Twisted Milnor torsion for finite group actions
The Milnor torsion is an invariant of unitary flat vector bundles on closed odd-dimensional manifolds. Its analytic counterpart was introduced by Ray and Singer and the equality of these torsions is the celebrated Cheeger-Müller theorem. In this talk, I will discuss how to extend the Milnor torsion to certain equivariant flat superconnections (or representations up to homotopy) for finite group actions and its relation to analytic torsion of the twisted de Rham complex.
Dioperads, Frobenius monoidal functors and integration along fibers
Dioperads encode algebraic structures with several input and output, generalizing operads. In the same way lax monoidal functors are exactly those preserving algebras over operads, I will explain that Frobenius monoidal functors are exactly those preserving algebras over dioperads.
In a second part, I shall describe how to construct (shifted) Frobenius monoidal structures given a certain orientation data, analogous to the procedure of integration along fibers induced by Poincaré duality. This construction arose from a question in derived Poisson geometry, which requires an ∞-categorical generalizing of the previous result. Time depending, I will discuss this motivation and the homotopical difficulties that stand in the way.
This is joint work with Valerio Melani.
Stable homotopy in low dimensional topology
I will give a survey of stable homotopy refinements of Floer homology theories for 3-manifolds and Khovanov homology of knots and links. The emphasis will be on structural features of the theories that become more transparent from the stable homotopy perspective. I will also point out some open problems related to the functoriality with respect to 4-dimensional cobordisms in Seiberg-Witten theory.
Moduli of sheaves on hyper-Kähler manifolds
On the homology classes defined by closed geodesics on the modular curve
Closed geodesics on the modular curve define certain homology classes of SL(2,Z). These homology classes are very interesting objects that are related to the arithmetic of real quadratic fields, half-integral weight modular forms, etc. . For example, it is known that the pairing between such homology classes and the Eisenstein class (a cohomology class of SL(2,Z) defined by Eisenstein series) gives special values of zeta functions of real quadratic fields, leading to many applications.
In this talk, I will discuss the size of the subgroup (in the homology of SL(2,Z)) generated by such homology classes defined by closed geodesics and its consequences. This talk is based on joint work in progress with Ryotaro Sakamoto.
WIQI topology seminar
Seminar webpage: https://guests.mpim-bonn.mpg.de/bianchi/wiqi.html
WIQI topology seminar
Seminar webpage: https://guests.mpim-bonn.mpg.de/bianchi/wiqi.html
Geometric Quantization associated to mixed toric polarizations
A crucial problem in geometric quantization is to understand the relationship among quantum spaces associated to different polarizations. Two types of polarizations on toric varieties, Kähler and real, have been studied extensively. In this talk, I will introduce the quantum spaces associated to mixed toric polarizations and explore their relationships with those associated with Kähler polarizations.
tba
Sub-Weyl bound for $GL(2)$ $L$-functions
We begin by briefly introducing the subconvexity problem for $L$-functions and the delta method, which has proven to be a powerful line of attack in this context. As an application, for a $SL(2,\mathbb{Z})$ form $f$, we obtain the sub-Weyl bound:
$$L(1/2+it,f)\ll_{f,\varepsilon} t^{1/3-\delta+\varepsilon}$$ for some explicit $\delta>0$, thereby crossing the Weyl barrier for the first time beyond $GL(1)$. The proof uses a refinement of the 'trivial' delta method.
Gamma class, total positivity and mirror symmetry
Mirror symmetry predicts that for any Fano manifold $X$ there should be a Landau-Ginzburg model $(X^{\vee},W)$ such that the quantum $D$-module of $X$ is isomorphic to the Gauss-Manin system of $(X^{\vee},W)$. In addition, the natural lattice structures on the spaces of flat sections of these $D$-modules, one coming from the image of the Chern character of $X$ and one from certain integral relative homology of $X^{\vee}$, should match, after the former is twisted by the Gamma class. These predictions have been verified for toric Fano manifolds.
In this talk, I will discuss the case when $X$ is a flag variety of arbitrary Lie group type, where $(X^{\vee},W)$ is known to be the Rietsch mirror. I will focus on $1=ch([\mathcal{O}_X])$ and explain the result that this element corresponds to the totally positive part of $X^{\vee}$ in the sense of Lusztig. If time permits, I will explain how to apply this result to prove Gamma conjecture I for these varieties.
Congruences and the Galois representations of classical cusp forms
Modular forms are central to modern number theory for many reasons, one of which being that they are a rich source of 2-dimensional Galois representations. But what information about the modular form is contained in the Galois representation? And how does one extract this information, for example about the Fourier coefficients of f?
In this expository talk, we will explore under what conditions the p-th Fourier coefficient of a classical normalised eigen cusp form f vanishing modulo some fixed prime ell is a congruence condition on the prime p, illustrating how to analyse and work with these kinds of Galois representations. This talk will focus on examples and will assume little background knowledge, so students and researchers from various mathematical disciplines are warmly invited to attend.
Jensen polynomials and inequalities related to partition statistics.
In this talk, I will discuss current developments on hyperbolicity of Jensen polynomials that began with the seminal work of Griffin, Ono, Rolen, and Zagier. Furthermore, I will present family of inequalities for certain partition statistics. This is an ongoing joint work with Kathrin Bringmann and Larry Rolen.
WIQI topology seminar
Seminar webpage: https://guests.mpim-bonn.mpg.de/bianchi/wiqi.html
WIQI topology seminar
Seminar webpage: https://guests.mpim-bonn.mpg.de/bianchi/wiqi.html
Rank growths of elliptic curves over some Galois extensions and Markov operators
I would like to give an overview of the insights from the work by Swinnerton-Dyer and Klagsbrun--Mazur--Rubin, who established relations between Markov operators over countable state spaces and changes in Mordell-Weil rank of elliptic curves with respect to base change to cyclic Galois extensions over number fields. If time allows, we will use this insight to generalize the technique to understand changes in Mordell-Weil ranks of elliptic curves with respect to families of S3-cubic extensions over number fields with fixed quadratic resolvents. My hope is to make the talk be accessible and explicit for non-specialists as well. This talk is based on joint works in progress with Daniel Keliher.
tba
tba (Geometric Langlands Seminar)
Zoom link:
https://eu02web.zoom-x.de/j/66594302263?pwd=6XqRNiAADoXfsLNrwCIji5UZgyh2jG.1
Meeting ID: 665 9430 2263
Passcode: 740104
Approximation of perfectoid rings by Noetherian rings and prisms
The theory of perfectoid towers, introduced by Ishiro-Nakazato-Shimomoto, provides an axiomatic approach to perfectoid theory in commutative algebra through tower-theoretic approximations. In contrast, Bhatt and Scholze introduced prisms as a "deperfection" of perfectoid rings. Our main result shows that a "gradual perfection" of a prism becomes a perfectoid tower. As a consequence, we prove that any p-torsion-free p-adically complete delta-ring that is reduced modulo p admits a perfectoid tower. This approach allows for more systematic construction of perfectoid rings and towers from Noetherian rings than previously possible. We also provide new examples of perfectoid towers arising from certain singularities, which were inaccessible by previous methods. In this talk, we will review the concepts of perfectoid towers, explain our construction from prisms, and demonstrate its applications through these novel examples.
CANCELLED -- Refined Chabauty–Kim for the thrice-punctured line
If X is a curve of genus at least two defined over the rational numbers, we know by Faltings's Theorem that the set X(Q) of rational points is finite, but how to systematically compute it is still an open problem. In 2005, Minhyong Kim proposed a new framework for studying rational (or S-integral) points on curves, called the Chabauty–Kim method. It aims to produce p-adic analytic functions on X(Q_p) containing the rational points X(Q) in their zero locus. I will give a brief introduction to Chabauty–Kim theory and present some applications to the S-unit equation.
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