For more information: https://scgp.stonybrook.edu/archives/35496
Title: Combinatorial and metric properties of flow polytopes
Speaker: Alejandro Morales
Abstract: Flow polytopes of graphs are a rich family of polytopes of interest in probability, optimization, representation theory, and algebraic combinatorics. Computing their volumes and
enumerating lattice points of some particular flow polytopes turn out to be combinatorially interesting problems that involve beautiful enumeration formulas and many familiar combinatorial
objects like permutahedra and associahedra. Baldoni and Vergne found a series of formulas for both of these purposes, which they call Lidskii formulas, that are combinatorially powerful and
pleasant and have log-concavity properties. Flow polytopes also have interesting subdivisions by Postnikov and Stanley that are recursive and triangulations by Danilov-Karzanov-Koshevoy related
to cluster algebras. I will give an overview of recent work on these polytopes including formulas that relate their volume to the number of lattice points, the geometry and combinatorics of their
triangulations, and some open questions.
Speaker: Tanvi Karwal
Title: Cosmic Tensions
Speaker: Robert McGehee
Title: Directly Detecting Light Dark Matter
Title: Almost colored f-vectors for generalized Associahedra
Speaker: Theo Douvropoulos
Abstract: After seminal works of Fomin, Reading, Zelevinsky, and Chapoton, the (generalized) W-Associahedra D(W) have become prominent members in the world of Coxeter-Catalan combinatorics. Fomin and Reading computed their f-vectors in a case-by-case way for all finite reflection groups W and observed that they are often given in terms of product formulas generalizing Kirkman-Cayley-type numbers.
During the last twenty years the community has developed this theory extensively, building further realizations for the dual cluster complexes, enhancing connections to the noncrossing partition lattice NC(W), the representation theory of W and more. Still however we were missing a uniform understanding of the f-vectors of D(W) and an explanation for their structure.
We present recent work, in part joint with Matthieu Josuat-Verges, where we give case-free proofs of product formulas for an almost colored refinement of the f-vectors of D(W). This refinement counts faces with respect to parabolic type, gives a justification for the Fomin-Reading counts, and in type A reveals connections with formal power series inversions and the nabla operator from symmetric functions.
Hashtag: #workshop
Title: Rigidity expanders
Speaker: Eran Nevo
Abstract: The d-dimensional algebraic connectivity a_d(G) of a graph G=(V,E) is a quantitative measure of d-dimensional rigidity introduced by Jordan and Tanigawa, in particular a_d(G)>0 if and
only if G is generically d-rigid. It extends Fiedler's notion of algebraic connectivity, corresponding to the d=1 case. We prove the existence of (2d+1)-regular "d-rigidity spectral expanders". That
is, we show that there exists a constant c(d)>0 and a family of (2d+1)-regular graphs G_n with increasing number of vertices satisfying a_d(G_n) > c(d). We conjecture that no such positive
constant exists for 2d-regular graphs. All the relevant background will be given during the talk. Based on joint work in progress with Alan Lew, Yuval Peled and Orit Raz.
Hashtag: #workshop
Title: Instanton Floer homology and Heegaard diagrams
Speaker: Zhenkun Li [Stanford]
Abstract: Instanton Floer homology was introduced by Floer in the 1980s and has become a power invariant for three manifolds and knots since then. It has led to many milestone results, such as the approval of Property P conjecture. Heegaard diagrams, on the other hand, is a combinatorial method to describe 3-manifolds. In principle, Heegaard diagrams determine 3-manifolds and hence determine their instanton Floer homology as well. However, no explicit relations between these two objects were known before. In this talk, for a 3-manifold Y, I will talk about how to extract some information about the instanton Floer homology of Y from the Heegaard diagrams of Y. Additionally, I will explain some of the applications and future directions of this work. This is a joint work with Baldwin and Ye.
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Title: Polyhedral diagonals and beyond
Speaker: Daria Poliakova
Abstract: Algebraists (or at least some of them) are interested in cellular diagonals of operadically meaningful polyhedra. I will remind some known formulas - "magic formula" for associahedra,
Saneblidze-Umble/Laplante-Anfossi formulas for permutahedra. I will explain how a slight generalization of these ideas from polyhedra to subdivisions should lie behind homotopy monoidality.
Title: Pivots, polytopes, and combinatorics
Speaker: Raman Sanyal
Abstract: A pivot rule is the mechanism that tells the simplex algorithm which path to take on a linear program from a given vertex to an optimal one. We introduced pivot polytopes as a mean
to capture the behaviour of certain classes of pivot rules on a given linear program. While this gives a new perspective on pivot rules, it turns out that pivot polytopes are also of interest to
combinatorialists. I this talk I will discuss - how pivot polytopes relate flag matroid polytopes to nestohedra, - how pivot polytopes encode associative structures on 2-dimensional noncrossing
structures (subsuming permutahedra, associahedra, multiplihedra), and - how pivot polytopes give a new perspective on fiber polytopes of cyclic polytopes. The talk is based on joint work with Benjes, Black, De Loera, Lutjeharms, and Poullot.
Title: Sets that Support a Joint Distribution
Speaker: Peter Winkler [Dartmouth College]
Abstract: Given a closed set on the plane and two probability distributions
on the real line, when are there random variables with the given
distributions whose joint distribution is the given set?
We consider both discrete and continuous distributions; in the
latter case, the problem is equivalent to asking which sets
in the unit square can support a permuton.
Joint work (in progress) with Chris Coscia and Martin Tassy.
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Title: Radon and fractional Helly type theorems
Speaker: Zuzana Patakova
Abstract: Radon theorem plays a basic role in many results of combinatorial convexity. It says that any set of d+2 points in R^d can be split into two parts so that their convex hulls intersect.
It implies Helly theorem and as shown recently also its more robust version, so-called fractional Helly theorem. By standard techniques this consequently yields an existence of weak epsilon nets and a (p,q)-theorem. We will show that we can obtain these results even without assuming convexity, replacing it with very weak topological conditions. More precisely, given an intersection-closed family F of subsets of R^d, we will measure the complexity of F by the supremum of the first d/2 Betti numbers over all elements of F. We show that constant complexity of F guarantees versions of all the results mentioned above. Partially based on joint work with Xavier Goaoc and Andreas Holmsen.
Hashtag: #workshop
Speaker: Rashmish Mishra
Speaker: Itay Bloch
Title: A Proof of Grunbaum's Lower Bound Conjecture for polytopes,
lattices, and strongly regular pseudomanifolds.
Speaker: Lei Xue
Abstract: If we fix the dimension and the number of vertices of a polytope, what is the smallest number of faces of each dimension? In 1967, Grunbaum made a conjecture on this lower bound problem for d-dimensional polytopes with at most 2d vertices. In the talk, we will discuss the proof of this conjecture and the equality cases. We will then extend our results to lattices with diamond property (the inequality part) and to strongly regular normal pseudomanifolds (the equality part). We will also talk about recent results on d-dimensional polytopes with 2d+1 or 2d+2 vertices.
Title: A Positive Answer to Barany's Question on Face Numbers of
Polytopes
Speaker: Joshua Hinman
Abstract: Although the face numbers of simple and simplicial polytopes are well understood, we still know frustratingly little about the face numbers of polytopes in general. In the late nineties, Imre Barany asked a fascinating question: for every convex polytope, does the number of k-dimensional faces is no less than the minimum of the numbers of vertices and the number of
facets? In practice, the answer always seemed to be "yes", but for no obvious reason. In this talk, we will add a piece to the puzzle of understanding face numbers by answering Barany's question in the affirmative. We will also prove a stronger statement in the form of linear inequalities on the face numbers of P.
Title: Obstructions for exact submanifolds with symplectic applications
Speaker: Kevin Sackel [University of Massachusetts-Amherst]
Abstract: Suppose a closed oriented manifold comes with a fixed real cohomology class. This cohomology class singles out those submanifolds for which the restriction of the given cohomology class is trivial. We may ask in what integral homology classes we find such "exact submanifolds." We build an infinite sequence of obstructions which are readily computable by (finite-dimensional) linear algebra performed on the de Rham complex. As the impetus for this work arose out of (locally conformal) symplectic geometry, which we shall partially describe, our main applications are symplectic in nature. For example, the following symplectic manifolds admit no non-separating (a fortiori contact-type) hypersurfaces: Kähler manifolds, symplectically uniruled manifolds, and the Kodaira-Thurston manifold.
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Title: Minimal sets of regular Denjoy-like maps
Speaker: Sergiy Merenkov [City College of New York and CUNY Graduate Center]
Abstract: I will discuss how regularity of invertible self-maps of certain manifolds influences the geometry of the corresponding minimal sets. A particular attention will be given to smooth and quasiconformal homeomorphisms of $n$-tori, $n\geq 2$, and hyperbolic surfaces. The results to be presented are higher dimensional versions of Denjoy's theorem on topological conjugacy for circle diffeomorphisms.
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For more information please visit: http://scgp.stonybrook.edu/archives/35652
Students can begin to submit major/minor changes effective Fall 2023 Semester
Title: Towards a mathematical description of superstring perturbation theory
Speaker: Giovanni Felder
Abstract: Superstring amplitudes, describing the scattering of superstrings, have perturbative power series expansions in the coupling constants, whose coefficients are integrals over moduli spaces of supercurves. These integrals over non-compact superspaces are expected to
(conditionally) converge, in contrast with the case of Feynmann diagrams of Quantum Field Theory, where integrals diverge and require renormalization. In this talk I will review the definition of these objects, present the challenges for a mathematical description (integration over middle
dimensional cycles, GSO projection, dependence on the regularization), and explain how to address them in the simplest non-trivial case of genus 2 contribution to the vacuum amplitude of the type II superstring in 10 dimensional Minkowski space. The talk is based on joint work with David Kazhdan and Alexander Polishchuk.
Hashtag: #workshop
Title: Super-Teichmueller spaces: coordinates and applications
Speaker: Anton Zeitlin
Abstract: The Teichmüller space parametrizes Riemann surfaces of fixed topological type and is fundamental in various mathematics and physics contexts. It can be defined as a component of the moduli space of flat G=PSL(2,R) connections on the surface. Higher Teichmüller space extends these notions to appropriate higher rank classical Lie groups G, and N=1 super Teichmüller space likewise studies the extension to the super Lie group G=OSp(1|2). In this talk, I will review the solution to the problem of producing Penner-type coordinates on super-Teichmüller space and its higher analogs. I will also talk about some applications of these coordinates.
Title: Cutoff profile of the colored ASEP: GOE Tracy-Widom
Speaker: Lingfu Zhang [UC Berkeley]
Abstract: In this talk, I will discuss the colored Asymmetric Simple Exclusion Process (ASEP) in a finite interval. This Markov chain is also known as the biased card shuffling or random Metropolis scan, and its study dates back to Diaconis-Ram (2000). A total-variation cutoff was proved for this chain a few years ago using hydrodynamic techniques (Labbé-Lacoin, 2016). In this talk, I will explain how to obtain more precise information on its cutoff, specifically to establish the conjectured GOE Tracy-Widom cutoff profile. The proof relies on coupling arguments, as well as symmetries obtained from the Hecke algebra. I will also discuss some related open problems.
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Title: Nilpotence Varieties, Pure Spinors Superfields and Supersymmetry
Speaker: Simone Noja
Abstract: In this talk I will introduce a (super-)mathematical perspective on the pure spinor superfield formalism, showing how to recover supersymmetry multiplets from geometric data related to the nilpotence variety of a certain Poincaré superalgebra. After discussing some lower dimensional examples, I will focus on the case of supersymmetry in six dimensions, where the nilpotence variety is a Segre manifold, and I will hint at a generalization of the formalism in the direction of derived algebraic geometry. If time permits, I will discuss how nilpotence varieties of some classical Lie superalgebras are related to the superconformal field theories.
Title: On the Relationship Between Super-Riemann Surfaces and PCOs
Speaker: Charles Wang
Abstract: We will discuss the relationship between the two formulations of superstring perturbation theory: the SRS formalism, based on integration over the supermoduli space of super-Riemann Surfaces, and the PCO formalism, based on integration over bosonic moduli space with insertions of picture changing operators and vertical integration. I will show that the picture changing operator approach arises from a specific construction of the supermoduli integration contour. If time permits, I will also discuss an explicit construction in genus two.
Title: Stable supermaps and SUSY Nori motives
Speaker: Daniel Hernandez Ruiperez
Abstract: We define stable supercurves and stable supermaps, and based on these notions we develop a theory of Nori motives for the category of stable supermaps of SUSY curves with punctures. This will require several preliminary constructions, including the development of a basic theory of supercycles.
Title: Worldsheet supersymmetry and uniform transcendentality of string amplitudes
Speaker: Oliver Schlotterer
Abstract: The low-energy expansion of string scattering amplitudes involves special numbers and functions such as polylogarithms and multiple zeta values which are informally assigned a notion of transcendental weight. For superstring tree amplitudes and certain building blocks of one-loop amplitudes, the order in the inverse string tension alpha’ matches the transcendental weight of the accompanying (elliptic) zeta values. This property is known as uniform transcendentality (UT) and also occurs in the context of Feynman integrals and perturbative field-theory computations. Since amplitudes of bosonic and heterotic string theories violate UT, one may wonder if UT originates from worldsheet supersymmetry which is supported by the short-distance properties of worldsheet-supersymmetric vertex operators. This talk aims to give a gentle introduction into the above ideas to a mixed audience of mathematicians and physicists.
Speaker: Eric D'Hoker (zoom)
Title: Supermoduli and superstring amplitudes
Abstract: Perturbative superstring amplitudes are given by integrals over the moduli spaces of super Riemann surfaces, summed over genera. We concentrate on the case of genus two which is the smallest genus at which supermoduli enter superstring amplitudes in a non-trivial way. Using the special properties of genus-two supermoduli space, we will show that superstring amplitudes may be evaluated for an arbitrary number of Neveu-Schwarz string states, thanks to new identities between modular tensors of Sp(4,Z).
Title: Extended super Mumford form on the Sato Grassmannian
Speaker: Katherine Maxwell
Abstract: The super Mumford form is a section over the moduli space of super Riemann surfaces, characterized by invariance under the action of the Neveu-Schwarz algebra action. In light of difficulties in performing integrals in superstring theory arising from the super Mumford form, it was suggested in the 80s that the relationship of the moduli space of super Riemann surfaces to the super Sato Grassmannian may be fruitful. Based on joint work with A. Voronov, I will discuss possible approaches to extending the super Mumford form, including our results on the proposed formula by A. Schwarz.
Speaker: Sebastien Boucksom, Sorbonne Universite
Title: From complex to non-Archimedean geometry - and back: Lecture 1: Spaces of norms and GIT
Abstract: The general purpose of this series of lectures is to illustrate the use of non-Archimedean geometry in the study of complex geometric objects in the large (logarithmic) scale, first in the classical finite-dimensional setting of Geometric Invariant Theory (GIT), and then in the infinite-dimensional context of the Yau-Tian-Donaldson (YTD) conjecture.
The first lecture will be devoted to the large scale geometry of spaces of norms, and their relation to GIT and the classical Hilbert-Mumford criterion. The second lecture will review the fundamental ingredients in the variational approach to the YTD conjecture, emphasizing the geometric properties of the space of KÃhler potentials and its completion. Finally, the third lecture will introduce the non-Archimedean counterparts to these spaces, their relation to K-stability, and their use in the recent progress accomplished towards the general case of the YTD conjecture.
Note: This lecture also serves as this week's Geometry/Topology seminar.
Title: From complex to non-Archimedean geometry - and back: Lecture 1: Spaces of norms and GIT
Speaker: Sebastien Boucksom [Sorbonne Universite]
Abstract: The general purpose of this series of lectures is to illustrate the use of non-Archimedean geometry in the study of complex geometric objects in the large (logarithmic) scale, first in the classical finite-dimensional setting of Geometric Invariant Theory (GIT), and then in the infinite-dimensional context of the Yau-Tian-Donaldson (YTD) conjecture.
The first lecture will be devoted to the large scale geometry of spaces of norms, and their relation to GIT and the classical Hilbert-Mumford criterion. The second lecture will review the fundamental ingredients in the variational approach to the YTD conjecture, emphasizing the geometric properties of the space of Kähler potentials and its completion. Finally, the third lecture will introduce the non-Archimedean counterparts to these spaces, their relation to K-stability, and their use in the recent progress accomplished towards the general case of the YTD conjecture.
Note: This lecture also serves as this week's Geometry/Topology seminar.
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Title: Localization of integrals on supermanifolds with application to representation theory of supergroups.
Speaker: Vera Serganova
Abstract: We compute volumes of supergrassmannians and odd symmetric grassmannians using Schwarz-Zaboronsky localization formula which expresses a Berezin integral as a sum of local contributions at all singular points of an odd vector field. To generalize this computation to other classical supermanifolds, we need a CS analogue of localization, since manyclassical supergroups don't have compact real forms. We prove an analogue of the Schwarz-Zaboronsky localization formula for complex smooth supermanifolds. Let X be a compact CS manifold and Q an odd vector field on X such that [Q,Q] is compact. Assume that Q has isolated singular points on X and preserves a volume form w. Then the integral of w over X equals the sum of local contribution at all singular points. We apply the localization formula in the case of homogeneous superspace X=G/K which admits a G invariant volume form. For specific choices of G and K we show that the integral of w over X is not zero. This allows us to use the unitary trick and show that K is a splitting subgroup of G, i.e. the restriction functor from Rep G to Rep K induces injection of Ext groups. In particular, we prove that a defect subgroup is splitting in the case when Lie G is any basic classical or exceptional superalgebra. This has several applications in support theory for supergroups. For example, we show that the DS associated variety detects projectivity in the category of finite-dimensional G-modules. The talk is based on joint papers with A. Sherman and D. Vaintrob.
Title: Generalized Root System
Speaker: Rita Fioresi
Abstract: In this talk (joint work with I. Dimitrov, Queens U.) we introduce the category of generalized root systems. The notion of ordinary root systems is the key to understand Lie theory and its many generalizations (contragredient superalgebras, affine, Kac-Moody (super) algebras etc). However, such notion is ``rigid'', it does not behave reasonably under quotients and moreover lacks of a unified treatment, that is definitions and results are usually confined to the realm of application. The rigidity of ordinary root systems stems from their invariance under the action of the Weyl group. Once we abandon the notion of Weyl group as we know it, we can look for another definition of root systems that is able to take into account all examples mentioned above and more.For example, the systems stemming from the eigenspace decompositionwith respect to a non maximal toral subalgebra (Kostant root systems). They play a key role in the classification of the complex structures on the symmetric space G/K, for K and non maximal torus. This is a generalization of the hermitian symmetric spaces theory. In this talk we give an effective way to compute bases for generalized root systems, which are quotients of Lie algebra ones and we classify all root systems of rank two up to combinatorial equivalence finding 16 such.
Speaker: Sebastien Boucksom, Sorbonne Universite
Title: From complex to non-Archimedean geometry - and back: Lecture 2: Pluripotential theory and the YTD conjecture
Abstract: The general purpose of this series of lectures is to illustrate the use of non-Archimedean geometry in the study of complex geometric objects in the large (logarithmic) scale, first in the classical finite-dimensional setting of Geometric Invariant Theory (GIT), and then in the infinite-dimensional context of the Yau-Tian-Donaldson (YTD) conjecture.
The first lecture will be devoted to the large scale geometry of spaces of norms, and their relation to GIT and the classical Hilbert-Mumford criterion. The second lecture will review the fundamental ingredients in the variational approach to the YTD conjecture, emphasizing the geometric properties of the space of Kähler potentials and its completion. Finally, the third lecture will introduce the non-Archimedean counterparts to these spaces, their relation to K-stability, and their use in the recent progress accomplished towards the general case of the YTD conjecture.
Title: From complex to non-Archimedean geometry - and back: Lecture 2: Pluripotential theory and the YTD conjecture
Speaker: Sebastien Boucksom [Sorbonne Universite]
Abstract: The general purpose of this series of lectures is to illustrate the use of non-Archimedean geometry in the study of complex geometric objects in the large (logarithmic) scale, first in the classical finite-dimensional setting of Geometric Invariant Theory (GIT), and then in the infinite-dimensional context of the Yau-Tian-Donaldson (YTD) conjecture.
The first lecture will be devoted to the large scale geometry of spaces of norms, and their relation to GIT and the classical Hilbert-Mumford criterion. The second lecture will review the fundamental ingredients in the variational approach to the YTD conjecture, emphasizing the geometric properties of the space of Kähler potentials and its completion. Finally, the third lecture will introduce the non-Archimedean counterparts to these spaces, their relation to K-stability, and their use in the recent progress accomplished towards the general case of the YTD conjecture.
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Title: Body and soul decompositions in supergeometry
Speaker: John Huerta
Abstract: Given a supermanifold with a projection map to the reduced manifold, elements of the structure sheaf decompose into a ``body'', a function on the reduced manifold, and a complementary part called the ``soul''. This induces, in turn, a decomposition of various kinds of geometric structure one could put on a supermanifold, including de Rham forms, line bundles with connection, as well as higher degree classes in Deligne cohomology.
Title: The birational geometry of the moduli of curves: geometric and tropical aspects.
Speaker: Gabriel Farkas
Abstract: It is one of the landmark results in algebraic geometry of the 20th century that the moduli space M_g of curves of genus g is a variety of general type when g>23. I will discuss joint work with Jensen and Payne proving that both moduli spaces M_22 and M_23 are of general type, highlighting both the geometrical and the novel tropical aspects related to this circle of ideas. Time permitting, I will also discuss how novel ideas from non-abelian Brill-Noether theory can be used to prove that the moduli space of genus 16 is uniruled.
Title: The birational geometry of the moduli of curves: geometric and tropical aspects.
Speaker: Gavril Farkas [Humboldt University, Berlin]
Abstract: It is one of the landmark results in algebraic geometry of the 20th century that the moduli space M_g of curves of genus g is a variety of general type when g>23. I will discuss joint work with Jensen and Payne proving that both moduli spaces M_22 and M_23 are of general type, highlighting both the geometrical and the novel tropical aspects related to this circle of ideas. Time permitting, I will also discuss how novel ideas from non-abelian Brill-Noether theory can be used to prove that the moduli space of genus 16 is uniruled.
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Title: Gelfand-Fuchs cohomology for super-manifolds
Speaker: Slava Pimenov
Abstract: Let $X$ be a smooth super-manifold of super-dimension $(m, n)$ and consider the Lie super-algebra $V_X$ of vector fields on $X$.We are interested in the Lie algebra cohomology of $V_X$ with trivial coefficients. Using a local to global construction the question can be reduced to calculation of cohomology of Lie algebra of super-vector fields in a formal neighborhood
of a point on an affine super-space $\mathbb{A}^{m,n}$. Previously known results covered the classical case $\mathbb{A}^m = \mathbb{A}^{m,0}$ as well as the case $\mathbb{A}^{m,n}$ with $m \leq n$. I will speak about my recent work where I extended these results to the case of super-dimension $(m, 1)$. I will also propose a conjectural answer in the general case and speculate on how one might try to prove it.
Title: TBA
Speaker: Nadia Ott
Abstract: Artin’s theorems on approximation and algebraization of formal deformations and stacks give general criteria for functors to be, in various senses, described by algebraic objects. It has long been expected that analogous results hold in supergeometry, however proofs of the full suite of Artin theorems have remained absent from the supergeometry literature. One reason for this may be the sense that establishing the Artin theorems in supergeometry would require the tedious repetition of various difficult arguments in commutative algebra, deformation theory, and algebraic geometry. I have recently proved the Artin theorems in the super case. Moreover, at several key points I was able to reduce to the (known) bosonic case by an argument which is significantly simpler than the original bosonic argument. In my talk I will show how to use the Artin theorem to construct some moduli spaces of interest in supergeometry, e.g.,the Picard (super)stack.
Title: Odd torus actions on moduli spaces of super stable maps of genus zero
Speaker: Enno Kessler
Abstract: Super stable maps are supergeometric generalizations of stable maps from a Riemann surface in an almost Kähler manifold and appear naturally in the compactification of the moduli space of super J-holomorphic curves. Super J-holomorphic curves are maps from a super Riemann surface to an almost Kähler manifold satisfying a Cauchy-Riemann equation. In this talk we will explain the construction of moduli spaces of super stable maps of genus zero of fixed tree type and show that they carry a torus action that leaves the even directions invariant. The invariant manifolds are then the corresponding moduli spaces of stable maps and the normal bundles are described as holomorphic sections of twisted spinor bundles.Based on joint work with Artan Shesmani and Shing-Tung Yau
Title: Genus zero super Gromov Witten invariants via odd torus localization
Speaker: Artan Sheshmani
Abstract: A major challenge in construction of supergeometric analogue of Gromov-Witten invariants is the suitable generalization of intersection theory. We propose to circumvent this difficulty by assuming a virtual torus localization theorem for the odd directions. That is, we construct a super virtual normal bundle to the torus-fixed loci of the moduli space of super stable maps, and compute the super Gromov-Witten invariants, via dividing by the equivariant Euler class of the super virtual normal bundle and intersecting with the virtual class of the torus fixed superstable maps. We define the super Gromov-Witten invariants of genus zero which satisfy generalized Kontsevich-Manin axioms. Furthermore, we present a recipe for calculation of super Gromov-Witten invariants of projective space. Based on joint work with Enno Keßler and Shing-Tung Yau.
Title: Cohomology and Combinatorics for Supertori
Speaker: Jeffrey Rabin
Abstract: A supertorus (“elliptic supercurve”) of dimension 1|1 is a simple example showing that sheaf cohomology groups of supermanifolds are generally non-free modules over the ring of Grassmann constants. I will generalize this example to supertori of dimension 1|n, computing H0(X,O) and H1(X,O). These groups simply reflect combinatorial invariant-theoretic properties of Grassmann algebras, and exhibit Serre duality and Lefschetz properties. This is a first step toward a general theory of Abelian supervarieties and Jacobians of supercurves.
Title: From complex to non-Archimedean geometry - and back: Lecture 3: Non-Archimedean pluripotential theory and K-stability
Speaker: Sebastien Boucksom [Sorbonne Universite]
Abstract: The general purpose of this series of lectures is to illustrate the use of non-Archimedean geometry in the study of complex geometric objects in the large (logarithmic) scale, first in the classical finite-dimensional setting of Geometric Invariant Theory (GIT), and then in the infinite-dimensional context of the Yau-Tian-Donaldson (YTD) conjecture.
The first lecture will be devoted to the large scale geometry of spaces of norms, and their relation to GIT and the classical Hilbert-Mumford criterion. The second lecture will review the fundamental ingredients in the variational approach to the YTD conjecture, emphasizing the geometric properties of the space of Kähler potentials and its completion. Finally, the third lecture will introduce the non-Archimedean counterparts to these spaces, their relation to K-stability, and their use in the recent progress accomplished towards the general case of the YTD conjecture.
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