Mathematics Area - SISSA - Algebraic Geometry

Computations in Algebraic Geometry

Guglielmo Padula — Wed, 03 Dec 2025 18:33:33 +0000

Lecturer:

Michele Graffeo

Course Type:

PhD Course

Academic Year:

2025-2026

Period:

January-February

Duration:

10 h

Description:

The course aims to teach the classical techniques for parameterising certain locally closed subschemes of the Hilbert and Quot schemes of points, namely Hilbert–Samuel strata. There will be a particular focus on using the software Macaulay2 in algebraic geometry. Five topics will be covered in five lectures during the course. An additional lecture will be devoted to explicit computations using the Macaulay2 software. Lecture notes written in collaboration with Dott. Gessica Alecci are available via the following link NotesCiAG.pdf.

Organisation of the course

(1) Classical projective and birational geometry
In the first lecture I will recall the definitions of classical maps in algebraic geometry such as blow-ups, Veronese/Segre/Plucker embeddings and the Cremona transformation. These classical constructions are essential for describing Hilbert–Samuel strata.

(2) Monomial ideals and partitions
In the second lecture, I will introduce some useful tools for working with ideals in the polynomial ring. These include the notions of socle and Macaulay duality. Along the way, we will see many examples concerning monomial ideals.

(3) Commutative algebra and families of ideals
In this lecture I will introduce the concept of a (flat) family of subschemes and its relation to that of a Hilbert polynomial. I will then present explicit constructions of families, which can be understood as morphisms to the Hilbert scheme.

(4) Hilbert schemes
This lecture will focus on the Hilbert scheme of points on a smooth variety. More specifically, we will study its tangent space and the corresponding Bialynicki-Birula decomposition. This tool provides a way to certify the presence of certain irreducible components of the Hilbert scheme, such as reduced elementary ones.

(5) The Grothendieck ring of varieties
In this lecture I will introduce the concept of a motive of an algebraic variety. Although motives are insensitive to many geometric properties of the corresponding object, I will explain how information can be extracted from them through explicit computations.

(6) Computations via Macaulay2
This lecture will serve as a proof of concept of the previous ones. I will show how to solve some of the exercises presented during the course. Finally, I will explain how to perform computations using the Macaulay2 package HilbertAndQuotSchemesOfPoints.m2 by Paolo Lella.

Research Group:

Geometry and Mathematical Physics

Next Lectures:

Algebraic Geometry

Ariel Surya Boiardi — Thu, 11 Sep 2025 09:03:41 +0000

Lecturer:

Barbara Fantechi

Course Type:

PhD Course

Master Course

Academic Year:

2025-2026

Period:

October-February

Duration:

60 h

Description:

Timetable

Tuesday 14-16 (Sissa)
Wednesday 9-11 (Sissa)
Thursday 10-11 (Aula D Edificio A Università s di Trieste)

The course will start October 7, 2025.

Research Group:

Geometry and Mathematical Physics

Next Lectures:

Cluster algebras

Ariel Surya Boiardi — Thu, 11 Sep 2025 09:02:44 +0000

Lecturer:

Mikhail Bershtein

Course Type:

PhD Course

Academic Year:

2025-2026

Period:

February-April

Duration:

40 h

Description:

All informations about the course can be found on the webpage https://people.sissa.it/~mbersht/Cluster2025.html

Research Group:

Geometry and Mathematical Physics

Next Lectures:

Topics in representation theory

Ariel Surya Boiardi — Thu, 11 Sep 2025 08:58:50 +0000

Lecturer:

Mikhail Bershtein

Course Type:

PhD Course

Academic Year:

2025-2026

Period:

October-December

Duration:

40 h

Description:

All informations on the course can be found on the webpage https://people.sissa.it/~mbersht/RT2025.html

Research Group:

Geometry and Mathematical Physics

Next Lectures:

Introduction to rigid analytic geometry-Adic spaces and applications

Valentin Eric Brice Nkana Ngan — Mon, 23 Jan 2023 14:22:21 +0000

External Lecturer:

Alberto Vezzani

Course Type:

PhD Course

Academic Year:

2022-2023

Duration:

20 h

Description:

The course is an introduction to some of the newest approaches to non-archimedean analytic geometry including:- Huber's adic spaces;- Raynaud's formal schemes and blow-ups;- Clausen-Scholze's analytic spaces.We will focus on specific examples arising from algebraic geometry, Scholze's tilting equivalence of perfectoid spaces and the Fargues-Fontaine curve.We will also show how to define (motivic) homotopy equivalences in this setting, with the aim of defining a relative de Rham cohomology for adic spaces over $\mathbb{Q}_p$ and a relative rigid cohomology for schemes over $\mathbb{F}_p$.

Research Group:

Geometry and Mathematical Physics

Location:

A-136

Location:

The alternative lecture room is A-005.

Next Lectures:

Singularities of smooth maps

Valentin Eric Brice Nkana Ngan — Fri, 18 Nov 2022 12:50:34 +0000

External Lecturer:

Ilia Bogaevsky

Course Type:

PhD Course

Academic Year:

2022-2023

Period:

January- March

Duration:

40 h

Research Group:

Geometry and Mathematical Physics

Next Lectures:

Introduction to stochastic matrices and orthogonal polynomials

Valentin Eric Brice Nkana Ngan — Wed, 21 Sep 2022 13:37:08 +0000

Lecturer:

Marco Bertola

Course Type:

PhD Course

Academic Year:

2022-2023

Duration:

30 h

Description:

The course aims at introducing the notion of Random MAtrices and the analysis of their spectral statistical properties. We will study the classical Wigner ensemble with the proof of the celebrated Wigner semicircle law for the eigenvalues. We wil then move on to the definition of more general Unitary Ensembles (where the underlying symmetry is given by the Unitery group) and prove fundamental structural results of Dyson on how to relate their statistical properties to the study of orthogonal polynomials. We will conclude with the study of general properties and the many applications of the theory of orthogonal polynomials, with hints at the asymptotic analysis and how this provides the first proof of Universality in the bulk and at the edge of the spectrum.

Research Group:

Geometry and Mathematical Physics

Next Lectures:

C*-ALGEBRAS THAT ONE CAN SEE

Valentin Eric Brice Nkana Ngan — Mon, 05 Sep 2022 13:01:36 +0000

External Lecturer:

P.M. Hajac (IMPAN, Warsaw)

Course Type:

PhD Course

Academic Year:

2022-2023

Period:

October

Duration:

20 h

Description:

C*-algebras are operator algebras forming the conceptual foundation of noncommutative geometry. Since commutative C*-algebras yield categories anti-equivalent to categories of locally compact Hausdorff spaces by the celebrated Gelfand-Naimark equivalence theorem, noncommutative C*-algebras are viewed as function algebras on quantum spaces. Their study from this point of view is referred to as noncommutative topology. Here KK-theory and index theory are among prime tools leading to significant applications. Graph C*-algebras form a narrow but particularly inspiring class in which the noncommutative topology of quantum spaces is given by the combinatorics of oriented graphs. Graphs provide a very tangible grip on their C*-algebras allowing one to obtain deeper and more explicit results than are attainable in a general abstract setting. In particular, graph C*-algebras are formidably good in instantiating general theory of C*-algebras. The aim of this lecture course is to introduce the basics of C*-algebras while enjoying educating fun with oriented graphs.

Research Group:

Geometry and Mathematical Physics

Location:

A-136

Next Lectures:

Introduction to Determinantal Point Processes And Intergrable Probability

Valentin Eric Brice Nkana Ngan — Mon, 05 Sep 2022 10:10:09 +0000

Lecturer:

Pierre Karim Emile Lazag

Course Type:

PhD Course

Academic Year:

2022-2023

Duration:

20 h

Description:

The course will present an introduction to the theory of determinantal point processes (DPP) and its use for the solution to the problem of the length of the longest increasing sub- sequence in a large random permutation. A celebrated result, belonging to what is now called ”integrable probability” and first proved by Baik-Deift-Johansson in 1999, asserts that the fluctuation of this length around its average is asymptotically distributed according to the Tracy-Widom distribution, similarly to the largest eigenvalue of a random Hermitian Gauss- ian matrix. The proof of this result will be the common thread of the course that will cover the following topics: definition and properties of DPP, trace class operators and Fredholm determinants, Macchi-Soshnikov/Shirai-Takahashi existence Theorem for DPP; orthogonal polynomial ensembles from random matrix theory; Robinson-Schensted correspondence and poissonized Plancherel measure; symmetric functions, Okounkov’s theory of Schur measures, fermionic Fock space and Boson-Fermion correspondance; Wiener-Helson classification of shift invariant subspaces; asymptotic analysis of the discrete Bessel kernel and proof of the Baik-Deift-Johansson Theorem. The course requires basic knowledge in probability theory, functional analysis and complex analysis, while a substantial part of its content consists of algebraic combinatorics, for which no prerequite is needed.

Research Group:

Geometry and Mathematical Physics

Next Lectures:

Singular points of complex algebraic plane curves

Valentin Eric Brice Nkana Ngan — Mon, 05 Sep 2022 10:07:11 +0000

Lecturer:

Miruna-Stefana Sorea

Course Type:

PhD Course

Academic Year:

2022-2023

Period:

November-December

Duration:

20 h

Description:

Abstract:
The aim of this course is to provide a local study of the singular points of plane curves. The theory of singularities of complex algebraic plane curves is situated at the crossroads of many interesting areas of mathematics, making the study of curve singularities particularly fruitful up to this day.

We start with a short introduction to the subject, where we briefly review results about manifolds and plane algebraic curves. Moreover, we focus on polar curves, Puiseux's theorem and resolution of singularities. Next, we show how combinatorial tools such as the Eggers tree can be used in the study of the contact of two branches of a curve. One of the core concepts in the first part of the course is equisingularity of curves.

In the second part of the course we study the geometry of the link of a singularity, Milnor's fibration theorem and the Milnor number. In addition, we also see a proof of Klein's equation (using Euler characteristics of constructible functions), which relates invariants of a projective curve with invariants of its dual curve. Time permitting, we will also tackle the decomposition of the link complement and the computation of the monodromy of the Milnor fibration.

Bibliography:
Wall, C. T. C. Singular points of plane curves. London Mathematical Society Student Texts, 63. Cambridge University Press, Cambridge, 2004. xii+370 pp.

Research Group:

Geometry and Mathematical Physics

Next Lectures: