducibles, and the plethysm coefficients for compositions of GL modules. Answering those questions could help Geometric Complexity Theory towards establishing lower bounds for the far-reaching goal to show that \(P \neq NP\).
In this minicourse, we will discuss how Computational Complexity Theory provides a theoretical framework for understanding what kind of formulas or rules we could have. We will introduce the basic concepts from computational complexity theory (including complexity classes, complete problems, reductions etc) and the important objects from Algebraic Combinatorics (Specht modules, Schur functions, Littlewood-Richardson and Kronecker coefficients etc). We will formulate our problems in terms of computational complexity and discuss some known results. Finally, as a proof of concept we show that the square of a symmetric group character could not have a combinatorial interpretation.

Vic Reiner: The Koszul Property in Algebraic Combinatorics
Many graded algebras appearing in Algebraic Combinatorics turn out to have the Koszul Property, which is defined algebraically. This property has interesting combinatorial consequences for the Hilbert series of the algebra, and naturally leads one to study its partner, called its Koszul dual algebra.
Examples of Koszul algebras include polynomial algebras, exterior algebras, Stanley-Reisner rings of posets and flag complexes, Orlik-Solomon algebras for supersolvable hyperplane arrangements, Veronese and Segre rings, Hibi rings, Chow rings of matroids, and partial commutation monoid algebras.
In addition to these examples, this course will discuss basic theory of Koszul algebras: their combinatorial properties, constructions, and their interaction with topics such as affine semigroup rings, walks in digraphs, representation stability, unimodality, log-concavity, the Polya frequency property and the Charney-Davis-Gal conjecture.

Speakers

Chris Eur

Harvard University Homepage

Greta Panova

University of Southern California Homepage

Vic Reiner

University of Minnesota Homepage

Program

Participants

Mudit Aggarwal

University of British Columbia

Ibrahim Ahmad

RWTH Aachen University

Mahmud Akelbek

Weber State University

Robert Angarone

University of Minnesota

George Balla

Technische Universität Berlin

Esther Banaian

Aarhus University

Barbara Betti

Max Planck Institute for Mathematics in the Sciences

Alessio Borzì

Max Planck Institute for Mathematics in the Sciences

Sarah Brauner

University of Quebec at Montreal

Anouk Brose

UC Davis

Elisabeth Bullock

Massachusetts Institute of Technology

Veronica Calvo Cortes

University of Oxford

Elise Catania

University of Minnesota

Herman Chau

University of Washington

Xiangying Chen

Otto von Guericke Universität Magdeburg

Ariana Chin

University of California, Los Angeles

Annika Christiansen

University of Oregon

Giulia Codenotti

FU Berlin

Richard (Rick) Danner

University of Vermont

Spencer Daugherty

North Carolina State University

Sebastian Debus

TU Chemnitz

Sebastian Degen

University of Bielefeld

Chinmay Dharmendra

University of Connecticut

Yassine El Maazouz

RWTH Aachen University

Sofía Errázuriz

Pontifical Catholic University of Chile

Maria Esipova

University of British Columbia

Chris Eur

Harvard University

Mieke Fink

University of Bonn

Luca Fiorindo

Università di Genova

William Frendreiss

Georgia Insitute of Technology

Raj Gandhi

Cornell University

Darij Grinberg

Drexel University

SUBBARAO VENKATESH GUGGILAM

Arctic University of Troms\o

Thiago Holleben

Dalhousie University

Elena Hoster

Ruhr-Universität Bochum

Linda Hoyer

RWTH Aachen University

Andy Hsiao

University of British Columbia

Daniel Iľkovič

Masaryk University

Aryaman Jal

KTH Royal Institute of Technology

Fareeha Jamal

United Arab Emirates University (UAEU)

Yuhan Jiang

Harvard University

Leo Jiang

University of Toronto

Katerina Kalampogia-Evangelinou

University of Athens

Iqra Khan

Phillips University Marburg, Germany

Aditya Khanna

Virginia Tech

Soyeon Kim

University of California, Davis

Viktória Klász

University of Bonn

Filip Kučerák

Masaryk University

Tom Kuhmichel

University of Siegen

Kevin Kühn

Goethe Universität Frankfurt

Arne Kuhrs

Goethe University Frankfurt

Vadym Kurylenko

SISSA

Seungkyu Lee

University of Bonn

John Lentfer

UC Berkeley

Hsin-Chieh Liao

University of Miami

Emiliano Liwski

KU Leuven

Rene Marczinzik

University of Bonn

Thomas Martinez

UC Los Angeles

Alex McDonough

University of California, Davis

Temesgen Molla

Addis Ababa University

Dania Morales

The University of Kansas

Alessio Moscariello

Università di Catania

Vassilis Dionyssis Moustakas

Università di Pisa

Leonie Mühlherr

University of Bielefeld

Yulia Mukhina

Laboratoire d'informatique de l'École polytechnique

Hanna Mularczyk

Massachusetts Institute of Technology

Anastasia Nathanson

University of Minnesota

Tongyu Nian

Osaka University

Apondi Otieno

Technical University of Kenya

Greta Panova

University of Southern California

Digjoy Paul

IISc Bangalore

Dmitrii Pavlov

MPI MiS

Enrico Piccione

University of Bergen

Lotfallah Pourfaraj

Central Tehran Branch, Islamc Azad University

Elizabeth Pratt

UC Berkeley

Hassan Raza

Wenzhou-Kean University

Sophie Rehberg

FU Berlin

Vic Reiner

University of Minnesota

Felix Röhrle

University of Tübingen

Leonardo Saud Maia Leite

KTH - Royal Institute of Technology

María Alejandra Schild

Pontifical Catholic University of Chile

Alec Schmutz

EPFL

Amanda Schwartz

University of Michigan

Olha Shevchenko

University of California Los Angeles (UCLA)

Mia Smith

University of Michigan

Qiuye Song

Beihang University

Andreas Spomer

University of Cologne

Christian Stump

Ruhr-Universität Bochum

Bernd Sturmfels

Max Planck Institute for Mathematics in the Sciences

Sheila Sundaram

University of Minnesota

Eliana Tolosa Villarreal

Università degli studi di Genova

Prajwal Dhondiram Udanshive

University of Western Ontario

Gabe Udell

Cornell University

Emil Verkama

KTH Royal Institute of Technology

Emanuele Verri

University of Greifswald

Julian Weigert

University of Konstanz

Martin Winter

Warwick University

Kai Hsiang, Jonathan Yang

University of British Columbia

Bailee Zacovic

University of Michigan

Yuhuai Zhou

RWTH

Scientific Organizers

Sarah Brauner

University of Quebec at Montreal

Christian Stump

Ruhr-Universität Bochum

Bernd Sturmfels

Max Planck Institute for Mathematics in the Sciences

Administrative Contact

Mirke Olschewski

Max Planck Institute for Mathematics in the Sciences Contact via Mail