One formulation of mirror symmetry predicts (omitting a few adjectives) a 1-1 correspondence between equivalence classes of certain lattice polygons and deformation families of certain del Pezzo surfaces.
Lattice polygons corresponding to smooth Del Pezzo surfaces are called T-polygons, and these have been classified by Kasprzyk-Nill-Prince using combinatorial methods. I will sketch a new geometric proof of their classification result.
Quasimaps provide an alternate curve counting system to Gromov-Witten theory, which are related by wall-crossing formulae. Relative (or logarithmic) Gromov-Witten theory has proved useful for constructions in mirror symmetry, as well as for determining ordinary Gromov-Witten invariants via the degeneration formula. Different versions of this theory rely on various technologies, including expansions (or accordions) as well as logarithmic structures. I will discuss how to use a hybrid of these approaches to produce a proper moduli space parametrizing quasimaps relative a smooth divisor in any genus.
Many of the algebraic structures we construct from geometric data are represented 'motivically', that is, their structure constants are obtained by pushing and pulling 'coefficients' (e.g. functions, sheaves) along diagrams like X ← W → Y. In this quasi-survey talk, I will explain how many of the convenient properties of the algebraic categories we like to work in (e.g. vector spaces, dg-categories) are already present in categories of correspondences themselves. This explains their frequent appearance in the study of universal homology theories.
While Gorenstein codimension 3 varieties are well understood from Buchsbaum-Eisenbud's structure theorem, the picture is less clear for codimension 4. In this talk we describe some constructions of stable curves corresponding to the possible Betti tables for Artin Gorenstein algebras of regularity and codimension four, as outlined in a paper of Schenck, Stillman and Yuan. These constructions use techniques from liaison theory and the Tom and Jerry formats of Brown and Reid.
In this talk, I will first introduce the background of Bridgeland stability condition. Then I will mention some existence result of Bridgeland stability. Next I will prove the Bogomolov-Gieseker type inequality of X(2,4), Calabi-Yau threefold of complete intersection of quadratic and quartic hypersufaces, by proving the Clifford type inequality of the curve X(2,2,2,4). Then this will provide the existence of Bridgeland stability condition of X(2,4).
I will present a four-term exact sequence relating the cohomology of a fibration to the cohomology of an open set obtained by removing the preimage of a general linear section of the base. This exact sequence respects three filtrations, the Hodge, weight, and perverse Leray filtrations, so that it is an exact sequence of mixed Hodge structures on the graded pieces of the perverse Leray filtration. I claim that this sequence should be thought of as a mirror to the Clemens-Schmid sequence describing the structure of a degeneration and formulate a 'mirror P=W' conjecture relating the filtrations on each side. Finally, I will present evidence for this conjecture coming from the K3 surface setting.
The set of weights of T-invariant curves to a Schubert variety at a T-fixed point admits a simple description due to Carrell and Peterson. The set of weights to the tangent space is much more complicated to describe, and has only been explicitly described in classical types. The main result is that although these two sets of weights are different, they generate the same cone in the dual of the Lie algebra of T.
This is a report on work in progress with my student Cristian Rodriguez. The mirror of a ℚ-Fano 3-fold with b2 = 1 is a rigid K3 fibration over ℙ1 such that Hodge bundle is degree 1 and some power of the monodromy at infinity is maximally unipotent. Although prior work focused on the maximally unipotent case (without base change), perhaps a classification of such Picard-Fuchs equations is possible.
In the smooth case these fibrations were described explicitly by Przyjalkowski, and Doran-Harder-Novoseltsev-Thompson showed that they are given by etale covers of the (1-dimensional) moduli of rank 19 K3 surfaces. In the case of a single 1/2(1,1,1) singularity they are given by rigid rational curves on the (2-dimensional) moduli of rank 18 K3 surfaces, and examples suggest they are Teichmuller curves in A2 (via the Shioda-Inose correspondence relating rank 18 K3s and abelian surfaces), as studied by McMullen.
If a Calabi-Yau threefold varies in a one-parameter family and aquires some double points, a small resolution will produce a rigid space. The local monodromy at such a 'conifold transition' is of infinite order. In the talk I report on some work done with S. Cynk (Krakow), which shows similar transitions to rigid Calabi-Yaus are possible with monodromy of finite order, in sharp distinction to what can happen for K3 surfaces.
Quandles are algebraic structures designed to mesh with the Reidemeister moves of knot theory. Joyce and Matveev showed that quandles give rise to a complete invariant of oriented knots. Since the Yang-Baxter equation resembles the third Reidemeister move, it is not surprising that quandles also form a class of set-theoretic solutions of the Yang-Baxter equation. In this talk I will explain how quandles and connected quandles can be enumerated up to isomorphism and list a few open problems. I will also present two additional classes (involutive and idempotent) of set-theoretic solutions of the Yang-Baxter equation with rich algebraic theory.
We extend Bourke and Garner's idempotent adjunction between monads and pretheories to the framework of ∞-categories, and exploit this to prove many classical theorems about monads in the ∞-categorical setting. Among other things, we prove that the category of algebras for an accessible monad on a locally presentable ∞ category is locally presentable. We also apply the result to construct examples of ∞-categorical monads from pretheories.
In the recent breakthrough on the uniform Mordell-Lang problem by Dimitrov-Gao-Habegger and Kuhne, their key result is a uniform Bogomolov type of theorem for curves over number fields. In this talk, we introduce a refinement and generalization of the uniform Bogomolov conjecture over global fields, as a consequence of bigness of some adelic line bundles in the setting of Arakelov geometry. The treatment is based on the new theory of adelic line bundles of Yuan-Zhang and the admissible pairing over curves of Zhang.
Let G be a complex, connected, reductive, algebraic group, and χ : ℂ× → G be a fixed cocharacter that defines a grading on 𝔤, the Lie algebra of G. In my first talk I have talked about the grading, derived category of equivariant perverse sheaves, bijection between the simple objects and some pairs that we are familiar with. In positive characteristic parity sheaves will play an important role. In this talk I will define parabolic induction and restriction both on nilpotent cone and graded setting. We will dive into the results of Lusztig in characteristic 0 in the graded setting. Under some assumptions on the field k and the group G we will recover some results of Lusztig in characteristic 0. These assumptions together with Mautner's cleanness conjecture will play a vital role. The main result is that every parity sheaf occurs as a direct summand of the parabolic induction of some cuspidal pair. Lusztig's work on ℤ-graded Lie algebras is related to representations of affine Hecke algebras, so a long term goal of this work will be to interpret parity sheaves in the context of affine Hecke algebras.
In a joint work with D. López N. and L.-F. Préville-Ratelle in 2015 we introduce a family of non-symmetric operads Dyckm, which satisfies that:
1. Dyck0 is the operad of associative algebras,
2. Dyck1 is the operad Dend of dendriform algebras, introduced by J.-L. Loday,
3. the vector space spanned by the set of m-Dyck paths has a natural structure of free Dyckm algebra over one element,
4. for any k ≥ 1, there exist degeneracy operators si : Dyckm → Dyckm-1 and face operators dj: Dyckm → Dyckm+1, which defines a simplicial complex in the category of non-symmetric operads.
The main examples of Dyckm algebra are the vector spaces spanned by the m-simplices of certain combinatorial Hopf algebras, like the Malvenuto-Reutenauer algebras and the algebra of packed words.
A well-known result on associative algebras states that, as an 𝒮-module, the operad of Ass of associative algebras is the composition Ass = Com ∘ Lie, where Com is the operad of commutative algebras and Lie is the operad of Lie algebras. The version of this result for dendriform algebras is that Dend = Ass ∘ Brace, where Brace is the operad of brace algebras.
Our goal is to introduce the notion of m-brace algebra, for m ≥ 2, and prove that there exists a Poincaré-Birkoff-Witt Theorem in this context, stating that Dyckm = Ass ∘ m-Brace.
I intend to overview classifications of simple Lie (super)algebras of finite dimension and of polynomial growth. Various properties of complex Lie superalgebras resemble same of modular Lie algebras. I will encourage to consider these classifications without fanaticism: certain non-simple Lie (super)algebras, "close" to simple ones, are often "better" for us than simple ones.
Interesting features of deformations: semi-trivial deformations and (in super setting) odd parameters.
I'll formulate classification of finite-dimensional simple complex Lie superalgebras, odd parameters including.
I'll formulate a definition of Lie superalgebra suitable for any characteristic and classification of simple (finite-dimensional) Lie superalgebras over algebraically closed fields of characteristic 2. With a catch: modulo (a) classification of simple (finite-dimensional) Lie superalgebras (over the same field) and (b) classification of their gradings modulo 2. I'll mention conjectures on classification of modular Lie algebras and superalgebras.
Is it feasible to classify simple filtered Lie (super)algebras of polynomial growth? Interesting examples: deforms of the Poisson Lie (super)algebras, Lie (super)algebras of "matrices of complex size", etc.
Examples. Double extensions of simple Lie (super)algebras are definitely "more interesting" than the simple objects they extend.
The contraction subgroup for x in the locally compact group, G,
con(x)={ g ∈ G ∣ xngx−n → 1 as n → ∞ },
and the Levi subgroup is
lev(x)={ g ∈ G ∣ {xngx−n}n∈ℤ has compact closure }.
The following will be shown. Let G be a totally disconnected, locally compact group and x ∈ G. Let y ∈ lev(x). Then there are x′ ∈ G and a compact subgroup, K ≤ G such that:
-K is normalized by x′ and y,
-con(x′)=con(x) and lev(x′)=lev(x) and
-the group ⟨x′,y,K⟩ is abelian modulo K, and hence flat.
If no compact open subgroup of G normalized by x and no compact open subgroup of lev(x) normalized by y, then the flat-rank of ⟨x′,y,K⟩ is equal to 2.
Let G be a complex, connected, reductive, algebraic group, and χ : ℂ× → G be a fixed cocharacter that defines a grading on 𝔤, the Lie algebra of G. Let G0 be the centralizer of χ(ℂ×). Here I will talk about G0-equivariant parity sheaves on the n-graded piece, 𝔤n. For the first half we will spend on building the background of derived category of equivariant perverse sheaves, bijection between the simple objects and some pairs that we are familiar with. In positive characteristic parity sheaves will play an important role. We want to study DbG0(𝔤n, k) for characteristic of k is positive. For that we will dive into the results of Lusztig in characteristic 0 in the graded setting. The main result from Lusztig is that every perverse sheaf occurs as a direct summand of the parabolic induction of the simple perverse sheaf associated to some cuspidal pair. The goal of the second talk will be to extend this result into positive characteristic.
We give recurrence formula for dimensions of Koszul operads. For example, dimensions of multi-linear parts of Lie-admissible operad satisfy the following recurrence relations dn=∑i=1n-1 μk Bn-1,k(d1…dn-1), where Bn,k are Bell polynomials and μk = k! ∑i=0k (k–i+1)i /i !. If p ≥ 5 is prime, then dp-1 ≡ 1 (mod p), dp ≡ -1 (mod p), dp+1 ≡ -1 (mod p), dp+2 ≡ -6 (mod p), dp+3 ≡ -56 (mod p), dp+4 ≡ -725 (mod p).
This video was produced by the Universidade de São Paulo, as part of the LieJor Online Seminar: Algebras, Representations, and Applications.
