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Abstract: Quiver Donaldson-Thomas invariants are integers determined by the geometry of moduli spaces of quiver representations. They play an important role in the description of BPS states of supersymmetric quantum field theories. I will describe a correspondence between quiver Donaldson-Thomas invariants and Gromov-Witten counts of rational curves in toric and cluster varieties. This is joint work with Hulya Arguz (arXiv:2302.02068 and arXiv:arXiv:2308.07270).

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Abstract:  I will review some of my recent work on moduli spaces of Higgs bundles, of connections and of t-connections (which subsums both). These moduli spaces are objects of interest in the Non Abelian Hodge Theory of projective manifolds over the complex numbers and they have been intensively studied in the last thirty five years. The situation over fields of positive characteristic is less explored and has also become the focus of what people call Non Abelian Hodge Theory in characteristic p. The moduli spaces of Higgs bundles and of connections are homeomorphic over the complex numbers. The situation over fields of positive characteristic is less clear. I will focus on explaining how a suitable compactification of these moduli spaces allows to bypass the lack of a homeomorphism to yield a canonical isomorphism of cohomology rings. Along the way, I will discuss some of the new phenomena, absent over the complex numbers, that emerge in positive characteristic. I will be short on technical details and my plan is to make the talk accessible to non-experts. For example, in illustrating the compactification technique, I will use as a guide for the intuition the Ehresmann Lemma from differential topology. Joint work with Siqing Zhang, with Andres Fernandez-Herrero, and with Davesh Maulik, Junliang Shen and Siqing Zhang.

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Abstract: The characteristic 0 Non Abelian Hodge Theory (0NAHT) entails that the moduli space of semistable Higgs bundles of degree zero on a curve is diffeomorphic to the moduli space of semistable flat connections. 
The characteristic p Non Abelian Hodge Theory (pNAHT) entails that the moduli stack of Higgs bundles is a twisted form of the moduli stack of flat connections. 
The technical cores of 0NAHT and pNAHT are very different (harmonic bundles vs Frobenius lifts), and the comparison between the two theories is intriguing. For example, 0NAHT respects semistability, does pNAHT also respect semistability?
In joint work with Mark de Cataldo and Michael Groechenig (vector bundle case), and joint work in progress with Andres Fernandez Herrero (principal bundle case), we answer this question affirmatively, and establish a version of pNAHT on the level of moduli spaces of semistable objects. Moreover, we show that the two moduli spaces have isomorphic intersection cohomologies.
No previous knowledge of NAHT is need for this talk. I hope to make this talk interesting also for number theorists and Lie theorists (our proof relies on detailed studies of Ngo’s presentation of the Hitchin system, and of Ogus-Vologodsky’s pNAHT isomorphisms).

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Abstract: Classical finiteness results of Arakelov and Parshin state that a fixed quasi-projective curve can only carry finitely many non-isotrivial families of smooth projective curves of fixed genus g. These results have been generalized by Faltings and Deligne for polarized variations of Hodge structure of arbitrary weight. In this talk, I will explain a further generalization which only depends on the topology of the base and not the algebraic structure, giving thus a partial answer to a question asked by Deligne. I will then explain an application proving the algebraicity of the non-abelian Hodge locus,  partially solving a conjecture of Simpson. The results in this talk are joint work with Philip Engel.

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Abstract: The SL(r)-invariants of a smooth projective complex surface is defined by means of the moduli spaces of rank r torsion free sheaves with fixed determinant. The PGL(r)-invariants are defined similarly using the moduli of twisted sheaves. In a joint work with Dirk Van Bree, Yunfeng Jiang and Martijn Kool we show that under some mild conditions, these two types of invariants coincide. There are some consequences of this including some applications in Vafa-Witten theory as well as in the existence of Azumaya algebras.

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Abstract: A phenomenon in topology is said to be stable if it occurs in all sufficiently high dimensions. As discovered by Quillen over five decades ago, such phenomena are closely related to number theory, and can often be described in terms of arithmetic objects known as formal groups. Unfortunately, in general this dictionary is not quite one-to-one, and many periodicities one sees on the arithmetic side become broken and more complex in the world of topology. In this talk, I will describe a solution to an old conjecture of Franke that the arithmetic - topology correspondence can be refined to an equivalence of categories when the ambient prime is sufficiently large.

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Abstract: A theorem of Donaldson and Sun asserts that the metric tangent cone of a smoothable Kähler–Einstein Fano variety underlies some algebraic structure, and they conjecture that the metric tangent cone only depends on the algebraic structure of the singularity. Later Li and Xu extend this speculation and conjecture that every klt singularity has a canonical “stable” degeneration induced by the valuation that minimizes the normalized volume. I’ll talk about some joint work with Chenyang Xu on the solution of these conjectures. If time permits, I will also discuss some further implications on the boundedness of singularities.

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Abstract: The moduli space of pointed rational curves exhibits a natural action of the symmetric group, permuting the marked points. This action induces a representation of the symmetric group on the cohomology of the moduli space. An intriguing question, inspired by the Manin-Orlov conjecture, is whether this representation admits an invariant basis under the action.We provide a combinatorial formula for the character of the representation, which reduces its computation to a purely combinatorial problem, along with introducing a new algorithm. Consequently, we partially affirm the question regarding the existence of an invariant basis, refining all previously known results.Our approach employs two types of birational models: (1) moduli of delta-stable quasimaps, which are certain triples of nodal curves, line bundles and sections, and (2) Hassett’s moduli of weighted stable curves.This is a joint work with Jinwon Choi and Young-Hoon Kiem.

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Abstract: Explicit descriptions of low degree K3 surfaces lead to natural compactifications coming from geometric invariant theory (GIT) and Hodge theory. The relationship between these two compactifications was studied by Shah and Looijenga, and revisited in the work of Laza and O’Grady. This latter work also provided a conjectural description for the case of degree four K3 surfaces. I will survey these results, and discuss a verification of this conjectural picture using tools from K-moduli. This is based on joint work with Kristin DeVleming and Yuchen Liu.

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Abstract: A metric on a category assigns lengths to morphisms, with the triangle inequality holding. This notion goes back to a 1974 article by Lawvere. We'll start with a quick review of some basic constructions, like forming the Cauchy completion of a category with respect to a metric.
And then will begin a string of surprising new results. It turns out that, in a triangulated category with a metric, there is a reasonable notion of Fourier series, and an approximable triangulated category can be thought of as a category where many objects are the limits of their Fourier expansions. And some other ideas, mimicking constructions in real analysis, turn out to also be powerful.

And then come two types of theorems: (1) theorems providing examples, meaning showing that some category you might naturally want to look at is approximable, and (2) general structure theorems about approximable triangulated categories.

And what makes it all interesting is (3) applications. These turn out to include the proof of an old conjecture of Bondal and Van den Bergh about strong generation, a representability theorem that leads to a short, sweet proof of Serre's GAGA theorem, a proof of a conjecture by Antieau, Gepner and Heller about the non-existence of bounded t-structures on the category of perfect complexes over a singular scheme, as well as (most recently) a vast generalization and major improvement on an old theorem of Rickard's.

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Abstract: An analogue of blow-ups are weighted blow-ups. Those are transformations, in nature similar to a blow-up, but which are a bit more flexible. For example, weighted blow-ups give better algorithms for resolving singularities of algebraic varieties, and often appear in moduli spaces of interest. The price that one has to pay for the extra flexibility is that the result of a weighted blow-up might no longer be a variety, but rather an algebraic stack. Therefore one natural question is: when is an algebraic stack a weighted blow-up of a simpler space? My coauthors and I give some criteria for when this question has a positive answer. This is a joint work with Arena, Di Lorenzo, Mathur, Obinna and Pernice.

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Abstract: Given a quasi projective family S of complex algebraic varieties, its Hodge locus is the locus of points of S where the corresponding variety admits exceptional Hodge classes (conjecturally: exceptional algebraic cycles). In this talk I will survey the many recent advances in our understanding of such loci, as well as the remaining open questions.

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Abstract: I will report on joint work with Hannah Larson, and joint work in progress with Jim Bryan, in which we try to make sense of Bott periodicity from a naively algebro-geometric point of view.

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Abstract: In the 80s, Beilinson and Lichtenbaum conjectured a general theory of motivic cohomology satisfying a list of expected properties. For example, one would expect an extension of a Grothendieck-Riemann-Roch theorem for schemes which are not necessarily smooth over a field and one would expect a relationship with syntomic cohomology in arithmetic situations. For equicharacteristic schemes, such a theory was offered last year in joint work with Matthew Morrow. I will explain aspects of this theory, emphasizing on when algebraic cycles can appear.

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Abstract: For any cubic surface over a field k, the number of its lines that are defined over k must be one of 0, 1, 2, 3, 5, 7, 9, 15, or 27. But which of these possibilities is actually realized by some cubic surface? I will talk about joint work-in-progress with Enis Kaya, Sam Streeter, and Happy Uppal on this question. The answer depends on the arithmetic of k and requires a thorough understanding of minimal del Pezzo surfaces and their coincidences under blowups.

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Abstract: Donaldson and Uhlenbeck-Yau established the classical result that on a compact Kahler manifold, an irreducible holomorphic vector bundle admits a Hermitian metric solving the Hermitian-Yang-Mills equation if and only if the vector bundle is Mumford-Takemoto stable. Motivated by the characterization of supersymmetric B-branes in string theory and mirror symmetry, Collins-Yau asked if a line bundle admits a solution of the deformed Hermitian-Yang-Mills (dHYM) equation is equivalent to it is stable for certain Bridgeland stability conditions. In this talk, we will discuss a partial answer to this question for a set of line bundles on a Weierstrass elliptic K3 surface. This is joint work with Tristan Collins, Jason Lo, and Shing-Tung Yau.

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Abstract: The classical question of determining which varieties are rational has led to a huge amount of interest and activity. On the other hand, one can consider a complementary perspective - given a smooth projective variety whose nonrationality is known, how "irrational" is it? I will survey recent developments, with an emphasis on surfaces and open problems.