Enumerative Geometry and Geometric Representation Theory

Jun 23, 2025, 10:00 AM → Jul 4, 2025, 5:30 PM Europe/Rome

Università di Pisa, Italy

The Summer School on "Enumerative Geometry and Geometric Representation Theory" will take place at the Centro Congressi Le Benedettine, in Pisa, from June 23 to June 29 and at the Aula G, Polo Fibonacci, of the University of Pisa, from June 30 to July 4, 2025. It can be seen as a second iteration of the 2021 IHÉS Summer School on "Enumerative Geometry, Physics and Representation Theory" organized by Neguţ, Sala, and Schiffmann. 

The school will offer six 5-hour lectures together with exercise/Q&A sessions and Advanced Research sessions. 

The school welcomes everybody but is primarily intended for Ph.D. students and young postdocs.

The Summer School will focus on the following main subjects: 

• logarithmic and tropical geometry and relations to enumerative geometry
• the geometry of the Hitchin moduli space
• Donaldson-Thomas invariants, BPS states, and their categorification
• Cohomological Hall algebras and their relation to enumerative geometry
• Coulomb branches in geometry and representation theory

 

We have limited funds. We may ONLY offer accommodations in shared rooms to a limited number of participants. If you need support, please let us know through the registration procedure.
Deadline for requesting support: February 15, 2025.

 

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Lecturers

Pierrick Bousseau (University of Georgia, USA, and CNRS, France)

Joel Kamnitzer (McGill University, Montreal, Canada)

Olivier Schiffmann (CNRS and Université de Paris-Saclay, France)

Junliang Shen (Yale University, USA)

Bernd Siebert (The University of Texas at Austin, USA)

Yukinobu Toda (Kavli IPMU, the University of Tokyo, Japan)     

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Advisors for the Afternoon Sessions

Hülya Argüz (University of Georgia, USA)

Camilla Felisetti (Università di Modena e Reggio Emilia)

Lucien Hennecart (CNRS and Université de Picardie Jules Verne, France)

Samuel Johnston (Imperial College London, UK)

Dinakar Muthiah (University of Glasgow, UK)

Tudor Pădurariu (CNRS and Sorbonne Université, France)

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Gong Show

Danil Gubarevich (Université de Paris-Saclay, France)

Mischa Elkner (IST, Austria)

Shivang Jindal (University of Edinburgh, UK)

Jingxiang Ma (University of Sheffield, UK)

Jeremy Peters (University of Toronto, Canada)

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Scientific Committee

Andrei Neguţ (EPFL, Switzerland)

Francesco Sala (Università di Pisa)

Olivier Schiffmann (CNRS and Université de Paris-Saclay, France)

Mattia Talpo (Università di Pisa)

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Organizing Committee

Pietro Leonardini (Università di Pisa)

Andrea Maffei (Università di Pisa)

Francesco Sala (Università di Pisa)  

Mattia Talpo (Università di Pisa)

Bernando Tarini (Scuola Normale Superiore, Pisa)

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The Summer School is supported by the Department of Mathematics, University of Pisa (Dipartimento di Eccellenza -- CUP I57G22000700001), Prin 2022 "Algebraic and geometric aspects of Lie theory" (2022S8SSW2), Prin 2022 "Geometry of algebraic structures: moduli, invariants, deformations" (2022BTA242), and by Scuola Normale Superiore.
 

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During the School, violence, harassment, or discrimination of any sort are not tolerated. In case you witness or are subject to such behaviors, contact the competent authorities. For further information, please ask the organizers.

poster.pdf

Monday, June 23

10:00 AM → 11:00 AM TBA

TBA

Speaker: Pierrick Bousseau (University of Georgia, USA, and CNRS, France)

11:00 AM → 11:30 AM Coffee Break

11:30 AM → 12:30 PM TBA

TBA

Speaker: Pierrick Bousseau (University of Georgia, USA, and CNRS, France)

12:30 PM → 3:00 PM Lunch break

3:00 PM → 4:00 PM Logarithmic Gromov-Witten invariants and its application to mirror symmetry

Speaker: Bernd Siebert (The University of Texas at Austin, USA)

4:00 PM → 4:30 PM Coffee Break

4:30 PM → 5:30 PM From quantum cohomology of symplectic resolutions to compactifications of families of Gaudin algebras

Speaker: Joel Kamnitzer (McGill University, Montreal, Canada)

Tuesday, June 24

10:00 AM → 11:00 AM From quantum cohomology of symplectic resolutions to compactifications of families of Gaudin algebras

Speaker: Joel Kamnitzer (McGill University, Montreal, Canada)

11:00 AM → 11:30 AM Coffee Break

11:30 AM → 12:30 PM Logarithmic Gromov-Witten invariants and its application to mirror symmetry

Speaker: Bernd Siebert (The University of Texas at Austin, USA)

12:30 PM → 3:00 PM Lunch Break

3:00 PM → 4:00 PM TBA

TBA

Speaker: Pierrick Bousseau (University of Georgia, USA, and CNRS, France)

4:00 PM → 4:30 PM Coffee Break

4:30 PM → 5:30 PM Exercise Session: TBA

TBA

Speaker: Hülya Argüz (University of Georgia, USA)

4:30 PM → 5:30 PM Exercise Session: TBA

TBA

Speaker: Samuel Johnston (Imperial College London, UK)

4:30 PM → 5:30 PM Exercise Session: The BFN construction of Coulomb branches, symplectic duality, and affine Grassmannians slices

Speaker: Dinakar Muthiah (University of Glasgow, UK)

Wednesday, June 25

10:00 AM → 11:00 AM From quantum cohomology of symplectic resolutions to compactifications of families of Gaudin algebras

Speaker: Joel Kamnitzer (McGill University, Montreal, Canada)

11:00 AM → 11:30 AM Coffee Break

11:30 AM → 12:30 PM Logarithmic Gromov-Witten invariants and its application to mirror symmetry

Speaker: Bernd Siebert (The University of Texas at Austin, USA)

12:30 PM → 12:40 PM Break

12:40 PM → 12:55 PM Gong Show: Wheel conditions for reductive groups

Speaker: Danil Gubarevich (Université de Paris-Saclay, France)

12:55 PM → 1:10 PM Gong Show: Mirror symmetry for ADE resolutions

Speaker: Jingxiang Ma (University of Sheffield, UK)

1:10 PM → 1:25 PM Gong Show: Parameter spaces for quantum cohomology

Speaker: Jeremy Peters (University of Toronto, Canada)

1:25 PM → 5:55 PM Free Afternoon

Thursday, June 26

10:00 AM → 11:00 AM TBA

TBA

Speaker: Pierrick Bousseau (University of Georgia, USA, and CNRS, France)

11:00 AM → 11:30 AM Coffee Break

11:30 AM → 12:30 PM Logarithmic Gromov-Witten invariants and its application to mirror symmetry

Speaker: Bernd Siebert (The University of Texas at Austin, USA)

12:30 PM → 3:00 PM Lunch Break

3:00 PM → 4:00 PM From quantum cohomology of symplectic resolutions to compactifications of families of Gaudin algebras

Speaker: Joel Kamnitzer (McGill University, Montreal, Canada)

4:00 PM → 4:30 PM Coffee Break

4:30 PM → 5:30 PM Exercise Session: TBA

TBA

Speaker: Hülya Argüz (University of Georgia, USA)

4:30 PM → 5:30 PM Exercise Session: TBA

TBA

Speaker: Samuel Johnston (Imperial College London, UK)

4:30 PM → 5:30 PM Exercise Session: The BFN construction of Coulomb branches, symplectic duality, and affine Grassmannians slices

Speaker: Dinakar Muthiah (University of Glasgow, UK)

Friday, June 27

10:00 AM → 11:00 AM From quantum cohomology of symplectic resolutions to compactifications of families of Gaudin algebras

Speaker: Joel Kamnitzer (McGill University, Montreal, Canada)

11:00 AM → 11:30 AM Coffee Break

11:30 AM → 12:30 PM TBA

TBA

Speaker: Pierrick Bousseau (University of Georgia, USA, and CNRS, France)

12:30 PM → 3:00 PM Lunch Break

3:00 PM → 4:00 PM Logarithmic Gromov-Witten invariants and its application to mirror symmetry

Speaker: Bernd Siebert (The University of Texas at Austin, USA)

4:00 PM → 4:30 PM Coffee Break

4:30 PM → 5:30 PM Research Session: Kac-Moody affine Grassmannian slices

Speaker: Dinakar Muthiah (University of Glasgow, UK)

4:30 PM → 5:30 PM Research Session: TBA

TBA

Speaker: Hülya Argüz (University of Georgia, USA)

4:30 PM → 5:30 PM Research Session: TBA

TBA

Speaker: Samuel Johnston (Imperial College London, UK)

Monday, June 30

10:00 AM → 11:00 AM Decomposition Theorem for abelian fibrations

Speaker: Junliang Shen (Yale University, USA)

11:00 AM → 11:30 AM Coffee Break

11:30 AM → 12:30 PM Decomposition Theorem for abelian fibrations

Speaker: Junliang Shen (Yale University, USA)

12:30 PM → 3:00 PM Lunch Break

3:00 PM → 4:00 PM Cohomological Hall algebras of sheaves on surfaces and applications

The aim of these lectures is to explain the construction of an associative algebra structure on the Borel-Moore homology of the stack of properly supported coherent sheaves on a smooth complex surface (the COHA) and to give some motivations and applications. More precisely, we hope to touch upon the following topics:

• Definition of the COHA of the preprojective algebra of a quiver, and the relation to Kac-Moody (and larger!) algebras and Kac polynomials.

• Definition of the COHA of zero-dimensional sheaves on S, and computation of that COHA (at least under some assumption on S, for instance if S is projective). Relation to W-algebras.

• Action of the above COHA of zero-dimensional sheaves on suitable moduli stacks / spaces of coherent sheaves on S. Examples related to Hilbert schemes on S, instanton spaces for the special case $S=\mathbb{A}^2$, moduli stacks of Higgs sheaves for $S=T^*C$, $C$ a smooth projective complex curve.

• Definition of the COHA for arbitrary (properly supported) coherent sheaves on a surface S, and some examples / applications (depending on time)

Speaker: Olivier Schiffmann (CNRS and Université de Paris-Saclay, France)

4:00 PM → 4:30 PM Coffee Break

4:30 PM → 5:30 PM TBA

TBA

Speaker: Yukinobu Toda (Kavli IPMU, the University of Tokyo, Japan)

Tuesday, July 1

10:00 AM → 11:00 AM TBA

TBA

Speaker: Yukinobu Toda (Kavli IPMU, the University of Tokyo, Japan)

11:00 AM → 11:30 AM Coffee Break

11:30 AM → 12:30 PM Cohomological Hall algebras of sheaves on surfaces and applications

The aim of these lectures is to explain the construction of an associative algebra structure on the Borel-Moore homology of the stack of properly supported coherent sheaves on a smooth complex surface (the COHA) and to give some motivations and applications. More precisely, we hope to touch upon the following topics:

• Definition of the COHA of the preprojective algebra of a quiver, and the relation to Kac-Moody (and larger!) algebras and Kac polynomials.

• Definition of the COHA of zero-dimensional sheaves on S, and computation of that COHA (at least under some assumption on S, for instance if S is projective). Relation to W-algebras.

• Action of the above COHA of zero-dimensional sheaves on suitable moduli stacks / spaces of coherent sheaves on S. Examples related to Hilbert schemes on S, instanton spaces for the special case $S=\mathbb{A}^2$, moduli stacks of Higgs sheaves for $S=T^*C$, $C$ a smooth projective complex curve.

• Definition of the COHA for arbitrary (properly supported) coherent sheaves on a surface S, and some examples / applications (depending on time)

Speaker: Olivier Schiffmann (CNRS and Université de Paris-Saclay, France)

12:30 PM → 3:00 PM Lunch Break

3:00 PM → 4:00 PM Decomposition Theorem for abelian fibrations

Speaker: Junliang Shen (Yale University, USA)

4:00 PM → 4:30 PM Coffee Break

4:30 PM → 5:30 PM Exercise Session: TBA

TBA

Speaker: Camilla Felisetti (Università di Modena e Reggio Emilia)

4:30 PM → 5:30 PM Exercise Session: TBA

TBA

Speaker: Lucien Hennecart (University of Edinburgh, UK)

4:30 PM → 5:30 PM Exercise Session: TBA

TBA

Speaker: Tudor Pădurariu (CNRS and Sorbonne Université, France)

Wednesday, July 2

10:00 AM → 11:00 AM Cohomological Hall algebras of sheaves on surfaces and applications

The aim of these lectures is to explain the construction of an associative algebra structure on the Borel-Moore homology of the stack of properly supported coherent sheaves on a smooth complex surface (the COHA) and to give some motivations and applications. More precisely, we hope to touch upon the following topics:

• Definition of the COHA of the preprojective algebra of a quiver, and the relation to Kac-Moody (and larger!) algebras and Kac polynomials.

• Definition of the COHA of zero-dimensional sheaves on S, and computation of that COHA (at least under some assumption on S, for instance if S is projective). Relation to W-algebras.

• Action of the above COHA of zero-dimensional sheaves on suitable moduli stacks / spaces of coherent sheaves on S. Examples related to Hilbert schemes on S, instanton spaces for the special case $S=\mathbb{A}^2$, moduli stacks of Higgs sheaves for $S=T^*C$, $C$ a smooth projective complex curve.

• Definition of the COHA for arbitrary (properly supported) coherent sheaves on a surface S, and some examples / applications (depending on time)

Speaker: Olivier Schiffmann (CNRS and Université de Paris-Saclay, France)

11:00 AM → 11:30 AM Coffee Break

11:30 AM → 12:30 PM TBA

TBA

Speaker: Yukinobu Toda (Kavli IPMU, the University of Tokyo, Japan)

12:30 PM → 12:40 PM Break

12:40 PM → 12:55 PM Gong Show: Some symmetries of Hausel’s big algebras

Speaker: Mischa Elkner (IST, Austria)

12:55 PM → 1:10 PM Gong Show: Quantum groups from Cohomological DT theory

Speaker: Shivang Jindal (University of Edinburgh, UK)

1:10 PM → 1:25 PM Gong Show: TBA

1:25 PM → 3:55 PM Free Afternoon

Thursday, July 3

10:00 AM → 11:00 AM Decomposition Theorem for abelian fibrations

Speaker: Junliang Shen (Yale University, USA)

11:00 AM → 11:30 AM Coffee Break

11:30 AM → 12:30 PM Cohomological Hall algebras of sheaves on surfaces and applications

The aim of these lectures is to explain the construction of an associative algebra structure on the Borel-Moore homology of the stack of properly supported coherent sheaves on a smooth complex surface (the COHA) and to give some motivations and applications. More precisely, we hope to touch upon the following topics:

• Definition of the COHA of the preprojective algebra of a quiver, and the relation to Kac-Moody (and larger!) algebras and Kac polynomials.

• Definition of the COHA of zero-dimensional sheaves on S, and computation of that COHA (at least under some assumption on S, for instance if S is projective). Relation to W-algebras.

• Action of the above COHA of zero-dimensional sheaves on suitable moduli stacks / spaces of coherent sheaves on S. Examples related to Hilbert schemes on S, instanton spaces for the special case $S=\mathbb{A}^2$, moduli stacks of Higgs sheaves for $S=T^*C$, $C$ a smooth projective complex curve.

• Definition of the COHA for arbitrary (properly supported) coherent sheaves on a surface S, and some examples / applications (depending on time)

Speaker: Olivier Schiffmann (CNRS and Université de Paris-Saclay, France)

12:30 PM → 3:00 PM Lunch Break

3:00 PM → 4:00 PM TBA

TBA

Speaker: Yukinobu Toda (Kavli IPMU, the University of Tokyo, Japan)

4:00 PM → 4:30 PM Coffee Break

4:30 PM → 5:30 PM Exercise Session: TBA

TBA

Speaker: Lucien Hennecart (University of Edinburgh, UK)

4:30 PM → 5:30 PM Exercise Session: TBA

TBA

Speaker: Tudor Pădurariu (CNRS and Sorbonne Université, France)

4:30 PM → 5:30 PM Exercise Session: TBA

TBA

Speaker: Camilla Felisetti (Università di Modena e Reggio Emilia)

Friday, July 4

10:00 AM → 11:00 AM TBA

TBA

Speaker: Yukinobu Toda (Kavli IPMU, the University of Tokyo, Japan)

11:00 AM → 11:30 AM Coffee Break

11:30 AM → 12:30 PM Decomposition Theorem for abelian fibrations

Speaker: Junliang Shen (Yale University, USA)

12:30 PM → 3:00 PM Lunch Break

3:00 PM → 4:00 PM Cohomological Hall algebras of sheaves on surfaces and applications

The aim of these lectures is to explain the construction of an associative algebra structure on the Borel-Moore homology of the stack of properly supported coherent sheaves on a smooth complex surface (the COHA) and to give some motivations and applications. More precisely, we hope to touch upon the following topics:

• Definition of the COHA of the preprojective algebra of a quiver, and the relation to Kac-Moody (and larger!) algebras and Kac polynomials.

• Definition of the COHA of zero-dimensional sheaves on S, and computation of that COHA (at least under some assumption on S, for instance if S is projective). Relation to W-algebras.

• Action of the above COHA of zero-dimensional sheaves on suitable moduli stacks / spaces of coherent sheaves on S. Examples related to Hilbert schemes on S, instanton spaces for the special case $S=\mathbb{A}^2$, moduli stacks of Higgs sheaves for $S=T^*C$, $C$ a smooth projective complex curve.

• Definition of the COHA for arbitrary (properly supported) coherent sheaves on a surface S, and some examples / applications (depending on time)

Speaker: Olivier Schiffmann (CNRS and Université de Paris-Saclay, France)

4:00 PM → 4:30 PM Coffee Break

4:30 PM → 5:30 PM Research Session: TBA

TBA

Speaker: Tudor Pădurariu (CNRS and Sorbonne Université, France)

4:30 PM → 5:30 PM Research Session: TBA

TBA

Speaker: Camilla Felisetti (Università di Modena e Reggio Emilia)

4:30 PM → 5:30 PM Research Session: TBA

TBA

Speaker: Lucien Hennecart (University of Edinburgh, UK)