Title: Crepant resolutions via stacks

Abstract: Consider an invariant that behaves nicely for smooth varieties, such as Euler number, Betti numbers, or Hodge numbers. Suppose we want a version of this invariant for singular varieties that sees interesting information about the singularities. I will discuss how this naturally leads to the notion of crepant resolutions of singularities. However, crepant resolutions by smooth varieties are rare in practice. I will discuss joint work with M. Satriano in which we show that crepant resolutions actually exist in broad generality, as long as one is willing to consider algebraic stacks. Specifically, any variety with log-terminal singularities admits a crepant resolution by a smooth algebraic stack. This talk will not assume familiarity with stacks.

Title: Rational torsion and reducibility for abelian varieties associated to newforms

Abstract: Let f be a newform and A its associated abelian variety. We have shown before that for certain primes r, if A is an optimal semistable elliptic curve with reducible torsion subgroup A[r], then A has rational r-torsion. In this talk we define necessary terms and how this result generalizes to other abelian varieties. We will sketch the proof and discuss the differences that arise when A is not necessarily an elliptic curve.

Title: 3-dimensional hypersurfaces and intermediate Jacobians.

Abstract: Clemens-Griffiths discovered in 1971 a surprising connection between two questions (all terms will be defined in the talk) 1. When is a 3-dimensional hypersurface rational (i.e. birational to P^3)? 2. When is a flat torus isometric to the Jacobian of an algebraic curve? I will explain this connection and how I used it recently to give a simple proof of the (already known) irrationality of the general quartic 3-fold. More importantly, we will see some fundamental classical ideas in action: resolution of singularities, the theory of periods, automorphism groups of varieties, and a beautiful example of Felix Klein. This talk will be aimed at beginners (like the speaker) in algebraic geometry.

Title: Relations between Frobenius eigenvalues of abelian varieties over finite fields

Abstract: Elliptic curves over a finite field F_q come in two flavours: ordinary and supersingular. As q varies over powers of a fixed prime p, the eigenvalues of Frobenius of an ordinary elliptic curve are uniformly distributed on a circle, while those of a supersingular elliptic curve are supported are finitely many places. For higher dimensional abelian varieties, this dichotomy branches out in a more interesting way. Further, the multiplicative relations between the Frobenius eigenvalues capture facets of the geometry of the underlying abelian variety. In joint work with Santiago Arango-Piñeros and Deewang Bhamidipati, we study this phenomenon and provide a classification of the possible scenarios in low dimension. In this talk, I will discuss some of our results and some open questions in this area.

Title: Principal minors, stable polynomials and tropical geometry

Abstract: In recent years, the fruitful approach of representing discrete phenomena using real multivariate polynomials, and studying them through the interplay of their coefficients, zeros, and exponents, has led to several breakthroughs. In this talk, I will demonstrate how this approach has been particularly effective in studying the principal minor map through determinantal representations. It has also helped in characterizing determinantal polynomials within the class of real stable polynomials, a multivariate generalization of real-rooted polynomials, thereby addressing a question posed by Borcea, Br\"and\'en, and Liggett in 2009. Tropicalization, on the other hand, is a powerful process that transforms algebraic objects into combinatorial ones, revealing underlying properties that are often difficult to detect otherwise. I will also show how the tropicalization of stable polynomials uncovers hidden matroidal and tropical Grassmannian structures within the principal minors of positive definite matrices. I will not assume familiarity with tropical geometry or stable polynomials. This is based on joint works with Felipe Rinc\'on, Cynthia Vinzant, and Josephine Yu.

Title: Uniform bounds on Sylvester-Gallai type configurations of polynomials

Abstract: The classical Sylvester-Gallai theorem says that if a finite set of points in the Euclidean plane has the property that the line joining any two points contains a third point from the set, then all the points must be collinear. More generally, a Sylvester-Gallai type configuration is a finite set of geometric objects with certain "local" dependencies. A remarkable phenomenon is that the local constraints give rise to global dimension bounds for linear SG-type configurations, and such results have found far reaching applications to complexity theory and coding theory. In this talk we will discuss non-linear generalizations of SG-type configurations which consist of polynomials. We will discuss how the commutative-algebraic principle of Stillman uniformity can shed light on low dimensionality of SG-configurations. I’ll talk about recent progress showing that these non-linear SG-type configurations are indeed low-dimensional as conjectured by Gupta. This is based on joint work with R. Oliveira.

Title: On the topology of the moduli space of tropical Z/pZ-covers

Abstract: We study the topology of the moduli space of (unramified) Z/pZ-covers of tropical curves of genus g≥2 where p is a prime number. By recent work of Chan-Galatius-Payne, the (reduced) homology of this tropical moduli space computes (with a degree-shift) the top-weight (rational) cohomology of the corresponding algebraic moduli space. We prove contractibility of certain subcomplexes of the tropical moduli space and use this result to show that it is simply connected and to fully determine its homotopy type for g=2 and all p.

Title: Spaces of flat surfaces

Abstract: A non-zero holomorphic one form on a Riemann surface endows it with a flat metric with conical singularities. Such an object is called flat surface. While spaces of flat surfaces with fixed types of singularities are well studied from different point of views, such as algebraic geometry, dynamics, and enumerative geometry, the global geometry of these so-called "strata of differentials" remains quite mysterious. In this talk we describe new results about the computation of topological and algebraic invariants of these spaces.

Title: Algebraically integrable foliations and canonical bundle formula

Abstract: Situations in which the canonical divisor of a smooth projective variety is the pullback of a divisor on a lower dimensional variety, are quite commonplace in higher dimensional algebraic geometry. In this set up, one can define a 'moduli' divisor on the base and results around its positivity are loosely referred to as canonical bundle formulas (CBF). CBF's are fundamental tools in modern birational geometry. A famous conjecture of Prokhorov and Shokurov predicts that on a sufficiently high birational model of the base, the moduli divisor becomes semiample. In this talk, I'll discuss how the birational geometry of algebraically integrable foliations gives a new way of looking at this conjecture and prove it in some special cases. This features work from a joint paper with Omprokash Das.

Title: Monodromy and vanishing cycles for curves in an algebraic surface

Abstract: This talk will be about the monodromy group associated to a family of algebraic curves in an algebraic surface as a subgroup of the mapping class group. I will start by surveying some older results in this area about the image of monodromy in the symplectic group. I will then discuss joint work with Nick Salter, where we describe the precise image of monodromy in the mapping class group in the special case of complete intersections.

Title: Homology of spaces of curves on blowups

Abstract: Let C be a smooth projective curve and X be a smooth projective variety. We will consider the space of degree d algebraic maps from C to X. When X is a projective space, Segal discovered an interesting phenomenon: as the degree increases, the homology of the space of algebraic maps approximates that of the space of continuous maps. Recently, Ellenberg-Venkatesh observed that this phenomenon is related to Manin's conjectures about rational points on Fano varieties, suggesting it holds more generally. I will talk about joint work with Ronno Das considering the case where X is a blowup of a projective space at finitely many points (in particular the case of del Pezzo surfaces).

Title: Supersolvable posets and fiber-type arrangements

Abstract: We present a combinatorial analysis of fiber bundles of generalized configuration spaces on connected abelian Lie groups. These bundles are akin to those of Fadell--Neuwirth for configuration spaces, and their existence is detected by a combinatorial property of an associated finite partially ordered set. We obtain a combinatorially determined class of K(pi,1) spaces, and under a stronger combinatorial condition prove a factorization of the Poincar\'e polynomial when the Lie group is noncompact. In the case of toric arrangements, this provides an analogue of Falk--Randell's formula relating the Poincar\'e polynomial to the lower central series of the fundamental group. This is joint work with Emanuele Delucchi.

Title: Exotic tori and actions by SL_d(Z)

Abstract: One of the most distinctive features of the d-dimensional torus (i.e., the product of d circles) T^d is that it admits an effective smooth action of the group SL_d(Z) of d x d integer matrices with determinant 1. A natural question is whether such an action (or any non-trivial action) can also exist on exotic tori, which are smooth manifolds that are homeomorphic but not diffeomorphic to T^d. In my talk, I will first review some general concepts about exotic manifolds in dimensions greater than 4 and then discuss this question. This is joint work with M. Krannich, A. Kupers, and B. Tshishiku.

Title: Euler characteristic-like invariants and positivity patterns in geometric contexts predicted by combinatorics

Abstract: We will discuss intrinsic structures and positivity properties of certain Euler characteristic-like invariants which appear in a wide range of contexts in algebraic combinatorics. Note that the positivity properties lie somewhere between unimodality and real-rootedness. While these patterns/invariants were first studied in more purely combinatorial examples, they also predicted behavior of more "geometric" objects such as Chow rings of matroids. We will explore ideas from geometric combinatorics that tie these different threads together and suggest new perspectives on existing examples.

Title: Glueing invariants of Donaldson--Thomas type

Abstract: Donaldson--Thomas invariants are numerical invariants associated to Calabi--Yau varieties. They can be obtained by glueing singularity invariants from local models of a suitable moduli space. By studying the moduli of such local models, we will explain how to recover Brav--Bussi--Dupont--Joyce--Szendroi's perverse sheaf categorifying the DT-invariants, as well as how to glue more evolved singularity invariants, such as matrix factorizations (thus answering a conjecture of Kontsevich and Soibelman). This is joint work with M. Robalo and J. Holstein.

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Title: The zeros of the Riemann zeta function and its generalization to modular forms

Abstract: According to the Riemann hypothesis, the Riemann zeta function should have zeros only at negative even integers and complex numbers with real part ⁠1/2⁠. We will study the completed Riemann zeta function (the one with the Gamma factor) and discuss how it sheds some light on the location of the zeros. There is a generalization of the Riemann hypothesis to L-functions of modular forms, and we will discuss what can be said in this context.

Title: Even periodization

Abstract: In this talk, I will present work in progress on even periodization. This is an operation on spectral stacks, which roughly approximates them as closely as possible with affines corresponding to even periodic ring spectra. This turns out to have close connections to the even filtration of Hahn-Raksit-Wilson, the prismatization stacks of Bhatt-Lurie, as well as the chromatic affineness results for topological modular forms of Mathew-Meier.

Title: On the Mac Lane Q-Construction for Exact Infinity Categories

Abstract: One of the useful tools to compute the K-theory and stable homology of abelian groups, and more generally exact categories is "Mac Lane's Q-Constrcution". We extend this construction to exact infinity categories. This is achieved by constructing, for any functor from exact infinity categories to a fixed stable infinity category A, a coherent chain complex in A that is an immediate generalization of Mac Lane's cubical Q-complex.

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