Thursdays 16:30, IMECC
4 December 2025, 16:30, IMECC 151
Speaker: Simone Marchesi (Universitat de Barcelona)
Title: Linear spaces of matrices of constant rank from syzygy bundles
A long-standing problem in linear algebra is the construction of linear spaces of constant rank, whose classical origins are in the work of Kronecker and Weierstrass. We will see that, under mild assumptions, these matrices are always associated to a syzygy bundle that fits in a (partially linear) resolution. Furthermore, this construction allows us to list all indecomposable matrices of constant rank up to 7, as well as describing the moduli spaces of simple vector bundles naturally defined by families of constant rank matrices. This is joint work with Rosa Maria Miró-Roig.
27 November 2025, 16:30, IMECC 151
Speaker: Leonardo Silva (IMECC)
Title: The moduli space of torsion-free sheaves on P3 with quasi-maximal c3
The main goal of this talk is to characterize and understand the geometric structure of moduli spaces of semistable torsion-free sheaves on P3. By fixing Chern classes, we focus on the problem of finding the moduli spaces that parameterize semistable sheaves with these specific classes. This is a current problem in algebraic geometry, with certain cases already characterized in the literature, as can be read in "Two infinite series of moduli spaces of rank 2 sheaves on P3" and "Irreducible components of the moduli space of rank 2 sheaves of odd determinant on projective space". Our goal is to understand the moduli spaces and to study their geometric properties, such as their number of irreducible components, connectedness, and smoothness of quasi-maximal c3.
13 November 2025, 16:30, IMECC 151
Speaker: Waleed Noor (IMECC)
Title: A function space approach to primes and zeta-zeros
The Nyman-Beurling criterion for the Riemann hypothesis (RH) is a classical Hilbert space reformulation of RH from the 1950s. In this talk we describe how translating these results in the language of Hilbert spaces of analytic functions provides a new path to the study of the Riemann ζ-function and the distribution of primes. This new approach allows for the employment of recent tools from operator theory, complex analysis and linear dynamics to arrive at some interesting conclusions regarding the zero-free regions of the Riemann ζ-function and the distribution of primes.
6 November 2025, 16:30, IMECC 151
Speaker: Maria Michalska (ICMC)
Title: Asymptotic growth of polynomials on sets
Let f be a real polynomial in n variables and S a subset of Rn. Clearly, there always exist constants C and d such that |f| < C|x|d for all x, where d can be taken to be the degree of the polynomial f. On the other hand, the exponent d might not be optimal when considering x in S. For instance, if S is compact, then d=0. Moreover, optimal d can be expected to be negative for polynomials vanishing at infinity when restricted to S. We will give an overview of properties of the optimal exponent when S is assumed to be semialgebraic, and how this relates to constrained optimization problems.
30 October 2025, 16:30, IMECC 151
Speaker: Renato Vidal Martins (IMECC/UFMG)
Title: Stability conditions for coherent systems on integral curves
In this talk, we will provide a brief introduction to stability conditions and apply them to the category of coherent systems on an integral curve. We define a three-parameter family of stability conditions on its derived category and investigate when these are of Bridgeland type. Additionally, we study the semistability of certain objects with respect to these conditions; and we explore possible directions for applications. Joint work with Marcos Jardim and Leonardo Roa-Leguizamon.
23 October 2025, 15:00, IMECC 151
Speaker: Jonatan Gomez Parada (IMECC)
Title: Identidades graduadas e cocaracteres em uma álgebra de matrizes triangulares superiores
Seja K⟨X⟩ a álgebra associativa livre, livremente gerada sobre um corpo K por um conjunto contável X = {x1,x2,...}. Se A é uma K-álgebra associativa, diremos que um polinômio f(x1,...,xn) ∈ K⟨X⟩ é uma identidade polinomial, ou simplesmente uma identidade de A se f(a1,...,an) = 0 para todo a1,...,an ∈ A. Considere a subálgebra de UT3(K) dada por: A = K(e1,1 + e3,3) ⊕ Ke2,2 ⊕ Ke2,3 ⊕ Ke3,2 ⊕ Ke1,3, onde ei,j denota as matrizes unitárias. Estudaremos as identidades Z2-graduadas da álgebra A e daremos uma descrição dos cocaracteres graduados desta álgebra.
2 October 2025, 16:30, IMECC 323
Speaker: Tomás S.R. Silva (IMECC)
Title: From Theory to Computation: Unraveling Hyperplane Arrangements
A hyperplane arrangement is a finite collection of codimension-one affine subspaces within a finite-dimensional vector space. The study of these arrangements draws from various branches of mathematics, uncovering deep and often surprising connections between combinatorics, algebra, algebraic geometry, and topology. In this seminar, we will explore some of the intriguing properties of hyperplane arrangements and highlight the computational challenges that arise in their analysis.
25 September 2025, 16:30, IMECC 323
Speaker: João Paulo Guardieiro Sousa (IMECC)
Title: Contando pontos racionais com a teoria de Stöhr-Voloch e resultados de Tate e Shafarevich
Nesta apresentação, irei discorrer sobre alguns dos meus últimos tópicos de pesquisa, envolvendo o número de pontos racionais de curvas definidas sobre corpos finitos. A primeira parte deste trabalho envolve a teoria de Stöhr-Voloch. Nós classificamos as curvas trinomiais que são Frobenius não clássicas com respeito ao morfismo de retas, e obtivemos uma fórmula para o seu número de pontos racionais. Além disso, conseguimos classificar também as curvas quadrinomiais que são dadas por polinômios com variáveis separadas. A segunda parte deste trabalho envolve superfícies elípticas que podem ser vistas como curvas elípticas sobre corpos de funções de característica 3. A partir de uma curva elíptica supersingular fixada sobre F9, nós obtivemos fórmulas para o posto de Mordell-Weil de algumas twists delas em extensões de Artin-Schreier e Kummer do corpo de funções de uma curva. Este é um trabalho em colaboração com Tiago Aprigio e Herivelto Borges, e foi financiado, em partes, por CAPES, CNPq e FAPESP.
16 September 2025, 16:30, IMECC 324
Speaker: Matija Tapuskovic (University of Oxford)
Title: Motivic Galois theory of algebraic Mellin transforms
In this talk, I will introduce the theory of algebraic Mellin transforms and their local expansions as power series whose coefficients are periods in the sense of Kontsevich—Zagier. We will see how this theory relates to the classical motivic Galois theory of periods — such as multiple zeta values — and how this action can be naturally extended to the full power series expansions of algebraic Mellin transforms. Remarkably, this allows us to encode the motivic Galois action on infinitely many periods using finite formulas. This framework provides a unified perspective on diverse examples, including dimensionally regularised Feynman integrals, Lauricella hypergeometric functions, and beyond. This is joint work with Francis Brown, Clément Dupont, and Javier Fresán.
4 September 2025, 16:30, IMECC 324
Speaker: Daniel Fadel (ICMC-USP)
Title: Existence and energy concentration of SU(2) Yang-Mills-Higgs critical points on 3-manifolds
I will present recent joint work with Da Rong Cheng (Miami)
and Luiz Lara (Unicamp) on the SU(2) Yang-Mills-Higgs functional with arbitrary
positive coupling constant over oriented Riemannian 3-manifolds. This gauge-invariant
energy is defined on pairs consisting of a connection on an SU(2)-bundle and a section
of the adjoint bundle, and its critical points are solutions to a nonlinear
second-order elliptic system modulo gauge.
Motivated by the work of Pigati and Stern (2021) on the abelian case, we introduce
a scaling parameter and study sequences of critical points of the rescaled functional
under natural energy bounds. On 3-manifolds with bounded geometry, we show that,
as the parameter tends to zero, energy concentrates at a finite set of points,
while the remaining energy is accounted for by the formation of an L2
harmonic 1-form. Around each concentration point, a finite collection of “bubbles”
arises — non-trivial critical points on R3 for the rescaled functional
with parameter equal to one. An energy gap result, together with a no-neck property,
ensures that the total concentrated energy at each point is precisely the sum
of the energies of finitely many such bubbles.
On closed 3-manifolds, we adapt the min-max construction of Pigati and Stern to the
SU(2) setting and, for sufficiently small values of the scaling parameter,
establish the existence of non-trivial critical points satisfying energy bounds
natural from the scaling perspective. When the manifold is a rational homology
3-sphere, we ensure the occurrence of non-trivial bubbling, which in turn implies
the existence of non-trivial critical points on R3.
If time permits, I will conclude with open questions and potential extensions to higher dimensions.
26 June 2025, 16:30, IMECC 324
Speaker: Tiago Jardim da Fonseca (UNICAMP)
Title: The geometry of M1,3
Let M1,n be the moduli space of genus 1 curves with n marked points. Belorousski showed that these moduli spaces are rational algebraic varieties for n less or equal to 10. A cohomology computation, or a point count over finite fields, suggests another property: the motive of M1,n is expected to be mixed Tate for n in the same range. I'll describe explicit, and rather elementary, stratifications of M1,1, M1,2 and M1,3 which show in particular the mixed Tate property in these cases. This is joint work with Francis Brown.
12 June 2025, 16:30, IMECC 324
Speaker: Leonardo Leguizamon (UNICAMP)
Title: Brill-Noether theory for the moduli space of vector bundles
The moduli space of vector bundles on smooth projective varieties
were constructed using GIT in the 70’s. Since then, these moduli spaces have been
studied extensively by several authors. Nevertheless, it should be mentioned that
in spite of big efforts, there is still a huge number of very hard questions left open.
A crucial problem is to determine the geometry of the moduli space in terms of the
existence and structure of their subvarieties.
One of subvarieties that have been of great interest are the Brill-Noether subvarieties.
A Brill-Noether subvariety is a subset of the moduli space whose points correspond
to bundles having at least k independent global sections. The main goal of Brill-Noether
theory is the study of these subsets, in particular questions concerning non-emptiness,
connectedness, irreducibility, dimension, singularities, and topological and
geometric structures. In this talk, I will present a broad overview of Brill–Noether
theory in the classical setting of algebraic curves, I will show some results
for Brill-Noether theory on surfaces.
5 June 2025, 16:30, IMECC 324
Speaker: Arpan Saha (UNICAMP)
Title: Resurgence in enumerative geometry
Gromov–Witten and Donaldson–Thomas invariants are enumerative invariants that can be defined for any Calabi–Yau threefold. Roughly speaking, GW invariants count holomorphic curves together with a parametrisation while DT invariants count them together with the data of the equations defining them. These two different ways of interpreting a curve lead to different behaviour under degenerations and so different counts. Nevertheless, it has been conjectured by Maulik–Nekrasov–Okounkov–Pandharipande that knowing any one of these two sets of invariants is sufficient to completely determine the other. Motivated by physical intuition from topological string theory, we shall be exploring in this talk an elegant realisation of this equivalence in the case of the resolved conifold, a particular example of a noncompact Calabi–Yau threefold, within the framework of resurgence theory. This is based on joint work with Murad Alim, Jörg Teschner, and Iván Tulli.
29 May 2025, 16:30, IMECC 324
Speaker: Dmitry Korshunov (IMPA)
Title: Hölder potentials in hyperkähler dynamics
Cantat and Dinh-Sibony studied so called dynamic currents associated to hyperbolic automorphisms of Calabi-Yau manifolds. They are defined as eigenvectors in H1,1 with eigenvalue >1. They are of interest in dynamics, because they allow to construct dynamically interesting measures, and for algebraic geometry they provide examples of rigid currents. I will describe the circle of ideas around these themes and report on a work in progress with Misha Verbitsky generalizing a theorem of Dinh and Sibony stating that potentials of dynamic currents are Hölder functions.
15 May 2025, 16:30, IMECC 324
Speaker: Arpan Saha (UNICAMP)
Title: An invitation to the Gross–Siebert programme
The Strominger–Yau–Zaslow conjecture is a geometric shadow of mirror symmetry that interprets mirror manifolds as dual special lagrangian fibrations over the same base. The naive statement of the conjecture turns out to be false but may be rescued in certain large complex structure limits. The toric degeneration programme due to Mark Gross and Bernd Siebert is an algebro-geometric realisation of the above conjectural statement motivated by a class of such limits, namely toric degenerations, that captures the mirror map in terms of combinatorial data on the SYZ base. The goal of this talk is to provide an elementary overview of some of the motivation behind this seemingly technical approach to mirror symmetry and, if time permits, touch on analogies with certain differential-geometric constructions in hyperkähler geometry.
8 May 2025, 16:30, IMECC 324
Speaker: Ethan Cotterill (UNICAMP)
Title: 27 lines on a cubic surface, revisited
Among the most well-known results in classical algebraic geometry, due to Cayley and Salmon, is the fact that every smooth cubic surface in P3 over the complex numbers contains exactly 27 lines. Kass and Wickelgren proved a far-reaching generalization of this result for lines on cubics over an arbitrary field F. In their version, the "number" of lines is an element of GW(F), the Grothendieck--Witt group of quadratic forms over F. The aim of this talk is to motivate, and explain some of the key ingredients in, their work.
24 April 2025, 16:00, IMECC 324
Speaker: Andrey Soldatenkov (UNICAMP)
Title: Apollonian carpets and the boundary of the Kähler cone of a hyperkähler manifold
The ample cone of a compact Kähler n-manifold M is the intersection of its Kähler cone and the real subspace generated by integer (1,1)-classes. Its isotropic boundary is the set of all points η on its boundary such that ∫M ηn=0. We are interested in the relation between the shape of the isotropic boundary of the ample cone of a hyperkahler manifold and the dynamics of its holomorphic automorphism group G. In this case, the projectivization of the ample cone is realized as an open, locally polyhedral subset in a hyperbolic space H. The isotropic boundary S is realized as a subset of the hyperbolic boundary (the absolute) A of H, which is naturally identified with a Euclidean sphere. It is clear that the isotropic boundary S contains the limit set of G acting on its ample cone. We prove that, conversely, all irrational points on S belong to the limit set. Using a result of N. Shah about limiting distributions of curves under geodesic flow on hyperbolic manifolds, we prove that every real analytic curve in S is contained in a geodesic sphere in S, and in presence of such curves the limit set is the closure of the union of these geodesic spheres. We study the geometry of such fractal sets, called Apollonian carpets, and establish the link between the Apollonian carpet and the structure of the automorphism group.