Abstract: Let $k$ be an algebraically closed field and $X$ a projective variety over $k$. Given a property $P$ of varieties, such as being regular, rational, or log terminal, we say that Bertini’s Theorem holds for $P$ if a general hyperplane section $H \subseteq X$ also has property $P$. Bertini-type theorems are known to hold for singularities arising in the complex minimal model program, including rational, log canonical, and log terminal singularities. In prime characteristic, analogous results for singularities defined via the Frobenius endomorphism have seen rapid progress, motivated by their close connection to complex singularities through reduction to characteristic $p > 0$. A major open problem has been whether $F$-rational singularities, the characteristic $p$ analogues of rational singularities, satisfy Bertini’ Theorem. In this talk, we discuss recent developments culminating in the resolution of Bertini’s Theorem for both $F$-pure and $F$-rational singularities. This talk is based on join work with Alessandro De Stefani and Austyn Simpson.

Abstract: Given a geometric space X, how many curves of a certain shape lie in X? For instance, Euclid knew that there is a single line in the plane passing through two points. In the 1990s, Kontsevich, using ideas inspired from string theory, generalized Euclid’s count to determine the number of rational plane curves of degree d through 3d-1 points. This was the birth of Gromov-Witten (GW) theory, which offers a robust framework to think about counting curves, and which has seen an explosion of progress in the last three decades. However, the GW machine comes with a serious caveat: it often spits out a count that includes unwanted “excess” contributions that are less geometrically meaningful, and that are hard to control. I will outline a broad program to confront this problem in a particular class of examples, which forms a bridge between classical and modern enumerative geometry. The story is largely complete in the case of curves on projective spaces, but there are many interesting directions which lie beyond.

Abstract: In the sequel of two talks, we will sketch the proof of a theorem of Mustațǎ, which says that one can read off  the log canonical threshold of a pair consisting of a smooth algebraic variety and closed subvariety, using jet schemes. These talks will be expository and largely explain the techniques involved in proving the results, mainly jet schemes, arc spaces and motivic integration, following Mustațǎ's paper "Singularities of pairs via jet schemes".

We discuss some results on genus 1 Gromov-Witten invariants of Hilbert scheme of points on the affine plane, including a determination of multi-point series in terms of one-point series and a close formula for an one-point series.

Abstract: The moduli space M_{SL} of stable SL_r(C)-Higgs bundles and the moduli space M_{PGL} of stable PGL_r(C)-Higgs bundles over a smooth Riemann surface of genus bigger or qual to 2 form a SYZ mirror partner. In recent years there are many progresses on several conjectures of the moduli space of Higgs bundles on curves.   For instance  recently the Hausel-Thaddeus conjecture equating the stringy Hodge numbers of M_{SL} and M_{PGL} was proved by using the p-adic integration and Ngo's support properties of the Hitchin fibration map techniques.  Maulik-Shen proved the P=W conjecture.  In the talk I will introduce the basic notion of moduli space of Higgs bundles, and the open problems related to enumerative geometry such as Gromov-Witten theory, Donaldson-Thomas theory and the Gopakumar-Vafa invariants. 

 Abstract: If f is a polynomial, we can associate an invariant called the Bernstein-Sato polynomial b_f(s), whose roots encode deep information about the singularities defined by f. However, the computation of b_f(s) is difficult and has only been accomplished in special cases. Moreover, if we let f lie in a singular ring, the Bernstein-Sato polynomial may not exist.

In this talk we will discuss the computation and existence for some element f in a numerical semigroup ring. We will also compute cutoff exponents, a characteristic p>0 invariant that will first arise to aid our computation of b_f(s), but are important invariants in their own right.

Abstract: Given a Calabi-Yau 3-fold, it is a classical problem to count algebraic curves with a given curve class. In the late 1990s, Gopakumar and Vafa connected this problem to certain moduli spaces of 1-dimensional sheaves (“D-branes” in physics language). Mathematically, one expects to define curve counting invariants called BPS invariants from a perverse filtration on this moduli space associated to a natural support map. In this talk, we focus on the case of (local) P^2 and study this filtration. We propose a conjecture relating the perverse filtration with a natural filtration defined via tautological classes, which can be viewed as a del Pezzo analogue of the P=W conjecture. As a consequence, we obtain structural results and predictions for the BPS invariants of local P^2. This is joint work with Y. Kononov, W. Lim, M. Moreira, J. Shen, F. Si, and F. Zhang.

Syzygies of algebraic varieties have long been a topic of intense interest among algebraists and geometers alike. After the pioneering work of Mark Green on curves, numerous attempts have been made to extend these results to higher dimensions. It has been proposed that the syzygies of adjoint linear series L=K+mA, with A ample, as the natural analogue for higher dimensions. The very ampleness of adjoint linear series is not known for even threefolds. So the question that has been open for the last 30 years is the following (Question): If A is base point free and ample, does L satisfy property N_p for m>=n+1+p? Ein and Lazarsfeld  [EL] proved this when A is very ample. In a joint work with Justin Lacini [BL], we give a positive answer to the original question above in its full generality.

The References are as follows:

[EL] Syzygies and Koszul cohomology of smooth projective varieties of arbitrary dimension, Invent. Math. 111 (1993), pp. 51-67.

[BL] Syzygies of adjoint linear series of projective varieties (To appear in Duke Math J.)

Abstract: One of the central problems in arithmetic geometry is the Tate conjecture, which predicts a description of algebraic cycles of a given variety using linear algebraic data. In this talk, we will review the conjecture and the Kuga--Satake correspondence. Then we report on the progress of the conjecture for minimal surfaces of geometric genus one, where we introduce a strategy to deal with those surfaces whose natural models are singular, using birational geometry. This is a joint work with Ziquan Yang.

The study of orbits and orbit closures in the flag variety has a long and storied history with deep connections to algebraic combinatorics, Lie theory, and representation theory. The orbits of Borel subgroups and their Zariski closures, the Schubert varieties, have been of particular import. A central notion in this area is the complexity of a reductive group action on a variety, which equals the minimum codimension of a Borel subgroup orbit. In this talk we provide a type-uniform formula for the torus complexity of the usual torus action on a Richardson variety by developing the notion of algebraic dimensions of Bruhat intervals. Then, when a Levi subgroup acts on a Schubert variety, we exhibit a codimension preserving bijection between the Levi-Borel subgroup orbits in the big open cell of that Schubert variety and the torus orbits in the big open cell of a distinguished Schubert subvariety. This allows us to give a type-uniform formula for the Levi-Borel complexity of that Schubert variety.

The monodromy conjecture of Denef—Loeser is a conjecture in singularity theory that predicts that given a complex polynomial f, and any pole s of its motivic zeta function, exp(2πis) is a "monodromy eigenvalue" associated to f. I will formulate a "birational geometric" version of the conjecture, and briefly sketch ongoing work to reduce the conjecture to the case of Newton non-degenerate hypersurfaces. These are hypersurface singularities whose singularities are governed, up to a certain extent, by faces of their Newton polyhedra. The extent to which the former is governed by the latter is a key aspect of the conjecture.

Abstract: In joint work in progress with Maximilian Hauck and Tasos Moulinos, we study the étale realization functor from prismatic F-gauges to Galois representations and analyze the stack for which the latter is the associated category of quasi-coherent sheaves.

Abstract: The two objects in the title are the technical cores of the (Non Abelian) Hodge theories in complex geometry and prime characteristic geometry. The two theories appear to be roughly parallel to each other, but their relationship remains mysterious. In this talk, I will explain various differences and similarities between those two theories, emphasizing the joint work with Mark de Cataldo on the new phenomena when logarithmic poles are added.