The Algebraic Geometry Seminar takes place on Wednesdays 3:30–4:30PM. Pre-talks (which graduate students are particularly encouraged to attend) take place Wednesdays 3:00–3:25PM.
The in-person talks will take place in BRNG 1230. Some talks will be presented online via Zoom.
Please see below for the most up to date information. Here is the Zoom link for the online talks:
https://purdue-edu.zoom.us/j/99467898661?pwd=FOMrspNCAywfyh5VFOKDS80aWZLZuX.1
April 29: James Hotchkiss (Columbia University)
Location: BRNG 1230
Title: The period-index problem
Abstract: The period-index problem is a classical question about central simple algebras over a field. I will give an introduction to the problem, and discuss recent developments in the case of function fields of complex algebraic varieties, based on a mix of techniques from Hodge theory, derived categories, and enumerative geometry.
January 14: Alice Lin (Harvard)
Location: STON 215
Title: Finiteness of heights in isogeny classes of motives
Abstract: Using integral p-adic Hodge theory, Kato and Koshikawa define a generalization of the Faltings height of an abelian variety to motives defined over a number field. Assuming the adelic Mumford-Tate conjecture, we prove a finiteness property for heights in the isogeny class of a motive, where the isogenous motives are not required to be defined over the same number field. This expands on a result of Kisin and Mocz for the Faltings height in isogeny classes of abelian varieties.
January 21: Yujie Luo (National University of Singapore)
Location: Zoom (https://purdue-edu.zoom.us/j/99467898661?pwd=FOMrspNCAywfyh5VFOKDS80aWZLZuX.1)
Title: Essential dimensions of polarized endomorphisms of abelian varieties
Abstract: We explore the essential dimension of polarized endomorphisms on abelian varieties and its relation with algebraic dynamics. This talk is based on joint work with Keiji Oguiso and De-Qi Zhang.
February 4: Alexander Polishchuk (University of Oregon)
Location: Zoom (https://purdue-edu.zoom.us/j/99467898661?pwd=FOMrspNCAywfyh5VFOKDS80aWZLZuX.1)
Title: Some birational models of M_{g,n} and related algebras
Abstract: I will discuss examples of birational models of M_{g,n} that on the one hand are given as GIT-quotients for torus actions on explicit affine schemes, and on the other hand admit modular descriptions. I’ll focus mostly on cases g=0 and g=1.
February 18: Ying Wang (University of Michigan)
Location: STON 215
Title: Non-Archimedean perspectives on Calabi-Yau metrics
Abstract: The Calabi problem asks for canonical metrics on Calabi-Yau varieties. We will review the history of this problem and some recent progress in the affine case. Then we will explain how to use non-Archimedean geometry to approach this problem, using the notion of essential skeleton of Calabi-Yaus.
February 25: Donu Arapura (Purdue University)
Location: STON 215
Title: Smooth projective varieties with non residually finite fundamental groups.
Abstract: I heard that Domingo Toledo passed away recently. So perhaps it’s fitting to discuss one of his famous results. For reasons that I will explain, it was long hoped that fundamental groups of complex algebraic manifolds should be residually finite, i.e. that they would inject into their profinite completions. In 1993, Toledo constructed a counterexample. I will present a slightly easier counterexample due Catanese, Kollár and Nori. My talk will be entirely expository and accessible to students.
March 4: Deepam Patel (Purdue University)
Location: STON 215
Title: Local Monodromy of constructible sheaves
Abstract: This will be a survey on results around local monodromy. I will begin by recalling the classical local monodromy theorem, and discuss recent generalizations to the setting of arbitrary constructible sheaves, as well as `large’ sheaves. If there is time, I hope to discuss some open problems and future directions. This is partly based on joint work with Madhav Nori.
March 11: Marta Benozzo (Université Paris-Saclay)
Location: BRNG 1230
Title: Anti-Iitaka inequality in positive characteristic
Abstract: A guiding problem in algebraic geometry is the classification of varieties. In dimension 1, the main invariant for their classification is the genus. Similarly, in higher dimension we study positivity properties of the canonical divisor and a first measure of these is its Iitaka dimension.
A long-standing problem is how we can relate Iitaka dimensions in fibrations: the Iitaka conjectures. Recently, Chang proved an inequality for the Iitaka dimensions of the anticanonical divisors in fibrations over fields of characteristic 0. Both Iitaka’s conjecture and Chang’s theorem are known to fail in positive characteristic. However, in a joint work with Brivio and Chang, we prove that anti-Iitaka holds when the “arithmetic properties” of the anticanonical divisor are sufficiently good.
March 25: Karthik Vasisht (Purdue University)
Location: BRNG 1230
Title: A geometric proof of The Local Monodromy theorem over the complex numbers
Abstract: Given a local system of complex vector spaces on a topological space X, one obtains a representation of the fundamental group of X, called the monodromy representation. Now suppose X is a proper variety with a map to the unit disk that is smooth away from the origin. In this setting, the higher direct images of the constant sheaf on X form a local system on the punctured disk. The Local Monodromy Theorem states that the associated monodromy representation is quasi-unipotent, i.e., its eigenvalues are roots of unity. In this talk, I will describe Grothendieck’s geometric proof of the Local Monodromy Theorem in the complex analytic setting.
April 8: Mingjia Zhang (IAS)
Location: BRNG 1230
Title: p-adic Simpson correspondence
Abstract: Parallel to the complex Corlette-Simpson correspondence, Ogus—Vologodsky discovered that for a variety X in characteristic p, there is a similar picture involving D-modules on X and Higgs bundles on the Frobenius twist of X. This is due to the fact that the algebra of differential operators on X has a large center, and it is an Azumaya algebra over its center. Interestingly, this phenomenon deforms to varieties over p-adic fields, giving rise to a p-adic version of the Simpson correspondence. This correspondence takes the form that for a smooth proper rigid space X over a perfectoid field, there is an equivalence between the categories of pro-etale vector bundles on X and Higgs bundles. This result is first established by Faltings (for curves) and Heuer (in general). I will explain joint work in progress with Bhargav Bhatt, where we gave another proof of the p-adic Simpson correspondence via the "Simpson gerbe". This perspective makes a natural connection to the characteristic p story and suggests further generalizations.
April 15: Alexander Petrov (MIT)
Location: BRNG 1230
Title: Riemann-Hilbert correspondence for B_dR^+-local systems
Abstract: For a complex manifold, (analytic version of) Riemann-Hilbert correspondence identifies the category of holomorphic vector bundles with a flat connection with the category of local systems of complex vector spaces. In p-adic geometry, the relevant notion of local systems turns out to be the 'B_dR^+-local systems': the structure sheaf on the pro-etale site of any variety over a p-adic field has a natural non-trivial one-parameter deformation given by the period sheaf B_dR^+, and we refer to locally free sheaves over this period sheaf as 'B_dR^+-local systems'. For a smooth proper scheme X over the period ring B_dR^+ (which is non-canonically isomorphic to C_p[[t]]) we construct an equivalence between B_dR^+-local systems on X_{t=0} and vector bundles on with a flat t-connection on X. In particular there is a functor from etale Q_p-local systems on X_{t=0} to vector bundles with a t-connection on X; this generalizes the Riemann-Hilbert functor of Liu-Zhu (based on earlier constructions of Faltings and Scholze) that associates a vector bundle with a t-connection to a Q_p-local systems defined on some descent of X_{t=0} to a finite extension of Q_p. The construction of our functor relies crucially on the recent new perspective on the p-adic Simpson correspondence given by 'Simpson gerbe' introduced by Bhatt-Zhang. This is joint work with Ben Heuer and Vadim Vologodsky.
April 22: Carlos Alberto Agrinsoni (Purdue)
Location: BRNG 1230
Title: Resolution of the Gold Degree case of the Exceptional APN Conjecture and new Progress in the Kasami-Welch Degree case
Abstract: Almost Perfect Nonlinear (APN) functions are central objects in the study of cryptographic Boolean functions, as they provide optimal resistance to differential cryptanalysis. A polynomial \( f(x) \in \mathbb{F}_q \) is called APN if, for every \( a, b \in \mathbb{F}_q \) with \( a \neq 0 \), the equation
\[
f(x+a) - f(x) = b
\]
has at most two solutions. \( f(x) \in \mathbb{F}_q[x] \) is \emph{exceptional APN} if it remains APN over infinitely many extensions of \( \mathbb{F}_q \). A fundamental reduction, due to Janwa and Wilson and later refined by Rodier, shows that the classification of exceptional APN functions can be formulated in terms of the rational points on the algebraic variety
\[
\mathcal{X}_f : \phi_f(X,Y,Z) = 0,
\]
where
\[
\phi_f(x,y,z) = \frac{f(x) + f(y) + f(z) + f(x+y+z)}{(x+y)(y+z)(x+z)}.
\]
This reduces the problem to the study of the associated multivariate polynomial \( \phi_f \), in particular its factorization and absolute irreducibility. In this setting, the Lang--Weil and Ghorpade--Lachaud bounds imply that if \( \phi_f \) has an absolutely irreducible factor over the base field distinct from the trivial factors \( (x+y), (y+z), (x+z) \) then $f(x)$ is not exceptional APN.
A conjecture of Aubry, McGuire, and Rodier asserts that, up to CCZ equivalence, the only exceptional APN functions are the monomials of degrees \( 2^k + 1 \) and \( 2^{2k} - 2^k + 1 \), known respectively as the Gold and Kasami--Welch cases. This conjecture naturally decomposes into three principal cases: the Gold degree case, the Kasami--Welch degree case, and the even degree case. While several subcases have been resolved, the general conjecture remains open.
In this talk, we present our proof of the conjecture in the Gold degree case, thereby establishing this case in full generality. Moreover, we present new progress in the Kasami-Welch degree case.
April 29: James Hotchkiss (Columbia University)
Location: BRNG 1230
Title: The period-index problem
Abstract: The period-index problem is a classical question about central simple algebras over a field. I will give an introduction to the problem, and discuss recent developments in the case of function fields of complex algebraic varieties, based on a mix of techniques from Hodge theory, derived categories, and enumerative geometry.
If you are interested in giving a talk, please contact one of the organizers:
Donu Arapura