Conference: Elliptic Genera, Spectra and Generalized Geometry

Paolo Piccione, University of São Paulo: Bifurcation Phenomena in Geometric Variational Problems

I will provide an overview of classical Bifurcation Theory, followed by an exploration of its contemporary applications in Riemannian Geometry. Topics covered will include minimal and constant mean curvature surfaces, as well as the Yamabe problem.  Video

Eduardo Gonzalez, University of Massachusetts: Quantum cohomology,  Seidel Elements, Shift operators and Coulomb branches

I will discuss joint work with D. Pomerleano and C. Y. Mak on the construction of an action of the (pure Coulomb branch) semi-infinite equivariant cohomology of the based loop group of a Lie group $G$ on the equivariant symplectic homology of an symplectic $G$-manifold. We will start with an introduction to classic results on Seidel/shift operators for closed symplectic manifolds.  Video

Carlos Ruiz, Colegio de Matematicas Bourbaki: Tame neural networks

In this talk I will describe some inequalities arising in logically tame neural networks that have been proved useful in the mathematical understanding of deep learning.  Video

Bernardo Uribe, UniNorte: Bordism and Birational Invariants Offiniye Groups

I will present the results of Bogomolov related to birational invariants of finite groups and their relation to bordism invariants of equivariant manifolds.  Video

Enrique Becerra, University of Miami: Some remarks on the spectrum of finite groups and the Eckedahl invariant

In this talk I will make some observations concernig the stringy spectrum of an orbifold coming from the quotient of a faithful representation of a finite group and its relation with the Ekedahl invariant.  Video

Jaqueline Mesquita, Universidade de Brasilia  Video

Eduardo Gonzalez, University of Massachusetts: : Quantum cohomology,  Seidel Elements, Shift operators and Coulomb branches

I will discuss joint work with D. Pomerleano and C. Y. Mak on the construction of an action of the (pure Coulomb branch) semi-infinite equivariant cohomology of the based loop group of a Lie group $G$ on the equivariant symplectic homology of an symplectic $G$-manifold. We will start with an introduction to classic results on Seidel/shift operators for closed symplectic manifolds.  Video

Miguel Xicoténcatl, CINVESTAV: On mapping class groups of non-orientable surfaces

In this talk I will survey on my recent work about the structure and cohomology of mapping class groups  Mod(N_g)  of non-orientable surfaces  N_g. These results include: the cohomology of the mapping class groups (with marked points) of the projective plane and the Klein bottle and its relation to characteristic classes of surface bundles, the Nielsen realization theorem for non-orientable surfaces, the non existence of a section for the natural projection form  Diff(N_g)  to  Mod(N_g)  and a systematic study of the Farrell cohomology of  Mod(N_g ; k) for g>2. This is joint work with: M. Maldonado, C. Hidber, N. Colin and R. Jiménez.  Video

Ernesto Lupercio, CINVESTAV: Stringy Motives

In this talk, I will survey our localization principle for orbifolds. Joint work with de Farnex, Neuvns and Uribe.  Video

Pavel Safronov, University of Edinburgh: Cohomological Donaldson-Thomas invariants of 3-manifolds

In my talk I will explain an approach to a rigorous definition of Morse homology of the complex Chern-Simons functional. While it is defined on the infinite-dimensional space of connections, its critical locus is finite-dimensional and one can use techniques of derived geometry to analyze it. I will explain connections between this Morse homology and 4d supersymmetric gauge theories, skein modules and Donaldson-Thomas invariants. This is based on joint work with Sam Gunningham (in progress) and Florian Naef.  Video

IMSA & Abess Center Joint Event: Dr. Andrea Bonilla (Click here for more information)

The World Academy of Sciences (Click here for more information)

Anatoly Libgober, University of Illinois Chicago: Equivariant Genera, spectrum of isolated singularities and hybrid models

I will discuss an equivariant version of Hirzebruch and elliptic genera of GIT quotients appearing in Witten's construction of hybrid models. In particular, the relation between invariants of these quotients from McKay correspondence for the elliptic genus has as a special case an LG/CY correspondence between the invariants of CY hypersurfaces in weighted projective spaces and invariants of isolated singularities.

As part of this construction, we explain the interpretation of the Steenbrink spectrum of singularities generic for its Newton polytope as a motivic measure on K-theory of varieties with a torus action.  Video

Josef Svoboda, University of Miami: Q-series Invariants and Abelian covers

I will present two results about q-series invariants of plumbed 3-manifolds defined by Gukov--Putrov--Pei-Vafa. The first result shows that a certain linear combination of these invariants coincides for two manifolds with the same universal abelian cover. The second result gives a formula for the invariants in terms of a rational function in a single variable, in the special case of Seifert manifolds. As a corollary to the formula, we obtain vanishing results for the invariants.  Video

Sergei Gukov, CalTech: Floer theory and quantum groups

Although it has been long expected that Vafa-Witten and Marcus-Kapustin-Witten theories lead to the same Floer homologies of 3-manifolds, direct computation of these invariants based on PDEs and moduli spaces has not been attempted until recently. In this talk, we will start with a careful discussion of moduli spaces in these 4d gauge theories and, following recent joint work with Artan Sheshmani and Shing-Tung Yau, compute these moduli spaces and the corresponding Floer invariants for an infinite family of 3-manifolds relevant to various surgery operations (the Gluck twist, knot surgeries, log-transforms). We compare the results to skein modules of the same family of 3-manifolds and explain that the K-theoretic version of these moduli spaces is essential for relation to a three dimensional Spin^c TQFT associated with quantum groups at generic values of the parameter q.  Video

Richard Paul Horja, University of Miami: A web of spherical functors in toric birational geometry

I will present a categorical proposal for the construction of a web of spherical functors associated to a toric configuration. The proposal expands on a conjecture by Aspinwall, Plesser and Wang in string theory. We will offer numerical and mirror symmetric evidence for the conjecture. This is joint work with Ludmil Katzarkov.  Video

Pavel Safronov, Univeristy of Edinburgh: Cohomological Donaldson-Thomas invariants of 3-manifolds

In my talk I will explain an approach to a rigorous definition of Morse homology of the complex Chern-Simons functional. While it is defined on the infinite-dimensional space of connections, its critical locus is finite-dimensional and one can use techniques of derived geometry to analyze it. I will explain connections between this Morse homology and 4d supersymmetric gauge theories, skein modules and Donaldson-Thomas invariants. This is based on joint work with Sam Gunningham (in progress) and Florian Naef.  Video

Lino Grama, UNICAMP: Spherical T-duality meets exotic Spheres and Homotopy Hopf Manifolds

The concept of Spherical T-duality aiming to generalize the classical well-studied T-duality. Shortly saying, for spherical T-duality, the role of the circle $\mathrm{S}^1$, or the circle group $\mathrm{U}(1)$, is replaced by the 3-sphere $\mathrm{S}^3$, or the special unitary group $\mathrm{SU}(2)$. On the one hand, topological T-duality relates pairs consisting of total spaces of $\mathrm{U}(1)$-principal bundles equipped with a cocycle in degree-3 ordinary cohomology; on the other hand, spherical T-duality relates pairs consisting of $\mathrm{SU}(2)$-principal bundles equipped with cocycles in degree-7 cohomology.

In this talk, we compare the two concepts and explain how 7-dimensional homotopy spheres are inserted in the context of spherical T-duality. Particularly, we show that a class of examples of the manifolds in a spherical T-duality diagram consists in $\Sigma^7\times\mathrm{S}^1$, where $\Sigma^7$ is a homotopy sphere.

Furthermore, any complex structure on $\Sigma^7\times\mathrm{S}^1$ with a chosen holomorphic structure can be transported to a different choice of diffeomorphism class of $\Sigma^7\times\mathrm{S}^1$ -- such a phenomenon does not occur for $\mathrm{S}^3\times\mathrm{S}^1$, that is, this Hopf manifold does not admit two distinct holomorphic structure under T-duality.

The results come from a joint work with Leonardo Cavenaghi and Ludmil Katzarkov.  Video

Leonardo Cavenaghi, UNICAMP: A generalized logarithmic transformation and its manifestations through exotic Spheres and Homotopy Hopf Manifolds

In this talk we explain  the concept of $\star$-diagrams, presenting their geometric and topologic aspects. We show how logarithmic transformations can be thought of as a straightened angle procedure, and generalize it to higher dimensions. We then show how some spherical T-duality can be realized by this new kind of logarithmic transformation, applied in the context of spherical duality.

The results come from a joint work with Lino Grama and Ludmil Katzarkov.  Video

Gueo Grantcharov, Florida International University: Isotropic Killing vector fields and related structures on complex surfaces

In a 4-dimensional vector space with metric of signature (2,2), two isotropic vectors spanning an isotropic plane determine a canonical action of the split quaternions. We notice that on an oriented manifold with two isotropic Killing vector fields spanning an isotropic plane everywhere, the induced almost para-hypercomplex structure is integrable. The underlying complex surface admits many geometric structures including indefinite analog of bihermitian structures and twistor space construction.  Based on the classification of compact complex surfaces this allows to describe the topology of the compact 4-manifolds with such fields and structures.

In the talk I'll discuss the relation of the result with other geometric properties of split-siganture 4-manifolds as well as present examples of para-hyperhermitian structures admitting 2 null Killing vector fields on most of these manifolds. (Joint work with J. Davidov and O. Mushkarov)  Video

Jose Medel Torrero, Florida International University  Video