This (informal) talk will introduce and discuss various aspects of deformation quantization, including its motivation, existence and classification of star products, and Morita equivalence.

Apresentamos os fundamentos da teoria da informação: em uma primeira palestra trataremos da informação de Shannon, em uma segunda palestra da geometria da informação de Fisher. A teoria da informação tem aplicações importantes em áreas como as telecomunicações e a estatística paramétrica. As palestras serão acessíveis aos alunos de graduação (um conhecimento basico da geometria de superfícies no espaço Euclidiano seria conveniente para a segunda palestra).

In this talk I will discuss some speculative ideas (work in progress) about the study of mirror maps for Calabi-Yau hypersurfaces from the perspective of model theory (a branch of mathematical logic). I will propose that some of the arithmetic and geometric properties of those mirror maps can be understood from purely model theoretic concepts like categoricity. The case of elliptic curves will be discussed in detail since it is the motivation of my proposal. No knowledge of model theory will be assumed so I will introduce the notions needed for the talk.

Motivated by mirror symmetry for the moduli space of Higgs bundles, we discuss the construction of Lagrangian subvarieties that arise from symplectic representations. In particular, for the standard representation of the symplectic group, we show that this corresponds to a particular irreducible component of the nilpotent cone of the Hitchin system using an auxiliary (spinor) moduli space and Morse theory techniques.

a research and study group of students and professors from post-graduate mathematics programs of PUC-Rio, IMPA and UFF, interested in algebraic, symplectic, affine, tropical, information geometry, homological, homotopical, commutative algebra, quantum mechanics, quantum field theory, conformal field theory, gauge theory, physical mathematics, category theory and operads, number theory, spectral theory, (quantum) information theory, ordinary differential equations, enumerative combinatorics, their interfaces in quantum topology, mirror symmetry, Langlands correspondence, and related topics.

• 🕑 Fridays (sexta-feiras) between 14:00 and 18:00, either one long or two shorter talks
• DMAT PUC-Rio, room L863 in Leme building (left from main entrance)
• take an elevator to 8th floor near "Bar Nossa Senhora do Carmo" (the one served by nuns)
• When room L863 will be occupied or not suitable, we will use room L856
• 🥗 lunch before the seminar; after 18:00 -- supper in Planetário Gávea
• soulmate seminar "Geometric Structures on Manifolds" -- Thursday 17:00, IMPA room 236

• Make a pull request to join our group
• Please ⭣⭣⭣ add here ⭣⭣⭣ interesting themes and topics for next seminar talks, references, slides of the talks, etc

Holidays: May 31

• tba -- Sergey Galkin -- toric Kähler manifolds, Hessian (dually flat) statistical manifolds, torification, geometric quantum mechanics, and SYZ mirror symmetry
• tba -- Lucas Branco -- Gaiotto Lagrangians
• May 31 -- Recesso do Corpus Christi
• May 24 -- John Alexander Cruz Morales -- Revisiting mirror maps: model theory, arithmetic and geometry.
• May 17 -- tba
• May 10 -- Rafael Ruggiero -- Introdução à teoria da informação. / Introduction to Information Theory
• May 10 -- Rafael Ruggiero -- Information Geometry and thermodynamics. Part I: Introduction to Information Theory.
• May 3 -- Henrique Bursztyn -- On deformation quantization
• Apr 26 -- Altan Erdnigor -- An overview of Geometric Class Field Theory
• Apr 26 -- Alex Gomez -- Model structures
• Apr 19 -- Sergey Galkin -- Toric geometry primer
• Apr 12 -- Sergey Galkin -- Mirror symmetry and Fano manifolds
• Apr 5 -- Bruno Suassuna -- Log-volume preserving rational endomorphisms of (C^*)^n
• Apr 5 -- Altan Erdnigor -- Quantization commutes with Reduction, part II/II
• Mar 29 -- Sexta-Feira da Paixão
• Mar 27 (Wed) 13h -- Boris Khesin -- Dynamics of pentagram maps
• Mar 8, 16h -- Altan Erdnigor -- Geometric Quantization and Equivariant Topology, part I/II
• Mar 8, 14h -- Lucas Branco -- What is... a Higgs bundle?

• Mar 1 -- José Carlos Valencia Alvites -- On lattice Quantum Chromodynamics models in the strong coupling regime
• Feb 23 -- Misha Verbitsky -- Deformation theory of holomorphically symplectic manifolds
• Feb 9 -- Alex Gomez -- A real analogue of the Moore--Tachikawa category
• Feb 2 -- Bruno Suassuna -- Wigner--Weyl transform
• Jan 26, 16h -- Dmitrii Korshunov -- Flexible polyhedra and symplectic geometry
• Jan 26, 14h -- Orchidea Maria Lecian -- Some new theorems on scarred waveforms in desymmetrised PSL(2,Z) billiards and in those of its congruence subgroups
• Jan 19 -- Altan Erdnigor -- Quivers and Gabriel's theorem
• Jan 12 -- Bruno Suassuna -- Weight systems of Chern-Simons-Witten and Rozansky-Witten, and Rozansky-Witten invariants

• Dec 8 -- Nossa Senhora da Conceição
• Dec 1 -- Alex Gomez -- Delzant's theorem
• Nov 24 -- Victor el Adji -- Matrix factorizations of polynomials
• Nov 24 -- Lucas Castello Branco -- Higgs Bundles and Mirror Symmetry
• Nov 24 -- Sérgio Loiola -- hidden Markov models
• Nov 17 -- Vladimir Roubtsov -- Kontsevich and Buchstaber polynomials, multiplication kernels and Calabi-Yau diferential operators
• Nov 3, Nov 10 -- Misha Skopenkov: Feynman checkers as an intro to 2-dimensional lattice QED
• Oct 27 -- Oktobermat XX
• Oct 20 -- Dmitri Panov -- Spherical surfaces with conical singularities
• Oct 13 -- Recesso do Dia do Professor
• Oct 6 -- Ilia Gaiur -- Multiplication kernels: introduction and examples
• Sep 29 -- Sergey Galkin -- Tropicalizations of rational maps preserving logarithmic volume forms (tropical talk II)
• Sep 22 -- Bruno Suassuna -- $n$ bosonic modes
• Sep 15 -- Arthur Fidalgo -- Combinatorial species: species of structures under the view of enumerative combinatorics
• Sep 8 -- recesso do Dia da Independência
• Sep 1 -- Altan Erdnigor -- Intersection homology
• Sep 1 -- Sergey Galkin -- Logarithmic volume forms and special birational transformations (pre-tropical talk I)
• Aug 25 -- Eduardo Alves da Silva -- Log Calabi-Yau geometry and Cremona maps
• Aug 18 -- Altan Erdnigor -- Homology of the Hilbert scheme of points on a surface
• Aug 11 -- Artem Kalmykov -- Mirror symmetry for Jacobians of hyper-elliptic curves
• Aug 4 -- Dmitrii Korshunov -- Bernstein-Sato polynomials and the division of distributions

• Jul 7: summary of seminar in 2023.1, and free discussion
• Jun 30 - Andrey Soldatenkov - o-minimal structure
• Jun 23 - Sergey Sergeev - WKB method
• Jun 16 - Sergey Burkin - Operads
• Jun 9 - holiday - Recesso de Corpus Christi on June 8-10
• Jun 2 - Rodrigo Matos - Fermi surface
• Jun 2 - Miguel Batista - mutually orthogonal latin squares and finite projective planes
• May 26 - Antonio Vasconcellos - random matrices
• May 19 - Graham Smith - Morse homology
• May 12 - Veronika Treumova - dessins d'enfant
• May 5 - Victor el Adji - determinantal varieties
• Apr 28 - Sergey Galkin - pencils of quadrics
• Apr 21 - holiday - Tiradentes
• Apr 14 - Filipe Nóbrega - pencils of conics
• Apr 7 - holiday - Sexta-feira da Paixão
• Mar 31 - Jacques Pienaar - QBism
• Mar 24 - Bruno Suassuna - formal perturbation theory for stochastic paths
• Mar 22, 17:00 - Bruno, Graham, Sergey, Veronika, Victor - free discussion

• Dec 16 - Marcos Craizer - Improper Affine Spheres and Monge-Ampère equations
• Dec 9 - Alexander Guterman - Permanents

Motivated by mirror symmetry for the moduli space of Higgs bundles, we discuss the construction of Lagrangian subvarieties that arise from symplectic representations. In particular, for the standard representation of the symplectic group, we show that this corresponds to a particular irreducible component of the nilpotent cone of the Hitchin system using an auxiliary (spinor) moduli space and Morse theory techniques.

In this talk I will discuss some speculative ideas (work in progress) about the study of mirror maps for Calabi-Yau hypersurfaces from the perspective of model theory (a branch of mathematical logic). I will propose that some of the arithmetic and geometric properties of those mirror maps can be understood from purely model theoretic concepts like categoricity. The case of elliptic curves will be discussed in detail since it is the motivation of my proposal. No knowledge of model theory will be assumed so I will introduce the notions needed for the talk.

Apresentamos os fundamentos da teoria da informação: em uma primeira palestra trataremos da informação de Shannon, em uma segunda palestra da geometria da informação de Fisher. A teoria da informação tem aplicações importantes em áreas como as telecomunicações e a estatística paramétrica. As palestras serão acessíveis aos alunos de graduação (um conhecimento basico da geometria de superfícies no espaço Euclidiano seria conveniente para a segunda palestra).

This (informal) talk will introduce and discuss various aspects of deformation quantization, including its motivation, existence and classification of star products, and Morita equivalence.

In this talk we are going to give a brief introduction to model categories and present a non-trivial example.

The Artin reciprocity law for a function field of a curve over a finite field can be interpreted as an isomorphism between the pro-finite completion of its Picard group and its abelianized étale fundamental group.
I will explain the Deligne's proof of the Artin's reciprocity law in the unramified case via an equivalence of categories of l-adic local systems of rank one on the curve and its Picard variety with some compatibilities by taking the pullback of the Abel-Jacobi map. Time permitting, we will discuss ramified, local, and characteristic zero versions of geometric class field theory.

Toric geometry (discovered in 1970s-1980s) and its modern generalization in a theory of [Newton-Okounkov bodies] (https://en.wikipedia.org/wiki/Newton%E2%80%93Okounkov_body) serve as bridge between two mathematical "worlds"

— convex geometry, and in toric case —combinatorics and piecewise-linear geometry — "mature" geometries (projective, algebraic, Kähler, symplectic, ...)

The notions of algebraic geometry (projective varieties, affine varieties, schemes, non-normal varieties; ampleness, discrepancies and types of singularities (smooth, terminal, canonical, log-terminal, (Q-)factorial, (Q-)Gorenstein..., Betti numbers) have counterparts in the world of convex and piecewise-linear lattice geometry (rational convex polytopes, rational convex cones, fans, sets in a lattice, convexity, reflexivity, simpliciality or simpleness, Delzant, ..., f-polynomials).

The bridge has proved to be useful in both directions: — Stanley's proof that Dehn-Sommerville relations completely determine possible numbers of faces of a convex simplicial polytope: by associating to each such polytope P a family of toric varieties X and showing that the respective numbers correspond to Betti numbers of (intersection) homology of X. In this light Dehn-Sommervile relations are seen as Poincaré duality and Hard Lefschetz theorem. — Bernstein—Kushnirenko theorem (1975) for number of roots of a system of polynomial equations as mixed volume of their Newton polytopes — explicit toric MMP, and later the use of birational cobordism and torification techniques to reproof resolution of singularities and weak facotorization theorem (AKMW, 2002) — Batyrev's construction (and its generalization by Borisov) of trillions of mirror-dual pairs of Calabi-Yau threefolds as resolutions of singularities of hypersurfaces in toric fourfolds associated with reflexive polytopes in 4 dimensions (resp. complete intersections associated with with higher-dimensional polytopes and nef partitions)

I will give basic definitions and sketch a dictionary, proving (or outlining/explaining) the results above.

The two topics are modern and classical. Gino Fano studied these manifolds maybe from 1900s (or 1890s) to his death in 1952. Mirror symmetry started in late 1980s and was formulated for Fano manifolds implicitly in early 1990s, and explicitly in 1996.

On symplectic side Fano manifolds are known as positive monotone symlpectic manifolds, whose earliest study is Archimedes's construction of a first integrable system of S^2: the one (according to Cicero) depicted on his grave (sphere X cylinder). On the complex side we call the very same S^2 as Riemann sphere, the best studied of all compact Riemann surfaces. Its mirror is the function, z + 1/z, used by Joukowski to engineer the wings of airplanes. Algebraic geometry of the same object goes back to the earliest antiquity — Babylonian mathematics, the Plimpton 322 papyrus (1800 BCE?) with solutions of Pythagorean triplets a^2 + b^2 = c^2, and all that. Two-dimensional (real four-dimensional) Fano manifolds are named after Pasquale del Pezzo, who gave a description of them in 1887, and in 1960s Yuri Manin related them to exceptional lattices E_n. Fano threefolds, that Fano himself studied are related to K3 surfaces and canonical curves of small genus.

As seen through the looking glass, Fano manifolds do not look like spaces, but more like functions! Ginzburg-Landau theory of superconductivity (1950s) is often cited in this context of mirror symmetry.

I am going to explain what all (or some of) these words mean, what problems are there, why in some sense this always used to be main topics in algebraic and symplectic geomegry, and show a brief demo, so that seminar participants might start doing new discoveries in these fields.

In this short talk, the goal is to discuss rational maps from (C^)^n to itself that preserves the usual logarithmic-volume form. For n=1, the form is dz/z, and for (C^)^n we take the product over coordinates z_1,...,z_n. For n=1, the only rational map that preserves the log-volume form is f(z) = c z; there are much more interesting examples for n>1. We state the Scaling-Winding Theorem for rational maps which scale the log-volue form and discuss how we can prove it; for this we need to define the naive tropicalization of the map. In analogy with the famous (open) Jacobian Conjecture, we can ask: if a rational endomorphism of (C^*)^n that preserves the usual log-volume form must be bi-rational. This is not true! We will show a counter-example for n=2, which in light of the Scaling-Winding theorem means a rational map from 2 variables to itself, preserving the log-volume form and which has winding number 2.

The second talk in this series will focus on linearization in equivariant cobordism, culminating in the proof of the Quantization commutes with Reduction theorem.

The pentagram map on polygons in the projective plane was introduced by R.Schwartz in 1992 and is by now one of the most popular and classical discrete integrable systems. We survey definitions and integrability properties of the pentagram maps on generic plane polygons and their generalizations to higher dimensions. In particular, we define long-diagonal pentagram maps on polygons in $\mathbf{RP}^d$, encompassing all known integrable cases. We also describe the corresponding continuous limit of such pentagram maps: in dimension d it turns out to be the $(2, d + 1)$-equation of the KdV hierarchy, generalizing the Boussinesq equation in 2D. This is a joint work with F.Soloviev and A.Izosimov.

In Hamiltonian mechanics, the phase space is the cotangent bundle $T^* X$ of the configuration space X. In quantum mechanics, the state space is the Hilbert space of complex valued "wave functions" $L^2(X)$. The theory of geometric quantization attempts to understand the correspondence between $T^* X$ and $L^2(X)$.

Geometric quantization is intricately connected to complex and symplectic geometry, representation theory of Lie groups, the Atiyah-Singer theorem, and more. In this talk we will delve into these connections in detail following the Ginzburg-Guillemin-Karshon book "Moment maps, cobordisms, and Hamiltonian group actions".

The second talk in this series will focus on linearization in equivariant cobordism, culminating in the proof of the Quantization commutes with Reduction theorem.

This is the first of two talks about Higgs bundles over a curve. Designed with students in mind, our aim is to provide an overview of some aspects of the Higgs bundles and their moduli space. By recalling these concepts, we lay the groundwork for the subsequent talk.

Protons and neutrons are nucleons, and each nucleon is made of elementary particles; quarks and gluons. These elementary particles interact among themselves through strong nuclear force (strong force). Quantum Chromodynamics (QCD) is the branch of theoretical physics that describes the strong force. In this talk we will explain the mathematical construction of the lattice QCD model in the strong coupling regime. Since our goal is to understand the mathematical structure of the model, we will basically focus on the consequences of the thermodynamic limit of correlation functions and the Osterwalder-Schrader positivity.

It is not hard to see that a holomorphically symplectic form defines a complex structure uniquely. The notion of а C-symplectic structure is used to define the holomorphically symplectic manifolds with no reliance on complex structures. Deformations of complex structures are usually obtained using the solutions of the appropriate Maurer-Cartan equation. I would explain what equation plays its role in the C-symplectic geometry, and construct a recursive solution for the C-symplectic Maurer-Cartan, which is in fact much easier than the classical deformation theory. This gives a new and very simple proof of Bogomolov's local Torelli theorem for hyperkahler manifolds. I would explain how this construction is used to give a generalization of Voisin's famous theorem about deformations of holomorphic Lagrangian subvarieties. This is a joint work with Nikon Kurnosov.

In the realm of topological quantum field theories (TQFTs), Moore and Tachikawa's conjecture 1106.5689 posits the existence of a two-dimensional TQFT, $\eta_{G_{\mathbb{C}}}:Cob2 \to MT$, associated with each complex semisimple group $G_{\mathbb{C}}$. This TQFT is characterized by a target category, $\mathrm{MT}$, with complex Lie groups as objects and holomorphic symplectic varieties featuring Hamiltonian actions of the groups as morphisms. On this talk, we will discuss a real analogue $MT_{\mathbb{R}}$ of the target category $\mathrm{MT}$ proposed by Olivier Chiriac in 2111.13268.

The talk is about the Wigner-Weyl transform (Eugene Wigner and Hermann Weyl). This is an example of "quantization", in the physicist's sense: a way to define quantum observables (self-adjoint operators on a Hilbert space) from functions in the symplectic vector space R^{2n}. The goal is to introduce the idea of quantization through this very particular example, since later we will have lots of talks with "quantization" in the title!

We will discuss the relationship between the symplectic geometry of Kapovich-Millson space of polygons and the space of isometric realizations of a polyhedral surface in the three dimensional euclidean space. It turns out that the boundaries of all realizations of a generic polyhedral disk sweep out a Lagrangian subset in the space of polygons.

The scarred wavefunctions of the desymmetrised $PSL(2,\mathbb{Z})$ group and those of its congruence subgroups are newly studied. The construction of irrationals after the (Farey)-Pell method is compared with the qualities of the quadratic fields. The action of automorphisms on trees is recalled to explain the action of the Hecke operaors on the Maass waveforms. The Margulis measure (which acquires a multiplicative constant under the action of certain $U$ flows) is used. The opportune Kirkhoff reduced surfaces of section are chosen. Closed geodesics are newly proven to scar the waveforms according to the quadratic field they are constructed after: a) In the desymmetrised $PSL(2,\mathbb{Z})$ group, the scarred waveforms are newly proven to be obtained under the action of the $U$ flow on the Margulis measure which acts on the quadratics fields which define the (also, classes) of closed geodesics. b) In the congruence subgroups of the desymmetrised $PSL(2,\mathbb{Z})$ domain, the scarred wavefunctions are newly proven to occur under the effect of the action of the Bogomolny transfer operators on the Margulis measure.

By definition, a quiver is a directed graph.
In this talk, we will recall the notion of a quiver representation. The category of quiver representations can be highly non-trivial, e.g. the category of modules over any finitely-dimensional algebra over a closed field is equivalent to the category of quiver representations for some quiver.
However, for some quivers, their representations have only finitely many indecomposables (meaning they are much easier to study).
Famously, due to Gabriel, those are precisely the simply-laced Dynkin quivers: $A_n$, $D_n$, $E_6$, $E_7$, $E_8$. I will explain how the corresponding combinatorics of root systems arise from representation theory via Bernstein--Gelfand--Ponomarev reflection functors and prove Gabriel's theorem.

(Based on a Final Project at IMPA) Discovering symplectic shapes becomes clear through Delzant's work in symplectic geometry. His theorem helps sort and classify symplectic toric manifolds by using a special map that shows how these shapes fit together. It's like finding unique patterns called "Delzant" polytopes that match each manifold. This talk offers a simple crash course in symplectic geometry, focusing on explaining Delzant's Correspondence Theorem.

We study the category of matrix factorizations associated to the germ of an isolated hypersurface singularity. This category is shown to admit a compact generator which is given by the stabilization of the residue field. We deduce a quasiequivalence between the category of matrix factorizations and the dg-derived category of an explicitly computable dg algebra. Building on this result, we employ a variant of Töen’s derived Morita theory to identify continuous functors between matrix factorization categories as integral transforms. This enables us to calculate the Hochschild chain and cochain complexes of these categories. Finally, we give interpretations of the results of this thesis in terms of noncommutative geometry based on dg categories.

The moduli space $M(G)$ of Higgs bundles for a complex reductive group $G$ on a compact Riemann surface carries a natural hyperkahler structure and it comes equipped with an algebraically completely integrable system through a flat projective morphism called the Hitchin map. Motivated by mirror symmetry, I will discuss certain complex Lagrangians (BAA-branes) in $M(G)$ coming from real forms of $G$ and give a proposal for the mirror ($BBB$-brane) in the moduli space of Higgs bundles for the Langlands dual group of $G$.  In this talk, I will focus on the real groups $SU^(2m)$, $SO^(4m)$ and $Sp(m,m)$. Higgs bundles for these groups have non-reduced spectral curves and we are led to describe certain subvarieties of the moduli space of rank 1 torsion-free sheaves on ribbons. If time permits we will also discuss another class of complex Lagrangians in M$(G)$ which can be constructed from symplectic representations of $G$.

I will talk about the Markov Chains. This term refers to the sequence of random variables where there is a single series dependency between adjacent periods. From this, I will talk about Hidden Markov Model: a process where hidden parameters are determined from observable parameters.

We discuss several recent results of ongoing work (in collaboration with I. Gaiur and D. Van Straten and with W. Buchstaber and I. Gaiur) on interesting properties of multiplicative generalized Bessel kernels, which include the famous Clausen and Sonin-Gegenbauer formulas, examples of polynomials of Kontsevich discriminant locus, given as addition laws for special two-valued formal groups (Buchstaber-Novikov-Veselov), as well as connections with “period functions”, solving some Picard-Fuchs type equations for Calabi-Yau cases and associated with analogues of Landau-Ginzburg superpotentials.

Nov 3: We present a new completely elementary model that describes the creation, annihilation, and motion of non-interacting electrons and positrons along a line. It is a modification of the model known under the names Feynman checkers, one-dimensional quantum walk, or the Ising model at imaginary temperature. It can be viewed as a six-vertex model with certain complex weights of the vertices. We show that the discrete model is consistent with the continuum quantum field theory, namely, reproduces the known expected charge density as the lattice step tends to zero. Most of the talk is accessible to undergraduate students; no knowledge of physics is assumed.

Nov 10: This is a continuation of the previous talk. Last time we discussed a quantum mechanical model, and now we proceed to quantum field theory. We present a new completely elementary model that describes the creation, annihilation, and motion of non-interacting electrons and positrons along a line. It can be viewed as a six-vertex model with certain complex weights of the vertices. We show that the discrete model is consistent with the continuum quantum field theory, namely, reproduces the known expected charge density as the lattice step tends to zero.

joint work with A. Ustinov

A spherical surface is a surface that can be glued from a collection of spherical triangles by isometric identification of their sides. Such a surface has a metric of curvature one outside of a finite number of points, where the metric has a conical singularity. In particular, each spherical surface is naturally a Riemann surface. Contrary to the hyperbolic case, when the theory is identical to the theory of Riemann surfaces, the case of spherical surfaces is almost totally open. I will speak about recent results in the area, such as a full description of the moduli space of spherical metrics with one conical singularity on a torus (joint work with Gabrielle Mondello and Alex Eremenko) and the description of possible conical angles on a spherical metric on a 2-sphere (joint work with Gabirelle Mondello).

In the second talk of the series I explained a procedure of tropicalizations of rational maps preserving logarithmic volume form.

I would introduce these things and show some formal computations of exponentials of not so trivial Hamiltonians. Would be based on what I learned working in my first physics paper.

In 1981, André Joyal started the development of an interesting theory of species of structures. In his theory, combinatorial species (e.g. graphs, permutations) can be seen as endofunctors of the category of finite sets and bijections. This allows the study of operations, like addition and multiplication, of species, which became a key technique for enumeration. In this talk, we will present a brief introduction to this technique, and some interesting examples of its usage in enumerative combinatorics.

Let $f(x,y) = P(x,y)/R(x,y)$ and $g(x,y) = Q(x,y)/R(x,y)$ be a pair of rational functions, i.e. elements of a field $\mathbf{C}(x,y)$ of fractions of a ring of polynomials $\mathbf{C}[x,y] \ni P,Q,R$ (in the first talks of the series the variables x and y will commute, but later they will stop commuting). For (x,y) a pair of complex numbers not in zero locus of PQR the map $(x,y) \mapsto (f(x,y),g(x,y))$ is well-defined and takes non-zero values, so a pair of rational functions f,g detrmine a rational map $φ : T - - - > T$ from a two-dimensional (complex) torus (or any other rational surface) to itself. We say that rational map φ (almost) preserves logarithmic volume form $ω := dlog(x) \wedge dlog(y) = \frac{dx \wedge dy}{xy}$ if there exists some complex number δ such that φ^* ω = δ ω, that is $\frac{df \wedge dg}{fg} = δ \cdot \frac{dx \wedge dy}{xy}$, explicitly it means that a pair of rational functions f,g in $\mathbf{C}(x,y)$ satisfy the following homogeneous PDE of degree 2 $x y (f_x g_y - f_y g_x) = δ f g$ This is one of the simplest (maybe, the simplest) examples of volume-preserving maps between log-Calabi-Yau varieties as defined in Eduardo's talk.

In his preprint Symplectic automorphisms of CP^2 and the Thompson group T (math/0611604) Alexandr Usnich defined a tropicalization of such maps — a homomorphism from the group of birational (invertible) volume-preserving maps to the group of piecewise-linear automorphism of R^2.

Usnich also described some their stuctural properties, including a conjecture on generators and relations, that was proved by Jérémy Blanc in Symplectic birational transformations of the plane (1012.0706). Blanc in his proof used heavily theory of algebraic surfaces.

In other vein, Jeffrey Diller and Jan-Li Lin in Rational surface maps with invariant meromorphic two forms (1308.2567) motivated by study of birational dynamic also used theory of surfaces to prove Theorem E analogous to Usnich's construction, and then prove Degree Factorization Theorem: "topological degree" (which they denote $λ_2$) of a rational log-volume preserving map (x,y) —> f(x,y),g(x,y), i.e. the number of solutions of a system of equations f(x,y) = a, g(x,y) = b for generic pair (a,b) equals to the product S W, where numbers S=|δ| and W can be read off from tropicalization: W is the winding number of the respective continuous map from the circle to itself, and S is a scaling factor which is shown to be equal to δ. See Theorem 6.18 in loc.cit.

I will prove higher-dimensional generalization of Degree Factorization Theorem: let $φ = (f_1(x_1,...,x_n)),...,f_n(x_1,...,x_n))$ be a rational map that preserves a logarithmic volume form $ω = \frac{d x_1 \wedge ... \wedge d x_n}{x_1 ... x_n}$ up to rescaling it by a complex constant δ,i.e. $φ^* ω = δ ω$. Then the respective scaling constant $δ \in \mathbf{C}$ is in fact an integer S, and the topological degree of φ factors as S W with W being a winding number, the degree of a map from (n-1)-dimensional sphere to itself obtained from tropicalization. Unlike work of Blanc and Diller-Lin, both relying on geometry of algebraic surfaces, other than standard tropical techniques including amoebas that I will explain in detail, I have to resort to analytic techniques, namely Griffiths's characterization of logarithmic forms in terms of grows asymptotics of respective integrals.

Another related question is a rational logarithmic Jacobian conjecture — is it true that volume-preserving map is birational? In light of decomposition theorem above the counterexample shall involve log-volume-preserving maps with non-trivial winding numbers. I will explain geometric construction of some such maps already in dimension two. To the best of my knowledge no explicit formulas have been written down so far, despite theoretically this is being computable, and participants of the seminar shall be able to do it.

In the context of algebraic geometry, decomposition and inertia groups are special subgroups of the Cremona group which preserve a certain subvariety of $\mathbf{P}^n$ as a set and pointwise, respectively. These groups were and still are classic objects of study in the area, with explicit descriptions in several instances. In the particular case where this fixed subvariety is a hypersurface of degree n+1, we have the notion of Calabi-Yau pair which allows us to use new tools to deal with the study of these groups and one of them is the so-called volume preserving Sarkisov Program. Using this approach we prove that an appropriate algorithm of the Sarkisov Program in dimension 2 applied to an element of the decomposition group of a nonsingular plane cubic is automatically volume preserving. From this, we deduce some properties of the (volume preserving) Sarkisov factorization of its elements. Regarding now a 3-dimensional context, we give a description of which toric weighted blowups of a point are volume preserving and among them, which ones will initiate a volume preserving Sarkisov link from a Calabi-Yau pair $(\mathbf{P}^3,D)$ of coregularity 2. In this case, D is necessarily an irreducible normal quartic surface having canonical singularities. This last result enhances and extends the recent works of Guerreiro and Araujo, Corti and Massarenti in a log Calabi-Yau geometrical perspective, and it is a possible starting point to study the decomposition group of such quartics.

A Hilbert scheme of points on a scheme X parametrizes configurations of n points on X. If X is a non-singular surface, the Hilbert scheme $X^{[n]}$ is smooth and maps to the symmetric power $S^n X$ by Hilbert-Chow morphism. Hilbert schemes are connected to resolutions of singularities, hyperkähler geometry, quiver varieties, representation theory, and much more.
In this talk, we will prove Göttsche's formula for computing the Betti numbers of the Hilbert scheme $X^{[n]}$ of $n$ points on a surface. We show how it is a corollary of the general theory of intersection cohomology and the decomposition theorem of Beilinson, Bernstein, Deligne, and Gabber. The talk is based on Nakajima's book on Hilbert schemes.

In this talk I will present a mirror symmetry constructions for Jacobians of hyperelliptic curves. In particular, using the abelian/non-abelian correspondence by Ciocan-Fontanine--Kim--Sabbah arXiv:math/0610265, we will calculate its J-function which usually encodes the Gromov-Witten invariants of a variety, but turns out to be nontrivial in this case. Furthermore we will construct a certain family of complex tori that could be considered as the mirror family by some numerical evidences. Based on work "Mirror symmetry for Jacobians of hyperelliptic curves.

We will discuss some basic properties of left modules over the Weyl algebra ('algebraic D-modules on an affine space') up to the J. Bernstein's inequality and the notion of a holonomic module. We'll proceed to prove the existence of Bernstein-Sato polynomials and employ them to analytically continue complex powers of polynomials. As an application, a Hironaka-free solution to the problem of division of distributions by polynomials will follow. I will conclude with a remark about how all this provides a purely algebraic proof of theorems of Malgrange-Ehrenpreis and Lojasiewicz-Hormander about the existence of fundamental solutions of linear PDEs with constant coefficients.

O-minimal structures are collections of subsets in $\mathbb{R}^n$ that have certain finiteness properties that make them similar to semialgebraic sets. I will start by recalling the definition of the latter and discussing the Tarski--Seidenberg theorem claiming that a linear projection of a semialgebraic set is again semialgebraic. Then I will try to give an overview of how one can generalize this and obtain the notion of an o-minimal structure. I expect that the talk will mostly be informal, since I am not a specialist in o-minimal structures and got interested in them because of their recent applications to Hodge theory.

In many physical problems the small parameter $h$ appears in the PDE and sometimes such a parameter stands with the operator of differentiation, for example $h\partial / \partial x$. One can ask very natural question about asymptotics of the solution for such PDE while $h\to 0$. Traditionally one of the answers on this question is the WKB approximation, when the solution is presented in the form of fast-oscillating exponential. This method gives the formal asymptotic solution for the PDE. From the other hand it is well known that WKB approximation fails near the so-called turning points which means that such form of the solution is not valid. We will present the generalization of the WKB approximation called the Maslov's Canonical Operator which allows to present asymptotics near turning points. This generalization admits the very interesting geometrical interpretation and also involves the Hamiltonian mechanics. As an example we will consider the 1D equation for the wave propagation in crystall and discuss some open questions related to this example.

Operads are structures similar to categories. Morphisms in an operad can have any number of inputs and exactly one output. Each operad encodes a type of algebra. Such types include associative algebras, Lie algebras, BV-algebras and other structures appearing in mathematics and physics.

We will describe some of the main applications and ideas in operad theory, including recognition principle, Koszul duality, formality of little disks and deformation quantization, Grothendieck—Teichmuller theory, and theory of homotopy coherent structures.

The structure of the dispersion relation is one of the central aspects to the study of periodic Schrödinger operators.

Besides the intrinsic interest from the viewpoint of several complex variables and algebraic geometry, the dispersion relation also carries relevant information for the spectral theory of periodic media. In particular, for the structure of spectral boundaries, isospectrality, and existence of eigenvalues for locally perturbed operators.

I will discuss some of these connections as well as recent irreducibility theorems for the Bloch and Fermi varieties, focusing on two joint works with Jake Fillman and Wencai Liu:

• Algebraic Properties of the Fermi Variety for Periodic Graph Operators, 2305.06471
• Irreducibility of the Bloch Variety for Finite-Range Schrödinger Operators, 2107.06447

These recent papers cover a wide class of lattice geometries in arbitrary dimension and verify the discrete version of certain conjectures of Kuchment for various discrete models.

A comment from Sergey: Rodrigo will introduce us to an area at the intersection of algebraic geometry, spectral theory, and mathematical physics. The very same Laurent polynomials appear in the study of Fermi varieties of periodic Schrödinger operators, in mirror symmetry for Fano varieties and K3 surfaces, in Apery's approximation for zeta(3), in two-loops diagrams for Bhabha scattering, and many other places.

I will introduce the concept of a finite projective plane (FPP), their basic structure, and show the existence of some of them. Beside that, I will talk about Mutually Orthogonal Latin Squares (MOLS), a "categorical" way to visualize it and how they interact with FPP. If possible I will talk about more complex FPP existence theorems.

In this seminar, we present a combinatorial proof of Wigner's Semi-Circle Law, a fundamental result in random matrix theory. It describes the limiting behavior of the eigenvalue distribution of large Gaussian Orthogonal Ensemble (GOE) matrices. While the original proof of the Semi-Circle Law relies on advanced techniques from analysis, our approach provides an alternative perspective through combinatorial methods.

I will present the main ideas underlying Morse homology theory in the finite dimensional case and will discuss some applications and conjectures in the infinite dimensional setting.

Dessin d’enfant (child’s drawing, map) is a graph embedded into a two-dimensional surface. The term was introduced by A. Grothendieck in 1984 in his famous "Esquisse d’un Programme". The theory of dessins d’enfants in a beautiful way connects 19th century complex analysis with 20th century arithmetic geometry, and very simple and intuitive objects with very abstract concepts. Besides overview of the topic I will present some results in one specific area, which is enumeration of graphs of a certain type.

I will introduce and study (generic) determinantal varieties: the loci of matrices of rank bounded from above. In addition to being a central topic in both Commutative Algebra and Algebraic Geometry, these varieties have several connections with Invariant Theory, Representation Theory and Combinatorics. For example, many classical constructions such as rational normal curves, Veronese manifolds, Segre manifolds, rational normal scrolls, fall into the class of determinant varieties.

I will describe their fundamental properties (such as normality, irreducibility, singular locus, etc) and calculate their basic invariants (degree, dimension, topological Euler number, etc). To do so, I will establish the first and the second fundamental theorems of invariant theory as one of the main tools.

I will introduce higher-dimensional analogues of pencils of conics (introduced by Filipe on April 14) - namely pencils of quadrics, for example I will explain a classical construction that relates pencils of quadrics with hyper-elliptic curves (double covers of a projective line), and discuss how their linear algebra and geometry is related to various questions about moduli spaces of bundles on these curves, in particular how to construct some forms of Jacobians and moduli spaces of stable rank 2 bundles on a hyper-elliptic curve in terms of a pencil of quadrics and the respective matrix of linear forms (Miles Reid's thesis and Desale-Ramanan's theorem).

Just one dimension higher - if we consider pencils of two-dimensional quadrics in a three-dimensional projective space (i.e. 4-times-4 symmetric matrices with entries in homogeneous linear polynomials of two variables) we obtain models of elliptic (genus one) curves, and can also relate it to a version of Poncelet theorem in three-dimensional space. When we consider pencils of 4-dimensional quadrics in 5-dimensional projective space their base locus is an interesting Fano threefold, and the respective spectral curve is a curve of genus two (all of them are hyper-elliptic), with the base locus (Fano threefold) being one of the simplest moduli spaces of vector bundles, and the respective Jacobian of a genus two curve being isomorphic to the variety of lines isotopic with respect to all quadratic forms in the pencil.

This is a somewhat classical geometric topic that easily related to both well-known constructions as well as interesting open problems related to some of my research. So it could be useful to students interested in algebraic geometry or conformal field theory, as well as to those whose research is related to elliptic and genus two curves and their Jacobians, among many others.

A conic is an algebraic curve given by the zero set of a homogeneous polynomial of degree two in three variables, as $ax^2+2hxy+by^2+2fxz+2gyz+cz^2$. Such an expression is given by six coefficients, but multiples of a given equation correspond to the same curve. Therefore, the moduli space of real conics and of complex conics are $\mathbf{RP}^5$ and $\mathbf{CP}^5$ respectively.

A real (complex) pencil of conics is a one-parameter family of conics given by a real (complex) line in the suitable moduli space. Let $u$ and $v$ be two distinct conics, the pencil that contains them can be algebraically given by $\alpha u + \beta v$, where $[\alpha \colon \beta] \in \mathbf{RP}^1$ (or $\mathbf{CP}^1$). In this talk we will describe and classify all orbits of pencils of conics under the action of the projective group $\mathrm{PGL}(3,\mathbf{R})$ ( and $\mathrm{PGL}(3,\mathbf{C})$ ).

• Short abstract for mathematicians:

We discuss methods of perturbative QFT applied to certain stochastic differential equations (SDE), perturbations of the Ornstein-Uhlenbeck process. No knowledge of SDE or QFT is assumed.

Perturbative QFT amounts to some sort of generating function for formal power series expansions, and these methods have found many applications in pure and applied mathematics. This is the main motivation for this talk. The lack of mathematical proofs often comes together with the magical formulas derived from these methods, and this will also be examplified in this talk.

The application we discuss was the subject of joint work with Bruno Melo (ETH Zurich) and Thiago Guerreiro (PUC-RIO), published (PhysRevA) in January 2021.

• Fine-print for physicists:

The motivation comes from experiments with optical tweezers, which roughly are laser beams able to trap certain particles around a point of mechanical equilibrium. The particle does not sit in the mechanical equilibrium, but jiggles stochastically around it; this stochasticity is known as Brownian motion and is the reason one uses SDE instead of ODE in the mathematical models. Often the working assumption is that the optical forces are a linear function of the displacement from mechanical equilibrium, which as we know is simply a first order Taylor approximation. We provided a way to deal with optical forces modelled by a linear function perturbed by small non-linearities, so potentially this method would be useful for sufficiently precise experiments. The validity of our method was probed by some numerical experiments, and for mysterious reasons it seems to work.

Nesta aula vamos falar da relação entre esferas afins impróprias e equações de Monge-Ampère. Discutiremos a construção centro-corda para esferas afins impróprias de dimensão 2 e as singularidades genéricas destas superfícies. Mostramos também uma generalização desta construção para dimensões pares arbitrárias.

Slides of the talk in PDF: http://www.mat.puc-rio.br/Upload/Arquivo/2022/12/guterman22decrio.pdf The classes of $(-1,1)$-matrices are very important in algebra and combinatorics and in various their applications. For example, well-known Hadamard matrices are of this type.

Two important functions in matrix theory, determinant and permanent, look very similar: $$\mathrm{det} A = \sum_{\sigma\in \Sigma_n} (-1)^{\sigma} a_{1\sigma(1)}\cdots a_{n\sigma(n)}$$ and $$\mathrm{per} A = \sum_{\sigma\in \Sigma_n} a_{1\sigma(1)}\cdots a_{n\sigma(n)}$$ here $A=(a_{ij})\in M_n(\mathrm{F})$ is an $n\times n$ matrix and $\Sigma_n$ denotes the set of all permutations of the set ${1,\ldots, n}$.

While the computation of the determinant can be done in a polynomial time, it is still an open question, if there are such algorithms to compute the permanent.

In 1974 Wang posed a question to find a decent upper bound for $|\mathrm{per}(A)|$ if $A$ is a square $\pm 1$-matrix of rank $k$. In 1985 Kräuter conjectured a certain upper bound.

We prove the Kräuter's conjecture and thus obtain the complete answer to the Wang's question. In particular, we characterized matrices with the maximal possible permanent for each value of $k$.

We also plan to discuss other problems related to permanent for $(0,1)$ and $(-1,1)$-matrices.

The talk is based on the joint work with M.V. Budrevich.

• A.R. Kräuter, Recent results on permanents of (+1, -1)-matrices, Ber. No. 249, Berichte, 243-254, Forschungszentrum Graz, Graz, 1985.
• E.T.H. Wang, On permanents of (+1, -1)-matrices, Israel J. Math., 18, 1974, 353-361.

• Henrique Bursztyn: deformation quantization
• Rafael Ruggiero: Information Geometry and thermodynamics
• Lada: Riemann-Hilbert correspondence
• Filipe: affine differential geometry and projective differential geometry. Schwarzian derivative.
• Alex Gomez: Torelli's theorem for compact Riemann surfaces, Henrik H. Martens' proof
• Altan Erdnigor: a proof of Torelli theorem for curves via Fourier--Mukai transform
• Arthur Moreira: domino in 3 dimensions
• Mukhopadhyaya's Four-Vertex Theorem day: Samuel Pacitti Gentil, Petr Pushkar, Filipe, and others (hybrid)
• Iulia Gorginian: TBA
• Maxim Smirnov: TBA (online)
• Misha Verbitsky: special affine Hessian manifolds and Monge-Ampere equation (TBC/TBA)
• Israel Vainsencher: TBA
• Thiago Guerreiro: Unruh effect for mathematicians

• Holomorphic Floer Theory: Kapustin-Rozansky, Kontsevich-Soibelman, Doan-Rezchikov (Fueter equations).
• SUSY, Nahm's theorem on classification of supersymmetry
• Renormalization in QFT (Quantum Field Theory): Costello's book and Connes-Kreimer's Hopf algebra aproach
• Combinatorial Physics
• CFT (Conformal Field Theory). Wess-Zumino-Novikov-Witten model. Verlinde formula. Tsuchiya-Ueno-Yamada construction. Fusion categories.
• TQFT (Topological QFT), extended TQFT, fully extended TQFT and cobordism hypothesis
• more applications of holonomic D-modules, perverse sheaves, and Riemann--Hilbert correspondence
• Hitchin integrable system on moduli of Higgs bundles. Gaudin's model as parabolic genus 0 case. Beilinson-Drinfeld's quantization.
• Stability, GIT, moduli spaces, Narasimhan-Seshadri theorem
• Parabolic bundles and Mehta-Seshadri theorem. Relation to moduli spaces of spherical polygons.
• Moduli spaces of Euclidean polygons, Kapovich-Millson's system of bending flows as a limit of Hitchin system, e.g. after thesis of Fabiola Cordero: integrable systems on spaces of polygons, matrices and bundles: Hitchin, Gaudin, Garnier, Kapovich-Millson bending flow, Jeffrey-Weitsman, Gelfand-Zetlin, Nishinou-Nohara-Ueda, Manon-Belmans-Galkin-Mukhopadhyay, etc
• Additive Horn problem, cohomology of Grassmannian, Knutsen-Tao honeycombs and their polytopes, planar networks
• Multiplicative Horn problem, quantum cohomology of Grassmannian, and the respective polytopes
• Non-abelian Hodge theory of Carlos Simpson: three invarnations of representations of fundamental group
• Kapustin--Witten perspective on geometric Langlands as a combination of N=4 SYM, S-duality, compactification on Riemann surface and mirror symmetry. Wilson and t'Hooft operators.
• geometric Langlands: De Rham (Beilinson--Drinfeld, Arinkin--Gaitsgory) X Dolbeault (Donagi--Pantev) X Betti (Ben-Zvi--Nadler)
• Atiyah-Floer conjecture and Donaldson-Floer theories
• matrix factorizations of a polynomial, Landau-Ginzburg models, singularity category, Khovanov-Rozansky link invariants
• Gauged Linear Sigma Models and phase transition, VGIT and MMP, Goldstone and Higgs mechanisms of mass generation
• semiorthogonal decompositios
• Calabi--Yau theorem and its proof (Nikita Klemyatin, TBC)
• other research-related topics (add yours)