Alekseev Equivariant symplectic geometry or moment map theory is the field built around several extraordinary results: the Marsden-Weinstein symplectic reduction theory, moment map convexity theorems, Duistermaat-Heckman localization for oscillating integrals, and the "quantization commutes with reduction" principle. Program of the course: Basics of symplectic geometry: definition, examples Recall: group actions of compact groups, normal forms on a neighborhood of an orbit Recall: Morse theory, Morse inequalities, perfect Morse functions Hamiltonian G-spaces: definition, examples Symplectic reduction, examples.
Symplectic cuts Atiyah-Guillemin-Sternberg convexity theorem Duistermaat-Heckman localization I: Berline-Vergne approach Equivariant cohomology: Cartan model, examples Localization II: Atiyah-Bott approach Kirwan surjectivity theorem, intersection pairings on reduced spaces G non-abelian: what changes? Itenberg Algebraic geometry is geometry of varieties defined by polynomial equations. Mikhalkin see description in first semester Symmetries and moduli spaces I S. Galkin The purpose of the course is to give an introduction to the construction and geometric properties of moduli spaces from the point of view of algebraic geometry.
Program of the course: Projective and affine algebraic varieties, smooth points and singularities. Algebraic groups and representations. Vector bundles. First examples: projective spaces and grassmannians. Toric varieties: Polytopes and fans. Toric varieties: Chambers and quotient constructions.
Intersection theory and vector bundles. Symmetries and moduli spaces II A. Szenes continuation: More quotients: grassmannians and flag varieties. Configuration of points on the line: the Hilbert-Mumford criterion. Vector bundles on Riemann surfaces: classification. Quot schemes and the construction of the moduli spaces of vector bundles.
Intersection theory on quotients. Introduction to quantum topology I A. Introduction to quantum topology II R. Kashaev see description in first semester Quantum mechanics for mathematicians M. The program includes: Lagrangian mechanics Hamiltonan mechanics Foundations of quantum mechanics Quantum mechanics in the phase space Semiclassical methods Field theory for mathematicians A. Alekseev Quantum field theory is a source of inspiration for a number of important concepts in modern mathematics.
Program of the course: Lagrangian field theory: basics of the calculus of variations, examples: free and interacting scalar fields, gauge theories, Yang-Mills theory in 2 dimensions, Chern-Simons theory in 3 dimensions. Atiyah-Segal axioms of Quantum Field Theory. Classification of 2-dimensional TQFTs. Master Class in Mathematical Physics.
Functions on Manifolds: Algebraic and Topological Aspects by V.V. Sharko
This is closely related to the subject of my research at the moment, which I might comment a bit about at the end of the talk if time permits. The geometry of the word problem The word problem is the task of deciding, given a word in a fixed generating set of a group, whether it represents the identity element of the group or not. While this sounds very algebraic, it has real geometric meaning. In this talk, I will introduce Dehn functions, which provide a geometric quantification of the difficulty of the word problem.
After treating classical examples, I will give new examples from joint work with Daniel Woodhouse showing that one-relator groups have a rich collection of polynomial Dehn functions. When talking about maps between spaces of functions, the term locality often comes up. This could mean a map of sheaves, a map depending only on the value of the function at a point, or on some of its derivatives. In the later case we say that the map descends to a map from a jet bundle. In this talk we will learn what jet bundles are and how are they related to locality. These questions were answered by McCullough and Rajeevsarathy who derived numerical equations whose solutions are in bijective correspondence with the conjugacy classes of roots of Dehn twists about nonseparating circles.
Later, using similar techniques, this work was extended to the case of Dehn twists about separating circles Rajeevsarathy, , multicurves Rajeevsarathy, Vaidyanathan, and Dehn twists in the mapping class group of a nonorientable surface. During the talk I will be primarily focused on presenting the results of McCullough and Rajeevsarathy from If time permits, I will also give some remarks about the analogous investigation in the nonorientable case.
These structures are somewhat reminiscent of Euclidean structures and if one can draw a certain number of analogies with the Euclidean case that led to a certain number of beautiful theorems , many questions about these structures remain widely open. In particular, it is very hard to tell whether a given manifold carries an affine structure.
I will try to give an insight to this mysterious world by giving a vaguely historical account of developments in the field and by stating a number of fairly simple questions that are still open. Critical points of the area functional: where to find them, and how to use them In this talk we will explain a few ideas involved in the variational approach to the construction of minimal surfaces. Moreover, we will show a few instances where the information about the index of instability of the constructed minimal surface allows the derivation of beautiful geometric applications. Massey products in toric topology With a history stemming from symplectic and algebraic geometry, toric topology began as the study of topological spaces with m-torus actions.
One notable object of study in toric topology is the moment-angle complex, whose cohomology can actually be described combinatorially. In particular, this combinatorial structure provides an avenue for studying higher cohomology operations, such as Massey products. The goal of this talk is to give an introduction to these objects, and to discuss some combinatorial descriptions of Massey products in moment-angle complexes.
Having set up the general long term dynamical problem, we will turn our attention to a short term recurrence problem of our so-called critical orbit and get an overview of my recent research. We will finish by talking about the great open problems in the field with interesting relations to number theory and more. We describe a locally finite graph naturally associated to each knot type K, called the Reidemeister graph. We determine several local and global properties of this graph and prove that the graph-isomorphism type is a complete knot invariant up to mirroring.
Lastly time permitting , we introduce another object, relating the Reidemeister and Gordian graphs, and briefly present an application to the study of DNA. Joint work with Daniele Celoria.
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The n-stranded pure surface braid group of a genus g surface can be described as the subgroup of the pure mapping class group of a surface of genus g with n-punctures which becomes trivial on the closed surface. I am interested in the least dilatation of pseudo-Anosov pure surface braids. In this talk, I will describe the upper and lower bounds I have proved as a function of g and n. Why you shouldn't be scared of integrals This is the case for Feynman integrals, a class of integrals encountered in quantum field theory. The study of those including the art of finding a sensible interpretation of their divergences is not only important for perturbative calculations in high-energy physics but also provides a rich playing field for mathematics as there are numerous connections to problems in algebra, number theory, geometry and topology.
In this talk, I will give an informal introduction and a broad overview of the field and discuss some of these connections in detail. Morally speaking it measures the complexity of a class by embedded surfaces representing this class. Buildings and the free factor complex The core idea of geometric group theory is to study groups acting nicely on beautiful spaces, where both the definitions of "nicely" and "beautiful" can vary.
I will give different definitions of these simplicial complexes, show why the descriptions are in fact equivalent and try to give an idea of basic similarities and differences between the two complexes. Morse subsets in hierarchically hyperbolic spaces When dealing with geometric structures one natural question that arises is "when does a subset inherit the geometry of the ambient space"? In the case of hyperbolic space, the concept of quasi-convexity provides answer to this question. However, for a general metric space, being quasi-convex is not a quasi-isometry invariant. This motivates the notion of Morse subsets.
In this talk we will motivate the definition and introduce some examples. Then we will introduce the class of hierarchically hyperbolic groups HHG , and furnish a complete characterization of Morse subgroups of HHG. If time allows, we will discuss the relationship between Morse subgroups and hyperbolically-embedded subgroups. This is a joint work with Hung C. Tran and Jacob Russell. Curve graphs, disc graphs and the topology of 3-manifolds Given any closed, orientable 3-manifold M, we can always decompose M into a union of two handlebodies of the same genus, glued along their boundary surfaces by a homeomorphism.
This is called a Heegaard splitting, and can be described by sets of curves in the common boundary surface which bound discs in one or other of the handlebodies. The set of curves in the surface of a handlebody which bound essential discs in the handlebody gives a subgraph of the curve graph called the disc graph, and Hempel defined a distance for a Heegaard splitting using this inclusion. We will give some background on Heegaard splittings and Hempel distance, and, time permitting, present a result on how the disc graph sits in the curve graph.
Homological stability for Artin monoids Many sequences of groups satisfy a phenomenon known as homological stability. In my talk, I will report on recent work proving a homological stability result for sequences of Artin monoids, which are monoids related to Artin and Coxeter groups. This extends the known cases of homological stability for the braid groups and other classical examples.
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Boundaries of Hyperbolic Groups The jungle of infinite groups is vast and unwieldly, but with the machete of geometry and the bug-spray of topology, we can attempt to explore some of its tamer wilderness. Through this we can ask what kind of groups have hyperbolic geometry, or at least an approximation of it called Gromov hyperbolicity. Hyperbolic groups are quite a nice class of groups but a large one, so we introduce the Gromov boundary of a hyperbolic group and explain how it can be used to distinguish groups in this class.
Key words: Cayley graph, quasi-isometry invariants, Hyperbolic group, Gromov, boundaries, Conformal dimension. A core problem in the study of manifolds and their topology is that of telling them apart. That is, when can we say whether or not two manifolds are homeomorphic? In two dimensions, the situation is simple, the Classification Theorem for Surfaces allows us to differentiate between any two closed surfaces. As an early pioneer in the area of 3-manifolds Seifert carved out his own corner of the landscape instead of attempting to tackle the entire problem.
By reducing his scope to the subclass of 3-manifolds which are today known as Seifert fibred spaces, Seifert was able to use our knowledge of 2-manifolds and produce a classification theorem of his own. In this talk I will define Seifert fibred spaces, explain what makes them so much easier to understand than the rest of the pack, and give some insight on why we still care about them today.
Spatial graphs and minimal knottedness Spatial graph theory investigates embeddings of graphs in R3. Finally, we show that there exist no minimally knotted planar spatial graphs on the torus. One of the main themes in geometric group theory is Gromov's program to classify finitely generated groups up to quasi-isometry. We show that under certain situations, a quasi-isometry preserves commensurator subgroups.
Diffeomorphism Groups: algebra, topology, homology
Such groups can be thought of as coarse fibrations whose fibres are cosets of H; quasi-isometries of G coarsely preserve these fibres. This generalises work of Whyte and Mosher--Sageev--Whyte. Sphere Packings, Kissing Numbers, and Integers In its original form the sphere packing problem asks: "What is the most efficient way to stack cannonballs? By the mid 19th century the complex numbers were mostly accepted by mathematicians, and motivated by their utility Hamilton was led to discover the quaternions, their four dimensional brother.
Quaternions share a lot of properties with complex numbers, so even though they don't commute we can still think of them as numbers. Not long after this an eight dimensional cousin was found, which we now call the octonions, and this completes the family. The aim of this talk is to describe a link between "integers" for these number systems and solutions to both the packing and kissing number problems in the relevant number of dimensions.
The Curve Complex, a structure which encodes information about curves on a surface, is one of the most important construction in the field since its introduction in , thanks to W. This complex has both applications to the study of other geometric objects and a very interesting geometric structure by itself. We will introduce the definition of the Curve Complex, with as many examples as possible, along with some application and properties.
In particular we will seize the opportunity to introduce the tremendously important concept of Gromov hyperbolicity, and talk about how this is one of the main features of the Curve Complex. A fibre bundle is a more general version of a covering space. Heuristically, it can be seen as a continuous map that locally looks like a projection i.
Perhaps the simplest question is that of triviality: is this map globally just a projection if you move things around a bit or not? This talk will hopefully require no more of you than an understanding of the fundamental group and of covering spaces, so should be quite accessible! Such an action gives rise to two natural real valued functions on G.
These are the word length function and the displacement function. In this talk we discuss these two quantities and explore different ways in which to compare them. This will be a gentle introduction to the topic — no prior knowledge required! Given finitely-generated subgroups H and K of a free group F, Hanna Neumann conjectured the existence of a bound on the rank of their intersection coming from the individual ranks of H and K.
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After giving some context for the conjecture, I will describe how Mineyev proved it using properties of an ordering of the elements of F which are reflected in how the subgroups act on its Cayley graph. If time permits I will show how the same argument in fact gives a strengthened version of the result, which maximises the information obtained from the correspondence between free groups and the topology of graphs. Alternating quotients of RAAGs, RACGs and surface groups We say that a group has many alternating quotients if for every finite set of group elements there exists a surjection onto an alternating group, which is injective on this finite set.
I will show that every RAAG satisfies exactly one of the following: 1. Therefore, every RAAG is a direct product of groups with many alternating quotients and infinite cyclic groups. Along the way I will prove a similar result for right-angled Coxeter groups and I will show that the fundamental groups of hyperbolic, closed, orientable surfaces have many alternating quotients. What are spectra? This talk will be a brief and gentle introduction to stable homotopy theory: the study of those topological phenomena that they occur in essentially the same way independent of dimension.
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June 8-12, 2015 at UC Berkeley
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