Mathematics

Organizational Meeting

friends of A. Coble (UIUC)

Organizational Meeting

Abstract: We will discuss the structure and schedule of the seminar. We'll be taking speakers slots too, so if you'd like to give a talk or know someone who would, please come. There will be cookies.

Positively curved Alexandrov spaces with many symmetries

John Harvey (Notre Dame)

Abstract: I will introduce two new tools -- the ramified orientable double cover and the slice theorem -- for Alexandrov geometry. These will be used to classify positively curved Alexandrov spaces under certain symmetry conditions, shedding new light on similar Riemannian results. This is joint work with Catherine Searle.

The Hitchin fibration and real forms through spectral data

Laura Schaposnik (UIUC)

Abstract: The talk will be dedicated to the study of the moduli space of G-Higgs bundles and the Hitchin fibration through spectral data, where G is a real form of a complex Lie group. Through some examples we shall see applications of this new geometric way of understanding the moduli space and, time permitting, we will mention how the data approach relates to Langlands duality and (A,B,A)-branes.

CANCELLED

Cohomology of the moduli space of curves

Rahul Pandharipande (ETH Zürich)

Abstract: The moduli space of curves carries tautological cohomology classes. I will discuss the study of relations amongst these classes starting with ideas of Mumford in the 1980s. The subject advanced in the 1990s with conjectures of Faber and Faber-Zagier. I will explain the current state of affairs based on Pixton's conjectures related to cohomological field theories. The talk represents joint work with A. Pixton and D. Zvonkine.

Some interactions between classical, semiclassical, and random symplectic geometry

Alvaro Pelayo (Washington University Math)

Abstract: I will describe some recent results about classical and quantum integrable systems, emphasizing the interplay between symplectic geometry and semiclassical analysis. I will also briefly describe some random counterparts of classical results in symplectic geometry.

On elliptesque and hyperbolesque curves

Bruce Reznick (UIUC Math)

Abstract: Any five points in the plane (no four on a line) determine a unique conic section. What can be said about a curve $C$ with the property that any five points chosen from $C$ either always determine an ellipse (or circle) or always determine a hyperbola? Such a curve is "elliptesque" or "hyperbolesque". Non-trivial examples include $y = x^3, x \ge 0$, which is hyperbolesque and $y = x^{3/2}, 1 \le x \le 1.3$, which is elliptesque. We show that if a smooth closed curve $C$ satisfies either condition, then it must be elliptesque and bound a convex region; no unbounded smooth curve can be elliptesque. Proofs are elementary.

The opening act of this talk is a discussion of the remarkable differential equation $((y'')^{-2/3})'''=0$; Sylvester observed in 1886 that the solutions to this equation are precisely the non-degenerate conic sections, simplifying a result originally proved by Monge in 1809. Two proofs of this will be given, and both are readily accessible to undergraduate math majors who have had calculus as well as linear algebra.

Departmental veterans will recognize this as a "Potpourri" talk.

Symplectic Galois groups and Springer theory

Kevin McGerty (University of Oxford)

Abstract: One of the fundamental phenomena in geometric representation theory is Springer's action of the Weyl group on the cohomology of the fibres of the Springer resolution of the nilpotent cone. Recently there has been much interest in the geometry of symplectic resolutions, of which the Springer resolution is an example. We will discuss how Springer theory can be generalized to this setting.

Topological Hamiltonian and contact dynamics, part I: an introduction

Stefan Mueller (UIUC Math)

Abstract: In classical mechanics, the dynamics of a Hamiltonian vector field models the motion of particles in phase space, and the dynamics of a contact vector field play a similar role in geometric optics (in the mathematical model of Huygens' principle). Topological Hamiltonian dynamics and topological contact dynamics are relatively recent theories that explore natural questions regarding the regularity of such dynamical systems (on an arbitrary symplectic or contact manifold). In a nutshell, Hamiltonian and contact dynamics admit genuine generalizations to non-smooth dynamical systems with non-smooth generating (contact) Hamiltonian functions. The talk begins with examples that illustrate the central ideas and lead naturally to the key definitions. The main technical ingredient is the well-known energy-capacity inequality for displaceable subsets of a symplectic manifold. We use it to prove an extension of the classical 1-1 correspondence between isotopies and their generating Hamiltonians. This crucial result turns out to be equivalent to certain rigidity phenomena for smooth Hamiltonian and contact dynamical systems. We then look at some of the foundational results of the new theories. The end of the talk touches upon sample applications to topological dynamics and to Riemannian geometry, which will be explored further in a second talk.

Quasi-Regular Mappings of Lens Spaces

Anton Lukyanenko (UIUC Math)

Abstract: A quasi-regular QR mapping between metric manifolds is a branched cover with bounded dilatation, e.g. $f(z)=z^2$. In a joint work with K. Fassler and K. Peltonen, we define QR mappings of sub-Riemannian manifolds and show that:
1) Every lens space admits a uniformly QR (UQR) mapping $f$.
2) Every UQR mapping leaves invariant a measurable conformal structure.
The first result uses an explicit "conformal trap" construction, while the second builds on similar results by Sullivan-Tukia and a connection to higher-rank symmetric spaces.

An Introduction to Tropical Geometry

Nathan Fieldsteel (UIUC Math)

Abstract: This will be the first talk in a two-part series in which we will give a broad overview of the relatively young field of tropical geometry, aiming to introduce the central objects of study while providing motivation, examples and connections to other fields. Professors are welcome to attend.

Hamiltonian and contact dynamics, part II: applications

Stefan Mueller (UIUC Math)

Abstract: After recalling the precise definition of a topological Hamiltonian dynamical system, I will sketch the proof of the 1-1 correspondence between topological Hamiltonian isotopies and topological Hamiltonian functions. I also show that this result has non-empty content by constructing a non-smooth topological Hamiltonian dynamical system (with support in a Darboux chart). We then shift gears and focus on two sample applications to 1) hydrodynamics (topological character of the helicity invariant, which measures the average asymptotic linking number of the flow lines of a divergence-free vector field) and to 2) Riemannian geometry (C^0-rigidity of the geodesic flows associated to a sequence of weakly uniformly converging Riemannian metrics).

Divergence of Weil-Petrsson geodesic rays.

Babak Modami ((UIUC Math))

Abstract: The Weil-Petrsson (WP) geodesic flow is a non-uniformly hyperbolic flow on the moduli space of Riemann surface. We review some results about a kind of symbolic coding of the flow using laminations and subsurface coefficients. Then we apply some estimates on WP metric and its derivatives in the thin part of moduli space to show that the strong asymptotics of a class of WP geodesic rays is determined by the associated laminations. As a result we give a symbolic condition for divergence of WP geodesic rays in the moduli space.

On S-duality and T-duality and algebro-geometric proof of modularity conjectures in BPS counting theories

Artan Sheshmani (Ohio State)

Abstract: We construct an algebraic-geometric framework to calculate the partition functions of "massive black holes" enumerating invariants of supersymmetric D4-D2-D0 BPS states in type IIA string theory. Using S-duality, the entropy of such black holes can be related to a certain N=2, d=4 Super Yang-Mills theory on a divisor in a threefold. Physicists: Gaiotto, Strominger, Yin, Denef, Moore, via careful study of such S-duality, have conjectured that these partition functions have modular properties. We give a rigorous mathematical proof of their conjectures in different geometric setups. This is a report of joint project with Amin Gholampour and Richard Thomas. We also use an algebro-geometric analogue of the string theoretic D4/D2 T-duality to prove the modularity properties of certain PT stable pair invariants over threefolds given by smooth and Nodal surface fibrations over a curve. Here our strategy is to use combination of degeneration techniques, conifold transitions, and wall crossing of Bridgeland stability conditions. This is a report of joint project with Gholampour and Yukinobu Toda.

An Introduction to Tropical Geometry: Part II

Nathan Fieldsteel (UIUC Math)

Abstract: A continuation of last week's seminar, we will begin by tying up some loose ends from last time. We will then present more of the general theory of tropical geometry, and discuss connections to polyhedral geometry, hyperplane arrangements, and grassmanians, time permitting. Professors are welcome to attend.

Positive loops and orderability in contact geometry

Peter Weigel (Purdue University Math)

Abstract: Orderability of contact manifolds is related in some non-obvious ways to the topology of a contact manifold V. We know, for instance, that if V admits a 2-subcritical Stein filling, it must be non-orderable. By way of contrast, in this talk I will discuss ways of modifying Liouville structures for high-dimensional V so that the result is always orderable. The main technical tool is a Morse-Bott Floer theoretic growth rate, which has some parallels with Givental's nonlinear Maslov index. I will also discuss a generalization to the relative case, and applications to bi-invariant metrics on Cont(V).

Introduction to Grothendieck Topologies

Juan S. Villeta-Garcia (UIUC Math)

Abstract: We will introduce Grothendieck topologies, sites, sheaves on them, and their cohomology. Examples will be taken from scheme theory and commutative algebras. The exposition will be basic and aimed at beginners (such as the speaker). This will be the first of a two-part talk. Professors are welcome to attend.

Lie Algebroid Spray

Songhao Li (Washington University Math)

Abstract: Analogous to the spray in Riemannian geometry, we introduce the Lie algebroid spray, or A-spray. A special case is the Poisson spray as introduced by Crainic and Marcut. As an application, we show that the source-simply-connected symplectic groupoid of a log symplectic surface is diffeomorphic to the cotangent bundle in such a way that the source map coincide with the bundle projection. (Joint work in progress with Marco Gualtieri)

Stacky Resolutions of Singularities

Matthew Satriano (University of Michigan)

Abstract: We will discuss a technique which allows one to approximate singular varieties by smooth spaces called stacks. As an application, we will address the following question, as well as some generalizations: given a linear action of a group G on complex n-space C^n, when is the quotient C^n/G a singular variety? We will also mention some applications to Hodge theory and to derived equivalences.

Introduction to Grothendieck Topologies

Juan S. Villeta-Garcia (UIUC Math)

Abstract: We will introduce Grothendieck topologies, sites, sheaves on them, and their cohomology. Examples will be taken from scheme theory and commutative algebras. The exposition will be basic and aimed at beginners (such as the speaker). Professors are welcome to attend.

A generalization of the group of Hamiltonian homeomorphisms

Augustin Banyaga (Pennsylvania State University Math)

Abstract: The Eliashberg-Gromov rigidity theorem implies that Symplectic Geometry underlines a topology. This talk is about the automorphism groups of this "continuous" symplectic topology. The group of symplectic homeomorphisms (Sympeo) has a remarkable subgroup: the group of Hamiltonian homeomorphisms (Hameo), defined by Oh and Müller using the $L^{(1q,\infty)}$ Hofer norm. We introduce a generalization of Hameo, called the group of strong symplectic homeomorphisms (SSympeo), using a generalization of the Hofer norm from the group of Hamiltonian diffeomorphisms to the whole group of symplectic diffeomorphisms. Each group Hameo and SSympeo has also a $L^\infty$ version. The two versions coincide (Müller, Banyaga-Tchuiaga).

Genera and derived algebraic geometry

Nick Rozenblyum (Northwestern)

Abstract: We will describe an approach, motivated by quantum field theory, to describe invariants of algebraic varieties using derived algebraic geometry. In particular, we will describe a version of non-abelian duality that can be used to produce volume forms on derived mapping spaces. Integration of these volume forms produces interesting invariants such as the Todd genus, the Witten genus and the B-model operations on Hochschild homology.

The Geometry of Filtered Quiver Varieties

Mee Seong Im (UIUC Math)

Abstract: Invariant theory has connections to many areas of mathematics: to name a few, Higgs bundles, David Mumford's geometric invariant theory and Hilbert schemes in algebraic geometry, Nakajima's quiver variety in representation theory, the Hamiltonian reduction construction in symplectic geometry, combinatorics, graph theory, coding theory, DNA strand configuration, and fingerprint technology. Around 1990's, Aidan Schofield and a number of other mathematicians introduced and extended the study of classical invariant theory to quiver varieties. I will discuss the evolution of invariant theory, invariant theory in geometric representation theory, some results and conjectures, and interesting applications.

On the Topological Dynamics Arising from a Contact Form

Peter Spaeth (Pennsylvania State University Math)

Abstract: Stefan Müller and Yong-Geun Oh introduced the Hamiltonian metric on the group of Hamiltonian isotopies of a symplectic manifold, and with it defined the groups of topological Hamiltonian isotopies and homeomorphisms. With Augustin Banyaga we introduced the contact metric on the space of strictly contact isotopies of a contact manifold, and defined the groups of topological strictly contact isotopies and homeomorphisms in a similar manner. In the talk I will explain how the one to one correspondence between smooth strictly contact isotopies and generating contact Hamiltonian functions extends to their topological counterparts when the contact form is regular. I will also prove that the group of diffeomorphisms that preserve a contact form is rigid in the sense of Eliashberg-Gromov. This last result is joint with Müller.

Propaganda for Higgs Bundles

Brian Collier (UIUC Math)

Abstract: The goal of the talk is introduce the Hitchin System associated to the moduli space of Higgs bundles and the spectral data associated to it. The talk will introduce/review some facts about the moduli spaces of holomorphic vector bundles and Einstein metrics and illustrate how Higgs bundles generalize this picture.

All boundaries of contact type can keep secrets

Ely Kerman (UIUC Math)

Abstract: Let $(M, \omega)$ be a symplectic manifold with nonempty boundary, $W$. The restriction of $\omega$ to $W$, $\omega_W$, has a one dimensional kernel which defines the characteristic foliation of $W$. If $W$ is a boundary of contact type then it admits a tubular neighborhood comprised of hypersurfaces whose characteristic foliations are all conjugate to those of $W$. Since these hypersurfaces lie in the interior one might guess (or hope) that the interior of $(M, \omega)$ determines $omega_W$ or at least some of its symplectic invariants. Several questions in this direction were raised by Eliashberg and Hofer in the early nineties. In this talk I will describe the resolution of some of these questions. I will prove that neither $\omega_W$ or its action spectrum is determined by the interior of $(M, \omega)$. This involves the construction of a new dynamical symplectic plug. The construction uses only soft techniques (Moser's method) and so should hopefully be accessible to all.

Centers and traces of the categorified affine Hecke algebra (or, some tricks with coherent complexes on the Steinberg variety)

Anatoly Preygel (Berkeley)

Abstract: This is a talk on some tricks and constructions on categories of bounded coherent complexes on nice stacks. The goal of the talk will be to explain how "proper descent with singular-support conditions" gives a framework for getting interesting answers when computing (dg-categorical) invariants of the circle by chopping it into intervals. Our main application will be to the affine Hecke category in geometric representation theory: The Steinberg (derived) stack parametrizes G-local systems on an annulus with B-reductions on the boundary. Its dg-category of bounded coherent complexes is monoidal, and categorifies the affine Hecke algebra in representation theory. We'll see how to identify the trace of this monoidal category with a full subcategory of bounded coherent complexes on Loc_G(torus), cut out by a nilpotent micro support condition. This is joint work with Ben-Zvi and Nadler.

Moduli of Elliptic Curves

Peter Nelson (UIUC Math)

Abstract: I'll talk about various sorts of moduli things of elliptic curves, and how you compute things about them. There might even be some computations!

Fixed points of symplectic circle actions

Donghoon Jang (UIUC Math)

Abstract: The study of fixed points of maps is a classical and important topic in geometry and topology. During this talk, we focus on the fixed points of maps in the case where manifolds admit symplectic structures and circle actions on the manifolds preserve the symplectic structures. We discuss main theorems on fixed points of symplectic circle actions and discuss techniques to study, ABBV Localization formula and Atiyah-Singer index formula.

Semi-toric systems as Hamiltonian S^1-spaces

Daniele Sepe (Utrecht University Math)

Abstract: The classification of completely integrable Hamiltonian systems on symplectic manifolds is a driving question in the study of Hamiltonian mechanics and symplectic geometry. From a symplectic perspective, such systems correspond to Hamiltonian R^n-actions which are locally toric. The class of integrable Hamiltonian systems on 4-dimensional symplectic manifolds corresponding to Hamiltonian S^1 x R actions (with some extra assumptions on the singularities) is known as semi-toric: it was introduced by Vu Ngoc, and Pelayo and Vu Ngoc obtained a classification for `generic' semi-toric systems. From such a system one obtains a 4-dimensional manifold with a Hamiltonian S^1-action by restricting the action: when the underlying symplectic manifold is closed, Karshon classified these spaces in terms of a labelled graph. This talk aims at explaining how, starting from a semi-toric system on a closed 4-dimensional symplectic manifold, Karshon's invariants of the underlying Hamiltonian S^1-space can be recovered using the notion of `polygons with monodromy' introduced by Vu Ngoc. This should be thought of as analogous to the procedure to obtain Karshon's invariants from Delzant polygons in the case of symplectic toric manifolds. This is joint work with Sonja Hohloch (EPFL) and Silvia Sabatini (IST Lisbon), and part of a longer term project to study Hamiltonian S^1 x R actions on closed 4-dimensional manifolds.

Unicorns and Beyond

Sebastian Hensel

Abstract: In this talk, I will first present joint work with Piotr Przytycki and Richard Webb giving a new short proof of uniform hyperbolicity of curves and arc graphs. Namely, I will describe unicorn paths in arc and curve graphs and show that they form 1-slim triangles. Using this, one can deduce that arc graphs are 7-hyperbolic (and curve graphs are 17-hyperbolic) I will then overview some other results which, in a similar vein, give quick and purely topological-combinatorial proofs of curve graph results. If time permits, I will explain how such proofs can sometimes be adapted to work in the Out(F_n) setting.

Local cohomology with support in generic determinantal ideals

Claudiu Raicu (Princeton University)

Abstract: The space $Mat(m,n)$ of $m\times n$ matrices admits a natural action of the group $\textrm{GL}_m \times \textrm{GL}_n$ via row and column operations on the matrix entries. The invariant closed subsets are the closures of the orbits of constant rank matrices. I will explain how to describe the local cohomology modules of the ring $S$ of polynomial functions on $Mat(m,n)$ with support in these orbit closures, and mention some consequences of the methods employed to computing minimal free resolutions of invariant ideals in $S$. These ideals correspond to nilpotent scheme structures on the orbit closures, and their study goes back to the work of De Concini, Eisenbud and Procesi in the 80s. Joint work with Jerzy Weyman.

Equations of Parametric Curves and Surfaces via Syzygies

Eliana Duarte (UIUC Math)

Abstract: The problem of finding an implicit equation of a parametric curve or surface, known as the Implicitization Problem, dates back to 1862. The method of eliminating parameters and the use of resultants were the main tools to find implicit equations. In this talk I will explain Sederberg’s method(1997) of how to use syzygies to compute the implicit equation of a parametric curve or surface.

Intersection Multiplicity of Serre in the Unramified Case

Chris Skalit (University of Chicago)

Abstract: Let $A$ be a regular local ring whose completion is a power series ring over a DVR. For properly-meeting subschemes of complimentary dimension, $Y, Z \subseteq \operatorname{Spec} A$, we show that the Serre intersection multiplicity, $\chi(\mathcal{O}_{Y},\mathcal{O}_Z) = \sum{(-1)^i \ell (\operatorname{Tor}_i^A(\mathcal{O}_Y,\mathcal{O}_Z))}$, is bounded below by the product of the multiplicities of $Y$ and $Z$. For those cases in which this bound is achieved, we investigate the implications it has for $Y, Z$, and their strict transforms on the blowup.

Integration of Exact Courant algebroids

Xiang Tang (Washington University Math)

Abstract: In this talk, we will discuss some recent progress about the problem of integration of exact Courant algebroids. We construct an infinite-dimensional symplectic 2-groupoid as the integration of an exact Courant algebroid. We show that every integrable Dirac structure integrates to a ``Lagrangian" sub-2-groupoid of this symplectic 2-groupoid.

Elliptic Actions on Teichmüller Space

Matthew Durham (UIC Math)

Abstract: Kerckhoff's solution to the Nielsen realization problem showed that the action of any finite subgroup of the mapping class group on Teichmüller space has a fixed point. The set of fixed points is a totally geodesic submanifold. We study the coarse geometry of the set of points which have bounded diameter orbits in the Teichmüller metric. We show that each such almost-fixed point is within a uniformly bounded distance of the fixed point set, but that the set of almost-fixed points is not quasiconvex. In addition, the orbit of any point is shown to have a fixed barycenter. In this talk, I will discuss the machinery and ideas used in the proofs of these theorems.

To Be Announced

Chunyi Li (UIUC)

MMP for deformed Hilbert scheme of points on projective plane

Chunyi Li (UIUC)

Abstract: The idea of running the minimal model program for the moduli space of sheaves via the wall-crossing of Bridgeland stability conditions is, as far as I know, first introduced by Toda. In the Hilb^n P2 case, a strong form conjecture, which is about the correspondence between the base locus decomposition walls for the effective cone of Hilb^n P2 and the destabilizing walls on the stability condition plane, is posed by Arcara, Bertram, Coskun and Huizenga. In this talk, I will introduce the stability condition on D^b(coh P2) and the birational geometry of (deformed)Hilb^n P2. Also I would state our theorem which proves ABCH's conjecture and generalizes the result to deformed Hilb^n P2 case.

CANCELLED

A normal form theorem around symplectic leaves

Ioan Marcut (UIUC Math)

Abstract: In this talk, I will discuss a normal form result in Poisson geometry, which generalizes Conn's theorem from fixed points to arbitrary symplectic leaves. The local model, at least in the integrable case, coincides with the local model of a free and proper Hamiltonian action around the zero set of the moment map. The result is joint work with Marius Crainic.

Escape paths of Besicovitch triangles (revisited)

Yevgenya Movshivich (EIU Math)

Abstract: An escape path of an oval is the shortest path that does not fit in the interior of the oval. In 1965, A. S. Besicovitch conjectured that a certain symmetric unit $z$-arc is an escape path for the equilateral triangle of side $\sqrt{28/27}$. The conjecture was proven in "Besicovitch triangles cover unit arcs", Geom. Dedicata, 123 (2006) by P. Coulton and Y. M. for a family of Besicovitch isosceles triangular covers of unit arcs. The base angle, alpha, there ranged from about 52.2 degrees to 60 degrees. The low limit of this range was changed to 45 degrees in “Besicovitch triangles extended”, Geom. Dedicata, 159 (2012), by Y. M. Having just one escape unit arc, means that this cover of unit arcs is minimal (tight). In the Spring 2008 in two separate talks by P. Coulton and by Y. Movshovich, it was announced that a family of non-isosceles triangular covers of unit arcs (that contained all Besicovitch isosceles triangular covers) were found and the isosceles covers had infinitely many escape unit paths. A few months later we discovered that the pure non-isosceles covers are not minimal, all unit arcs fit in their interior, thus they have no escape unit paths. At the same time, each isosceles cover had a $Z$-arc as its only escape unit path. We will present a geometric argument supporting this last statement and conjecture on the sizes of the non-isosceles triangular covers of unit arcs that would make them minimal.

A classification of extremal Lagrangian planes

Benjamin Bakker (Courant Institute)

Abstract: Classically, an extremal class $R$ in the cone of effective curves on a K3 surface $X$ is representable by a smooth rational curve if and only if $R^2=-2$. Settling a conjecture of Hassett and Tschinkel, we prove the natural generalization to higher dimensions: for a holomorphic symplectic variety $M$ deformation equivalent to a Hilbert scheme of $n$ points on a K3 surface, an extremal effective curve class $R$ sweeps out a Lagrangian $n$-plane if and only if certain intersection-theoretic criteria are met, including $(R,R)=-(n+3)/2$. The proof uses recent work of Bayer and Macri to represent effective cycles in moduli spaces of sheaves using Bridgeland stability conditions.

Introduction to Grothendieck Topologies: Part II

Juan S. Villeta-Garcia (UIUC Math)

Abstract: We will continue our discussion of Grothendieck topologies, focusing on the etale site, and its associated cohomology. We'll begin with an introduction to etale morphisms and why we care about them. We will draw our examples from the cohomology of curves. The exposition will be basic and aimed at beginners (such as the speaker). Professors are welcome to attend.

Relative equilibria and vector fields on stacks

Eugene Lerman (UIUC Math)

Abstract: TBA

Construction of the second flip of $M_{g}$

David Smyth (ANU)

Abstract: I will discuss aspects of the construction of the second flip in the log minimal model program for $M_{g}$ (joint with Alper, Fedorchuk, van der Wyck). I will focus on the way in which formal local VGIT is used to construct the second flip as an algebraic space.

The Étale Fundamental Group

Matej Penciak (UIUC Math)

Abstract: The purpose of this talk is to introduce the étale fundamental group of a scheme. Taking the Galois theory of fields and the theory of covering spaces as our guides, we will explore their generalizations to the setting of schemes. After defining the étale fundamental group, we will give an idea of how these groups may be computed.

Morse Theory and the Moduli Space of Curves

Susan Tolman (UIUC Math)

Abstract: Based on joint work with Bott and Weitsman, we will explain how to use Morse theory to calculate the Betti number of reduced spaces for proper Hamiltonian loop-group actions, such as the moduli space of curves.

Almost purity theorem with applications to the homological conjectures - Part I

Kazuma Shimomoto (Meiji University)

Abstract: I will talk about almost purity theorem proved by Davis and Kedlaya with applications to the homological conjectures in local algebra. The almost purity theorem originates from p-adic Hodge theory by Faltings. I will also talk about its brief history and then construct a big Cohen-Macaulay algebra under some special condition.

An Introduction to Boij-S\"oderberg Theory

Matt Mastroeni (UIUC Math)

Abstract: Let $k$ be a field. The aim of the talk is to give sufficient background on free resolutions and graded Betti numbers over the polynomial ring $k[x_1, \dots, x_n]$ in order to state the Boij-S\"oderberg Conjectures, which were proved in 2008 by Eisenbud and Schreyer. I will also explain how this answers the Multiplicity Conjecture of Herzog, Huneke, and Srinivasan and give an example illustrating the conjectures. Time permitting, I might say a few words about the proof of the Boij-S\"oderberg Conjectures, but the details will be reserved for a future talk.

Almost purity theorem with applications to the homological conjectures - Part II

Kazuma Shimomoto (Meiji University)

Abstract: I will talk about almost purity theorem proved by Davis and Kedlaya with applications to the homological conjectures in local algebra. The almost purity theorem originates from p-adic Hodge theory by Faltings. I will also talk about its brief history and then construct a big Cohen-Macaulay algebra under some special condition.

Imaginary time flow in geometric quantization and in Kahler geometry, degeneration to real polarizations and tropicalization

Jose Mourao (Instituto Superior Técnico)

Abstract: We will recall the problem of dependence of quantization of a symplectic manifold on the choice of polarization and study its relation with geodesics in the space Kahler metrics. Complex one parameter subgroups of the "group" of complexified hamiltonian symplectmorphisms appear naturally in this context. For some classes of symplectic manifolds we will describe geodesic rays of Kahler structures degenerating to real polarizations and study the associated metric collapse. Each such ray selects a basis of holomorphic sections which converge to distributional sections supported on Bohr-Sommerfeld fibers as the geodesic time goes to infinity. The same geodesic rays lead to tropicalization of toric varieties and of hypersurfaces on toric varieties.

Real slices of the moduli space of Higgs bundles

Laura Schaposnik Massolo (UIUC Math)

Abstract: After introducing Higgs bundles and their moduli space, through the natural hyperkähler structure of the moduli space of Higgs bundles for complex groups we shall construct three anti-holomorphic involutions whose fixed points in the moduli space give branes in the A-model and B-model. After defining what those branes are, we shall attempt to relate them to log-symplectic structures and their invariants.

Mapping stacks and the notion of properness in algebraic geometry

Daniel Halpern-Leistner (Columbia University)

Abstract: One essential feature of a scheme X which is flat and proper over a base scheme S is that for any other finite type S scheme, there is a finite type algebraic space Map(X,Y) parameterizing families of maps from X to Y. There have been several extensions of these results to the setting where X is a proper stack, and Y is a stack satisfying various hypotheses. Unfortunately many of the stacks arising in nature, such as global quotient stacks X/G, have affine stabilizer groups and are about as far as possible from being proper. However, we will show that for many non proper X and a large class of Y, the mapping stack Map(X,Y) is still algebraic and finite type. This leads us to introduce new notions of "projective" and "proper" for morphisms between stacks such that "projective" => "proper", and flat and "proper" => Map(X,Y) is algebraic for reasonable X. We discuss a large list of examples of "projective stacks", including X/G where G is reductive and X is projective-over-affine with H^0(O_X)^G finite dimensional, as well as any quotient stack which admits a projective good moduli space. Based on these, we will come up with an even longer list of "proper" stacks, including stacks which are proper over a scheme in the classical definition. Along the way, we will discuss some surprising "derived h-descent" results in derived algebraic geometry.

Connections between the Geometry of Hyperplane Arrangements and their Combinatorics

Nathan Fieldsteel (UIUC Math)

Abstract: From the data of an arrangement $\mathcal{A}$ of hyperplanes, we can construct two toric varieties. The first is determined by the rational fan $\Sigma(\mathcal{A})$ which has as its maximal cones the sectors of the complement of $\mathcal{A}$. The second is determined by $\Sigma(\mathcal{L}(\mathcal{A}),G)$, a rational fan determined by intersection lattice of the arrangement, together with a choice of building set. This second construction follows the work of Feichtner and Yuzvinsky in which they associate a smooth toric variety to any atomic lattice. We are interested in finding a relationship between these two fans, especially when $\mathcal{A}$ is the arrangement of type $A_n$, $B_n$, or $D_n$.

Upper bounds for the Gromov width of coadjoint orbits of compact Lie groups

Alexander Caviedes Castro (University of Toronto)

Abstract: I will show how to find an upper bound for the Gromov width of coadjoint orbits with respect to the Kirillov-Kostant-Souriau symplectic form by computing certain Gromov-Witten invariants. The approach presented here is closely related to the one used by Gromov in his celebrated Non-squeezing theorem.

Circumcenter of Mass and the generalized Euler line

Sergei Tabachnikov (Penn State Math)

Abstract: I shall define and study a variant of the center of mass of a polygon, called the Circumcenter of Mass. The Circumcenter of Mass is an affine combination of the circumcenters of the triangles in a non-degenerate triangulation of a polygon, weighted by their areas, and it does not depend on the triangulation. For an inscribed polygon, this center coincides with the circumcenter. The Circumcenter of Mass satisfies an analog of the Archimedes Lemma, similarly to the center of mass of the polygonal lamina. The line connecting the circumcenter and the centroid of a triangle is called the Euler line. Taking an affine combination of the circumcenters and the centroids of the triangles in a triangulation, one obtains the Euler line of a polygon. The construction of the Circumcenter of Mass extends to simplicial polytopes and to the spherical and hyperbolic geometries.

Fusion products and a novel way to compute their characters

Bolor Turmunkh (UIUC Grad)

Abstract: We will introduce a graded tensor product of simple Lie algebras called the Fusion product and discuss the character of this module. This will be done through examples. Then we will see a novel way to compute the characters of Fusion products of $\mathfrak{sl}_2(\mathbb{C})$-modules using the quantum Q-system for $\mathfrak{sl}_2(\mathbb{C})$.

Convexity Theorems for Semisimple Symmetric Spaces

Dana Balibanu (Utrecht Math)

Morse Theory of D-Modules

Thomas Nevins (UIUC Math)

Abstract: Hamiltonian reduction arose as a mechanism for reducing complexity of systems in mechanics, but it also provides a tool for constructing complicated but interesting algebraic varieties from simpler ones. I will illustrate how this works via examples. I will explain a new structure theory, motivated by Hamiltonian reduction, for some categories (of D-modules) of interest to representation theorists, and, if there is time, indicate applications to the cohomology of (hyperkaehler) manifolds. The talk will not assume that members of the audience know the meaning of any of the above-mentioned terms. The talk is based on joint work with K. McGerty.

Implicitization Using Approximation Complexes

Eliana Duarte (UIUC Math)

Abstract: I will present the method of using approximation complexes to compute the image of a rational map from $\mathbb{P}^{n-1}$ to $\mathbb{P}^{n}$, under some hypotheses on the base locus and on the image. The method uses tools from commutative algebra such as Koszul complexes and Castelnuovo-Mumford regularity which I will introduce.

Folded Symplectic Reduction

Daniel Hockensmith (UIUC Math)

Abstract: The Marsden-Weinstein-Meyer reduction theorem is an indispensable tool for the study of Hamiltonian group actions on symplectic manifolds. It gives an explicit recipe for the construction of a symplectic reduced space using only regular values of the moment map and the group action. I will prove that if one replaces symplectic manifolds with oriented, folded-symplectic manifolds in the statement of the MWM reduction theorem then a reduced space with a natural folded-symplectic form is obtained in the same way. I will then argue that the assumptions of this generalized theorem are too strong, leading us towards a more robust set of assumptions for a folded-symplectic reduction theorem.

What does a "right" cohomology for rings look like?

Juan S. Villeta-Garcia (UIUC Math)

Abstract: Motivated by the aforementioned question, we introduce Andre-Quillen (co)-homology for commutative algebras using methods of homotopy theory. We connect the theory to the cotangent comples, and prove certain vanishing theorems characterizing classes of maps. We end with some examples in the rational case, and mention a topological characterization.

Local Rigidity & Nash-Moser Methods

Roy Wang (Utrecht University Math)

Abstract: J. Conn used analytic methods to prove his theorem on the linearization of Poisson structures. For some time that proof was heuristically interpreted as a local rigidity result for linear, compact, semi-simple Poisson structures. In his thesis I. Marcut made this interpretation rigorous, which lead to surprising new results. In collaboration we aim to isolate the method and formulate a local rigidity theorem, which we apply to other geometrical structures. As an example I sketch a proof of the Newlander-Nirenberg theorem.

A new proof of Bowen's theorem on Hausdorff dimension of quasi-circles

Andy Sanders (UIC Math)

Abstract: A quasi-Fuchsian group is a discrete group of Mobius transformations of the Riemann sphere which is isomorphic to the fundamental group of a compact surface and acts properly on the complement of a Jordan curve: the limit set. In 1979, Bowen proved a remarkable rigidity theorem on the Hausdorff dimension of the limit set of a quasi-Fuchsian group: it is equal to 1 if and only if the limit set is a round circle. This theorem now has many generalizations. We will present a new proof of Bowen's result as a by-product of a new lower bound on the Hausdorff dimension of the limit set of a quasi-Fuchsian group. This lower bound is in terms of the differential geometric data of an immersed, incompressible minimal surface in the quotient manifold. If time permits, generalizations of this result to other convex-co-compact surface groups will be presented.

Birational geometry of the moduli space of one-dimensional sheaves

Jinwon Choi (KIAS)

Abstract: We study the birational geometry of the moduli space of stable sheaves on $\mathbb{P}^2$ with Hilbert polynomial $dm+1$. We determine the effective/nef cone in terms of natural geometric divisors. We also present the birational model constructed from the locally free resolutions of the general sheaves. The two spaces are related by the Bridgeland-type wall-crossing. As corollaries, we compute the Betti numbers of the moduli spaces when $d \leq 6$. The results confirm the prediction from physics. This is joint work with Kiryong Chung.

Infinitesimal Algebraic Geometry and Infinitesimal Infinitesimal Algebraic Geometry

Peter Nelson (UIUC Math)

Abstract: Sometimes the more classical infinitesimal objects attached to a "smooth" group don't contain as much information as one would like, especially in an algebraic setting. I'll discuss one or two (still pretty classical) improvements on the situation. Since I like thinking about universal things, I'll try to say a few things about the moduli spaces of these improvements, and maybe even how they relate to the moduli of the original groups.

Legendrian Knots and Constructible Sheaves

Eric Zaslow (Northwestern)

Abstract: We study the unwrapped Fukaya category of Lagrangian branes ending on a Legendrian knot. Our knots live at contact infinity in the cotangent bundle of a surface, the Fukaya category of which is equivalent to the category of constructible sheaves on the surface itself. Consequently, our category can be described as constructible sheaves with singular support controlled by the front projection of the knot. We use a theorem of Guillermou-Kashiwara-Schapira to show that the resulting category is invariant under Legendrian isotopies, and conjecture it is equivalent to the representation category of the Chekanov-Eliashberg differential graded algebra of the knot. This sounds harder than it is. Briefly-- INPUT: Knot diagram, OUTPUT: Category. I will illustrate the above with simple examples. This work is joint with David Treumann and Vivek Shende.

Dynamical convexity and elliptic orbits for Reeb flows

Miguel Abreu (Instituto Superior Técnico)

Abstract: A classical conjecture states that any convex hypersurface in even-dimensional euclidean space carries an elliptic closed orbit of its characteristic flow. Dell'Antonio-D'Onofrio-Ekeland proved it in 1995 for antipodal invariant convex hypersurfaces. In this talk I will present a generalization of this result using contact homology and a notion of dynamical convexity first introduced by Hofer-Wysocki-Zehnder for contact forms on the 3-sphere. Applications include certain geodesic flows, magnetic flows and toric contact manifolds. This is joint work with Leonardo Macarini.

Cohomological characterization of products of theta-divisors

Sofia Tirabassi (University of Utah)

Abstract: We present a joint work with J. Jiang and M. Lahoz in which it is proven that any smooth complex projective variety of maximal Albenese dimension, with Euler characteristic 1 and Albanese image normal and of general type is a product of theta-divisors. We also generalize in higher dimension Hacon--Pardini classification of surfaces of maximal Albanese dimension with genus and irregularity equal 3. The techniques we use are based on Green--Lazarsfeld generic vanishing theorems and on the use of integral transforms.

Asymptotics of certain families of Higgs bundles

Brian Collier (UIUC Math)

Abstract: Higgs bundles are algebro-geometric objects that live over a Kahler manifold. Through nonabelian Hodge theory, the moduli space of Higgs bundles is homeomorphic to the space of reductive representations of the fundamental group (or a central extension) of the manifold. To get this homeomorphism one goes through two deep, nonconstructive existence theorems. In this talk I will sketch this correspondence, then consider a family of Higgs bundles of particular geometric interest, and talk about some new results on the asymptotics of certain families of Higgs bundles.

Integration of generalized complex structures

Michael Bailey (CIRGET/UQAM/McGill)

Abstract: Generalized complex geometry is a generalization of both symplectic and complex geometry, proposed by Nigel Hitchin in 2002, which is of particular interest in string theory and mirror symmetry. Modulo a parity condition, generalized complex manifolds locally "look like" holomorphic Poisson manifolds, though globally they may not admit a complex structure at all. Therefore, locally they should integrate to holomorphic symplectic groupoids. One can take the global integration if one passes to holomorphic "symplectic" stacks. Earlier work by Crainic defined an integration for generalized complex structures which did not capture the holomorphic nature.

Springer Theory for D-modules

Sam Gunningham (University of Texas-Austin)

Abstract: The Springer correspondence relates unipotent conjugacy classes in a reductive algebraic group G (e.g. GL_n), with representations of its Weyl group W (e.g. S_n). More precisely to every irreducible representation of W, one can attach an equivariant local system on a unipotent conjugacy class. Lusztig was able to account for all such local systems using his notion of cuspidal sheaves, together with certain relative Weyl groups. In this talk I will give a new perspective on Springer Theory using tools from sheaf theory and category theory, and I will explain how to generalize the Springer correspondence to give a description of the derived category of conjugation equivariant D-modules on G.

Transverse Geometry of Codimension one Foliations Calibrated by Closed 2-Forms

David Martinez Torres (PUC-Rio de Janeiro)

Abstract: A codimension one foliation is (topologically) taut if it admits a closed 1-cycle everywhere transverse to the foliation. The theory of taut foliations is extremely rich in dimension 3, however, it less satisfactory in higher dimensions. In this talk we will discuss a different generalization of 3-dimensional taut foliati- ons to higher dimensions inspired in symplectic geometry. These are codimension one foliations which admit a closed 2-form which makes every leaf a symplectic manifold. Our main result is that on an ambient closed manifold a foliation (of class at least C^1 in the transverse direction) admitting a 2-calibration has its transverse geometry encoded in a 3-dimensional foliated submanifold. This is joint work with Álvaro del Pino and Francisco Presas (ICMAT, Madrid)

Holomorphic one-forms on varieties of general type

Mihnea Popa (UIC Math)

Abstract: I will explain recent work with C. Schnell, in which we prove that every holomorphic one-form on a variety of general type has non-empty zero locus (together with a suitable generalization to arbitrary Kodaira dimension). The proof makes use of generic vanishing theory for Hodge D-modules on abelian varieties.

Schemes as Functors

Matej Penciak (UIUC Math)

Abstract: Replacing schemes with their functor of points offers a useful perspective to tackle moduli problems. In this talk I will explain this interpretation of schemes, characterize the functors that come from this construction, and try to motivate this viewpoint through various examples. Along the way I will discuss the Quot and Hilbert schemes--two schemes that represent common moduli problems.

Lagrangian correspondences - a toric case study

Ana Cannas da Silva (ETH)

Abstract: What lagrangians in a symplectic reduced space admit a (one-to-one transverse) lifting to the original symplectic manifold? I will discuss this question (going back to work of Werheim and Woodward) through examples and counterexamples (joint work with Meike Akveld).

On the geometry of the flip graph

Valentina Disarlo (Indiana U Math)

Abstract: Given an orientable finite type punctured surface, its flip graph is the graph whose vertices are the ideal triangulations of the surface (up to isotopy) and two vertices are joined by an edge if the two corresponding triangulations differ by a flip, i.e. the replacement of one diagonal of the a quadrilateral by the other one. The combinatorics of this graph is crucial in works of Thurston and Penner's decorated Teichmuller theory. In this talk we will explore the geometric properties of this graph, proving that it provides a coarse model of the mapping class group in which the mapping class groups of the subsurfaces are convex. Moreover, we will provide bounds on the growth of the diameter of the flip graph modulo the mapping class group, providing a partial answer to an open problem in combinatorics.

Counting curves on K3 surfaces: the Katz-Klemm-Vafa formula

Rahul Pandharipande (ETH Zurich)

Abstract: I will explain our recent proof (with R. Thomas) of the KKV formula governing higher genus curve counting in arbitrary classes on K3 surfaces. The subject intertwines Gromov-Witten, Noether-Lefschetz, and Donaldson-Thomas theories. A tour of these ideas will be included in the talk.

Regularity and Piecewise Polynomial Functions

Michael DiPasquale (UIUC Math)

Abstract: The algebra $C^r(\mathcal{P})$ of piecewise polynomial functions continuously differentiable of order $r$ over a polytopal complex $\mathcal{P}$ is a fundamental object in approximation theory. One of the fundamental questions in spline theory is to compute the dimension of the vector space $C^r_k(\mathcal{P})$ of splines of degree at most $k$. In the 1980s Billera pioneered an algebraic approach to spline theory using tools from homological and commutative algebra. We show how this approach, particularly the notions of the Hilbert polynomial and Castelnuovo-Mumford regularity, has interesting things to say about computing the dimension of $C^r_k(\mathcal{P})$.

Lie algebra cohomology and a degenerate cup product on the flag manifold

Sam Evens (Notre Dame)

Abstract: Belkale and Kumar introduced a degeneration of the usual cup product on $H^*(G/P)$ which gives an optimal solution to the geometric Horn problem. In this talk, I will explain joint work with Bill Graham where we realize the Belkale-Kumar product using relative Lie algebra cohomology. We do this using a family in the variety of Lagrangian subalgebras.

Wall-crossing in genus zero Landau-Ginzburg theory

Dustin Ross (University of Michigan)

Abstract: Given a quasi-homogeneous polynomial of degree d, Landau-Ginzburg theory studies certain intersection numbers on the moduli space of d-spin curves (parametrizing curves with d-th roots of the canonical bundle). I will describe a generalization of these intersection numbers obtained by allowing some of the points on the curves to be weighted in the sense of Hassett. As one changes the weights, the invariants thus obtained can be related by a wall-crossing formula. I will explain how the wall-crossing formula generalizes the mirror theorem of Chiodo-Iritani-Ruan, and in particular how it gives a completely enumerative (A-model) interpretation of the mirror phenomenon.

The symplectic nature of the fundamental group

Brian Collier (UIUC Math)

Abstract: Let $\pi$ be the fundamental group of a RIemann surface and $G$ be a real or complex reductive algebraic group. The goal of this talk is to understand the representation variety $Hom(\pi,G)//G$ from a algebraic geometry perspective. In particular, we will describe the symplectic structure on the representation variety in terms the cup product in group cohomology. The talk will very closely follow the wonderful paper of Bill Goldman with the same title as this talk. All concepts will be explained as if the audience has little or no experience with them, as this is the case for the speaker. Also, the relation of the above topic with HIggs bundles will only be briefly mentioned at the end.

Symplectic toric manifolds as centered reductions of products of weighted projective spaces

Milena Pabiniak (Instituto Superior Técnico)

Abstract: We prove that every symplectic toric orbifold is a "centered" symplectic reduction of a Cartesian product of weighted projective spaces. Reduction is centered if the level set contains central Lagrangian torus fiber of the product of weighted projective spaces. In that case one can deduce certain information about non-displaceable sets or existence of quasimorphisms. For example, a theorem of Abreu and Macarini shows that if the level set of the reduction passes through a non-displaceable set then the image of this set in the reduced space is also non-displaceable. Using this theorem and our result we reprove that every symplectic toric orbifold contains a non-displaceable fiber and identify this fiber. Joint work with Aleksandra Marinkovic.

Combinatorics and topology of toric maps

Mircea Mustata (University of Michigan)

Abstract: Toric varieties are algebraic varieties endowed with a ``nice" action of an algebraic torus. A remarkable feature is that their geometry can be fully described in terms of combinatorics of fans and polytopes. I will discuss some results concerning the topology of the fibers of toric maps and a combinatorial invariant that comes out of these considerations. This is based on joint work in progress with Marc de Cataldo and Luca Migliorini.