Abstract: Manin's Conjecture predicts that the behavior of rational curves on a Fano variety is controlled by certain geometric invariants. We verify this conjecture for "most" Fano threefolds X of Picard rank 1 by computing the dimension and the number of components of the moduli space of rational curves. In this talk I will focus on the index 1 case and the role played by Manin's Conjecture in formulating the argument. This is joint work with Sho Tanimoto.

Abstract: In the beginning of the last century, Emmy Noether wondered about the rationality of the field extension k(V)^G/k for any finite group G and any field k, where k(V)^G are the G-invariant rational functions over the regular representation V of G.

In 1969 Swan provided the first counterexample to such rationality for certain examples of cyclic groups and k being the field of rational numbers. In this talk I would like to relate this problem to the class of the classifying stack of a (finite) group, BG, in the Grothendieck ring of algebraic stack and summerize some results of Ekedahl and Totaro.

In 2016, I have shown that that the motivic class of BG is trivial if G is a finite subgroup of GL_3(k). Finally, I will present some recent work (in collaboration with R. Singh) on the underlying combinatorial structure behind these motivic problems.

Abstract: Segre classes are a fundamental ingredient in Fulton-Macpherson intersection theory. We will present a new approach to the computation of the Segre class of a subscheme of complex projective space, based on the construction of a suitable Newton-Okounkov body. The result may be viewed as a common generalization of results of Kaveh and Khovanskii and of an earlier result on Segre classes of monomial schemes.

Abstract: A question going back to Serre asks which groups arise as fundamental groups of smooth complex projective varieties, or more generally, compact Kaehler manifolds. These groups are called Kaehler groups. We first give a brief overview of this subject and some known results, then discuss Kaehler groups arising from surface bundles.

Abstract: Since the beginning of tropical geometry, a persistent challenge has been to emulate tropical versions of classical results in algebraic geometry. The well-know statement "any smooth surface of degree three in P^3 contains exactly 27 lines" is known to be false tropically. Work of Vigeland from 2007 provides examples of tropical cubic surfaces with infinitely many lines and gives a classification of tropical lines on general smooth tropical surfaces in TP^3.

In this talk I will explain how to correct this pathology by viewing the surface as a del Pezzo cubic and considering its embedding in P^4 via its anticanonical bundle. The combinatorics of the root system of type E6 and a tropical notion of convexity will play a central role in the construction. This is joint work in progress with Anand Deopurkar.

Abstract: In this talk I will discuss a family of classical field theories in the Batalin-Vilkovisky formalism. These theories take as input n-symplectic Lie algebroids which correspond to symplectic manifolds, Poisson manifolds, and (higher) Courant algebroids. I will also discuss possible boundary conditions for these theories, e.g., Dirac structures and multiplectic structures. Finally, I will discuss aspects of the quantum theory in low dimensions.

Abstract: By the Bogomolov-Mumford theorem, every projective K3 surface contains a rational curve. I will discuss some generalizations of this theorem to higher dimensions concerning rational curves and algebraically coisotropic subvarieties in HyperKähler varieties. I will further discuss some related open questions, examples, and the crucial role played by Gromov-Witten theory in the study of these topics. Based on joint work with Georg Oberdieck and Qizheng Yin.

Abstract: Venkatesh introduced a derived enrichment of the Hecke algebra of a p-adic group, which is a graded algebra that he shows has a graded action on the cohomology of the associated symmetric space. Much of the interesting higher derived structure comes from modular representation theory: k-valued representations of a finite group don't form a semisimple category when the order of the group is zero in k. In this expository talk, we'll introduce Venkatesh's construction, and discuss another situation where this non-semi-simplicity of the Hecke algebra plays a role in number theory. We'll spend the rest of the lecture discussing examples of (derived) Hecke algebras for the finite group GL_n(F_p).

Abstract: Kappa classes were introduced by Mumford as a tool to explore the intersection theory of the moduli space of curves. There is a close connection between the intersection theory of kappa classes on the moduli space of unpointed curves and the intersection theory of psi classes on all moduli spaces: we show that the potential for kappa classes is related to the Gromov-Witten potential of a point via a change of variables given by complete symmetric polynomials, rediscovering a theorem of Manin and Zokgraf from '99. In contrast to their methods, the starting point of our story is a combinatorial formula that relates intersections of kappa classes and psi classes via a graph theoretic algorithm. Further, this story is part of a large wall-crossing picture for the intersection theory of Hassett spaces, a family of birational models of the moduli space of curves. This is joint work with Renzo Cavalieri.

Abstract: Multiplicative Higgs bundles are an analogue of ordinary Higgs bundles where the Higgs field takes values in a Lie group instead of its Lie algebra. In this talk I'll discuss two contexts where multiplicative Higgs bundles appear in supersymmetric gauge theory. I'll explain how hyperkähler structures on these moduli spaces arise physically and mathematically and relate to the theory of Poisson Lie groups, and finally I'll introduce a speculative multiplicative analogue of the geometric Langlands conjecture. This is based on joint work in progress with Vasily Pestun.

Abstract: Let X be a projective variety with mild singularities and L an ample line bundle on X. In 1988 Fujita conjectured that for k≥dim⁡(X)+1, L^k⊗ω_X is base point free, and for k≥dim⁡(X)+2, L^k⊗ω_X is very ample. Fujita's conjecture is known to be true for small dimensions and for certain types of varieties, for example toric varieties. In this talk we'll focus on projectivized bundles P(E), where E is an ample toric vector bundle on a smooth toric variety, and give evidence for Fujita's conjecture to hold for P(E) and L=O_P(E)(1). This will entail looking at the parliament of polytopes for the toric vector bundle E and its symmetric powers, Sym^mE.

Abstract: The Grothendieck ring of algebraic stacks was introduced by Ekedahl in 2009. It may be viewed as a localization of the more common Grothendieck ring of varieties. If G is a finite group, then the class BG of its classifying stack BG is equal to 1 in many cases, but there are examples for which BG≠1. When G is connected, BG has been computed in many cases in a long series of papers, and it always turned out that BG∗G=1. We exhibit the first example of a connected group G for which BG∗G≠1. As a consequence, we produce an infinite family of non-constant finite étale group schemes A such that BA≠1.