Geometry Seminar at the University of Kansas

Spring 2022
02/04/2022 (Friday, 6:00pm)Tasuki Kinjo (University of Tokyo)Cohomological Donaldson--Thomas theory and Higgs bundlesIn this talk, I will introduce the BPS cohomology of the moduli space of Higgs bundles on a smooth projective curve of rank r and degree d. The construction is based on cohomological Doanldson--Thomas theory. The BPS cohomology and the intersection cohomology coincide when r and d are coprime, but they are different in general. I will prove that the BPS cohomology does not depend on d. This is a generalization of the Hausel--Thaddeus conjecture to non-coprime case and a cohomological version of Toda's chi-independence conjecture for Higgs bundles. This talk is based on a joint work with Naoki Koseki and another joint work with Naruki Masuda., Password: 485717
02/18/2022 (Friday, 3:00pm)Jan Manschot (Trinity college Dublin)Topological correlators of N=2* Yang-Mills theoryN=2* Yang-Mills theory is a mass deformation of N=4 Yang-Mills, which preserves N=2 supersymmetry. I will consider the topological twist of this theory with gauge group SU(2) on a smooth, compact four-manifold X. A consistent formulation requires coupling of the theory to a Spin-c structure, which is necessarily non-trivial if X is non-spin. I will discuss the contribution from the Coulomb branch to correlation functions in terms of the low energy effective field theory coupled to a Spin-c structure, and present how these are evaluated using mock modular forms. Upon varying the mass, the correlators can be shown to reproduce correlators of Donaldson-Witten theory as well as Vafa-Witten theory. Based on joint work with Greg Moore, arXiv:2104.06492., Password: 485717
02/25/2022 (Friday, 4:00pm)Ritvik Ramkumar (UC Berkley)Rational singularities of nested Hilbert schemesFor a smooth surface S the Hilbert scheme of points S^(n) is a well studied smooth parameter space. In this talk I will consider a natural generalization, the nested Hilbert scheme of points S^(n,m) which parameterizes pairs of 0-dimensional subschemes X \supseteq Y of S with deg(X) = n and deg(Y) = m. In contrast to the usual Hilbert scheme of points, S^(n,m) is almost always singular and it is known that S(n,1) has rational singularities. I will discuss some general techniques to study S^(n,m) and apply them to show that S^(n,2) also has rational singularities. This relies on a connection between S^(n,2) and a certain variety of matrices, and involves square-free Gröbner degenerations as well as the Kempf-Weyman geometric technique. This is joint work with Alessio Sammartano., Password: 485717
03/04/2022 (Friday, 4:00pm)Izzet Coskun (UIC)Brill-Noether Theorems for moduli spaces of sheaves on surfacesIn this talk, I will describe Brill-Noether type theorems for moduli spaces of sheaves on surfaces. I will give applications to Ulrich bundles, classification of stable vector bundles and classification of globally generated vector bundles. This is joint work with Jack Huizenga., Password: 485717
03/11/2022 (Friday, 4:00pm)Ziquan Zhuang (MIT)Finite generation and Kähler-Ricci soliton degenerations of Fano varietiesBy the Hamilton-Tian conjecture on the limit behavior of Kähler-Ricci flows, every complex Fano manifold degenerates to a Fano variety that has a Kähler-Ricci soliton. In this talk, I'll discuss the algebro-geometric analogue of this statement and explain its connection to certain finite generation results in birational geometry. Based on recent joint work with Harold Blum, Yuchen Liu and Chenyang Xu., Password: 485717
03/25/2022 (Friday, 4:00pm)Yuanqi Wang (University of Kansas)On topology of resolutions for non-isolated orbifold singularities.Expository talk, Password: 485717
04/08/2022 (Friday, 4:00pm)Sien Gong (University of Kansas)Lipschitz convergence of Riemannian manifolds.Expository talk, Password: 485717
04/15/2022 (Friday, 6:00pm)Zhixian Zhu (Capital Normal University in Beijing)$k$-jet ampleness of line bundles on toric varieties$k$-jet ampleness of line bundles generalizes globally generation and very ampleness by existence of enough global sections to separate higher order analogues of tangent vectors. We will give some sharp bounds guaranteeing that a line bundle on a projective toric variety, in terms of its intersection numbers with the invariant curves and its Seshadri constants. This is a joint work with Jose Gonzalez., Password: 485717
04/22/2022 (Friday, 4:00pm)Jeongseok Oh (Imperial College)Virtual cycles on projective completionsFor a compact quasi-smooth derived scheme M with (-1)-shifted cotangent bundle N, there are at least two ways to localise the virtual cycle of N to M via torus and cosection localisations, introduced by Jiang-Thomas. We produce virtual cycles on both the projective completion and projectivisation of N and show the ones on the former push down to Jiang-Thomas cycles and the one on the latter computes the difference. Using the idea we study the difference between quintic and formal quintic Gromov-Witten invariants., Password: 485717
04/29/2022 (Friday, 4:00pm)Anand Patel (Oklahoma State University)The number of times a particular hypersurface occursPick your favorite hypersurface and ask: Is there a systematic method for counting the number of times your hypersurface arises (up to projective equivalence) in a family? For instance, how many times does a general cubic surface arise as a hyperplane section of a fixed, general cubic threefold? As I will explain in the talk, the overall question can be better posed in terms of calculating a mysterious integer polynomial associated to the starting hypersurface. In the case of a general cubic surface, my collaborators and I recently uncovered this polynomial, and it unlocked the answer to the specific enumerative problem above: 42,120. The case of cubic surfaces is actually cutting-edge, as of Spring 2022. I intend to survey all that I know about this problem in the talk. This work is joint work (in clusters) with: Anand Deopurkar, Mitchell Lee, Hunter Spink, and Dennis Tseng., Password: 485717
05/06/2022 (Friday, 4:00pm)David Stapleton (University of Michigan)Studying the birational geometry of Fano varieties using holomorphic formsOne of the best invariants for studying the birational geometry of a variety is its holomorphic forms. In characteristic 0, low degree hypersurfaces (or more generally Fano varieties) do not have any holomorphic forms. For this reason, many problems about birational geometry of these varieties are quite difficult and interesting. E.g. (1) determining if the birational automorphism group is infinite or finite, (2) studying the possible rational endomorphisms of a Fano variety, and (3) understanding the rationality/nonrationality of a Fano variety. Surprisingly, Kollár showed that in characteristic p>0, there are Fano varieties that admit many global (n-1)-forms, and introduced a specialization method for using these forms in characteristic p to control the birational geometry of characteristic 0 Fano varieties. In this talk, we show how this method helps to study problems (1)-(3). This is joint work with Nathan Chen., Password: 485717

Fall 2021
09/03/2021 (Friday, 4:00pm)Yunfeng Jiang (University of Kansas)The Virtual fundamental class for the moduli space of surfaces of general typeSir Simon Donaldson conjectured that there should exist a virtual fundamental class on the moduli space of surfaces of general type inspired by the geometry of complex structures on the general type surfaces. In this talk I will present a method to construct the virtual fundamental class on the lci (locally complete intersection) covering Deligne-Mumford stack over the moduli stack of general type surfaces with only semi-log-canonical singularities. A tautological invariant is defined by taking the integration of the power of the first Chern class of the CM line bundle over the moduli stack of general type surfaces. This can be taken as a generalization of the tautological invariants on the moduli space of stable genus g curves to higher dimensions., Password: 165887
09/10/2021 (Friday, 4:00pm)Shaoyun Bai (Princeton University)Bifurcation of embedded curves and Gopakumar--Vafa invariantsThe mathematical definition of the BPS numbers for Calabi--Yau 3-folds is mysterious, in spite of various proposals. On the symplectic side, these invariants are conjecturally related to a certain count of embedded pseudo-holomorphic curves. I will present progress in this direction established in recent joint work with Mohan Swaminathan, which is built on Wendl's proof of the generic super-rigidity conjecture of Bryan--Pandharipande., Password: 165887
09/17/2021 (Friday, 4:00pm)Iacopo Brivio (NCTS)On deformation invariance of plurigenera in positive and mixed characteristicA famous theorem of Siu states that if X\to S is a smooth projective family of complex varieties, then the plurigenera of the fibers P_m(X_s) are independent of s for all m. In this talk I will construct smooth families of varieties X\to SpecR, where R is a DVR of positive or mixed characteristic, such that for all P_m(X_k)>P_m(X_K) sufficiently divisible m. I will also present some positive results on invariance of plurigenera for certain families of good minimal models., Password: 165887
10/01/2021 (Friday, 4:00pm)Qingyuan Jiang (University of Edinburgh)Derived projectivizations of two-term complexesFor a given two-term complex of vector bundles on a derived scheme (or stack), there are three natural ways to define its "derived projectivizations": (i) as the derived base-change of the classical projectivization of Grothendieck; (ii) as the derived moduli parametrizing one-dimensional locally free quotients; (iii) as the GIT quotient of the total space by $\mathbb{G}_m$-action. In this talk, we first show that these three definitions are equivalent. Second, we prove a structural theorem about the derived categories of derived projectivizations and study the corresponding mutation theory. Third, we apply these results to various moduli situations, including the moduli of certain stable pairs on curves and the Hecke correspondences of one-point modification of moduli of stable sheaves on surfaces. If time allowed, we could also discuss the generalizations of these results to the general derived Quot schemes of locally free quotients., Password: 165887
10/08/2021 (Friday, 2:00pm)Nick Kuhn (Stanford University) Blowup formulas for virtual sheaf-theoretic invariants on projective surfaces For a smooth projective surface X, natural objects of study are its moduli spaces of (semi-) stable coherent sheaves. In rank one, there are a lot of deep results, starting with Göttsche's famous formula for the Betti numbers of the Hilbert schemes of points of X in terms of the Betti numbers of X itself. Even for rank two, however, little is known. There are results for particular choices of X by Yoshioka and others, and a blowup formula for virtual Hodge numbers due to Li-Qin. In general, the moduli spaces are non-smooth and one often studies virtual analogues of invariants, which are better behaved and have connections to physics. For example, there is an elegant conjectural formula for the virtual Euler characteristics of rank 2 moduli spaces due to Göttsche and Kool. I will present joint work with Y. Tanaka on a blowup formula for virtual invariants of moduli spaces of sheaves on a surface, which presents a step towards proving an analogue of Li-Qin's blowup formula for the virtual Euler characteristic., Password: 165887
10/29/2021 (Friday, 4:00pm)John Sheridan (Princeton University)Transferring syzygy information from X to its punctual Hilbert scheme(s).Although the nature of the defining equations for curves in projective space are quite well understood in the most interesting cases, analogous statements for higher dimensional varieties (particularly along the lines of a very precise conjecture of Mukai) remain quite open in general. In this talk, with the perspective that punctual Hilbert schemes are a nice concrete construction of higher dimensional varieties from curves/surfaces X, we discuss some first instances of how syzygy information from X transfers to these higher dimensional partners., Password: 165887
11/05/2021 (Friday, 4:00pm)Payman Eskandari (University of Toronto)The unipotent radical of the Mumford-Tate group of a very general mixed Hodge structure with a fixed associated gradedhe Mumford-Tate group G(M) of a mixed Hodge structure M is a subgroup of GL(M) which satisfies the following property: any rational subspace of any tensor power of M underlies a mixed Hodge substructure if and only if it is invariant under the natural action of G(M). Assuming M is graded-polarizable, the unipotent radical U(M) of G(M) is a subgroup of the unipotent radical U_0(M) of the parabolic subgroup of GL(M) associated to the weight filtration on M. Let us say U(M) is large if it is equal to U_0(M). This talk is a report on a recent joint work with Kumar Murty, where we consider the set of all mixed Hodge structures on a given rational vector space, with a fixed weight filtration and a fixed polarizable associated graded Hodge structure. It is easy to see that this set is in a canonical bijection with the set of complex points of an affine complex variety S. The main result is that assuming some conditions on the (fixed) associated graded hold, outside a union of countably many proper Zariski closed subsets of S the unipotent radical of the Mumford-Tate group is large., Password: 165887
11/12/2021 (Friday, 4:00pm)Yuanqi Wang (University of Kansas) Complete Ricci-flat K\"ahler manifolds of generalized ALG asymptoticsThis is a work in progress. Under a technical assumption, following Tian-Yau’s classical work about canonical metrics on quasi projective varieties, we provide an geometry existence theorem of iso-trivial generalized ALG Ricci flat K\”ahler manifolds. They are non-compact, complete, and asymptotic to a group quotient of the Riemannian product C times D, where C is the flat 2-dimensional Euclidean space, and D is a closed Calabi-Yau manifold., Password: 165887
11/19/2021 (Friday, 4:00pm)Boyu Zhang (Princeton University)The smooth closing lemma for area-preserving surface diffeomorphismsIn this talk, I will introduce the proof of the smooth closing lemma for area-preserving diffeomorphisms on surfaces. The proof is based on a Weyl formula for PFH spectral invariants and a non-vanishing result of twisted Seiberg-Witten Floer homology. This is joint work with Dan Cristofaro-Gardiner and Rohil Prasad., Password: 165887
12/3/2021 (Friday, 2:00pm)Ruijie Yang (Institut für Mathematik at Humboldt-Universität zu Berlin) The cohomology of semisimple local systems and the Decomposition TheoremIn the first part of the talk, I will discuss two Hodge-theoretic aspects of cohomology of semisimple local systems: polarizations and the theory of weights. In particular, I will talk about how to use tools from harmonic analysis to give a generalization of Hodge-Riemann bilinear relations (which relate the topological Poincare pairing with a positive definite pairing) and a global invariant cycle theorem for semisimple local systems (which give extra constraints on possible maps between cohomology groups). In the second part, I will discuss how to apply previous results to give a relatively simple and geometric proof of the Decomposition Theorem for semisimple local systems, which was proved by Sabbah. It asserts that semisimple local systems remain semisimple under a topological operation, the so-called direct images under proper algebraic maps. This is based on the joint work with Chuanhao Wei., Password: 165887
12/17/2021 (Friday, 10:00am)Ben Davison (University of Edinburgh) The decomposition theorem for 2-Calabi-Yau categoriesThe decomposition theorem for 2-Calabi-Yau categories Abstract: Examples of 2CY categories include the category of coherent sheaves on a K3 surface, the category of Higgs bundles, and the category of modules over preprojective algebras or fundamental group algebras of compact Riemann surfaces. Let p:M->N be the morphism from the stack of semistable objects in a 2CY category to the coarse moduli space. I'll explain, using cohomological DT theory, formality in 2CY categories, and structure theorems for good moduli stacks, how to prove a version of the BBDG decomposition theorem for the exceptional direct image of the constant sheaf along p, even though none of the usual conditions for the decomposition theorem apply: p isn't projective or representable, M isn't smooth, the constant mixed Hodge module complex Q_M isn't pure... As applications, I'll explain a proof of Halpern-Leistner's conjecture on the purity of stacks of coherent sheaves on K3 surfaces, and if time permits, a (partly conjectural) way to extend nonabelian Hodge theory to Betti/Dolbeault stacks., Password: 165887

Spring 2021
5/18/2021 (Tuesday, 3:00pm)Ikshu Neithalath (UCLA)Skein Lasagna modules of 2-handlebodiesMorrison, Walker and Wedrich recently defined a generalization of Khovanov-Rozansky homology to links in the boundary of a 4-manifold. We will discuss recent joint work with Ciprian Manolescu on computing the "skein lasagna module," a basic part of MWW's invariant, for a certain class of 4-manifolds., Password: 377587
4/30/2021Dan Edidin (University of Missouri)Quadric bundles and Euler classesIn this talk we explain Edidin and Graham's construction of a characteristic class called the Euler class. This class is associated to the top Chern class of an isotropic bundle of a vector bundle with quadratic form. Whe then explain how this class can be localized to the zero locus of an isotropic section - a recent construction of Oh and Thomas., Password: 993752
4/16/2021Roberto Svaldi (EPFL)On the boundedness of elliptically fibered varietiesIn this talk, we will survey some ideas to address the boundedness of varieties admitting an elliptic fibration. After introducing general ideas, we will discuss how they apply concretely to special classes of varieties: n-folds of Kodaira dimension n-1, and elliptic Calabi--Yau varieties. Part of this talk is based on current work in progress joint with S. Filipazzi and C.D. Hacon., Password: 993752
4/9/2021Franco Rota (Rutgers University)Motivic semiorthogonal decompositions for abelian varieties A motivic semiorthogonal decomposition is the decomposition of the derived category of a quotient stack [X/G] into components related to the "fixed-point data". They represent a categorical analog of the Atiyah-Bott localization formula in equivariant cohomology, and their existence is conjectured for finite G (and an additional smoothess assumption) by Polishchuk and Van den Bergh. I will present joint work with Bronson Lim, in which we construct a motivic semiorthogonal decomposition for a wide class of smooth quotients of abelian varieties by finite groups, using the recent classification by Auffarth, Lucchini Arteche, and Quezada., Password: 993752
3/26/2021Gonçalo Oliveira (Universidade Federal Fluminense (UFF))Special Lagrangians and Lagrangian mean curvature flow(joint work with Jason Lotay) A standing conjecture of Richard Thomas, motivated by mirror symmetry, gives a stability condition supposed to control the existence of a special Lagrangian submanifold in a given Hamiltonian isotopy class of Lagrangians. Later, Thomas and Yau conjectured a similar stability condition controls the long-time existence of the Lagrangian mean-curvature flow. In this talk I will explain how Jason Lotay and myself have recently proved versions of these conjectures on all circle symmetric hyperKahler 4-manifolds., Password: 993752
3/19/2021Yefeng Shen (University of Oregon)Virasoro constraints in Landau-Ginzburg modelsIn this talk, we introduce Virasoro operators in Landau-Ginzburg models of nondegenerate quasi-homogeneous polynomials with nontrivial diagonal symmetries. These operators satisfy the Virasoro relations. Inspired by the famous Virasoro conjecture in Gromov-Witten theory, we conjecture that the (Fan-Jarvis-Ruan-Witten) generating functions arise in these Landau-Ginzburg models are annihilated by the Virasoro operators. We verify the conjecture in various examples and discuss the connections to mirror symmetry of LG models and Landau-Ginzburg/Calabi-Yau correspondence. This talk is based on work joint with Weiqiang He., Password: 993752
3/5/2021Travis Mandel (University of Oklahoma)Quantum theta bases for quantum cluster algebrasOne of the central goals in the study of cluster algebras is to better understand various canonical bases and positivity properties of the cluster algebras and their quantizations. Gross-Hacking-Keel-Kontsevich (GHKK) applied ideas from mirror symmetry to construct so-called "theta bases" for cluster algebras which satisfy all the desired positivity properties, thus proving several conjectures regarding cluster algebras. I will discuss joint work with Ben Davison in which we combine the GHKK techniques with ideas from the DT theory of quiver representations to quantize the GHKK construction, thus producing quantum theta bases and proving the desired quantum positivity properties., Password: 993752
2/26/2021Kristin DeVleming (UCSD)Moduli of surfaces in P3I will discuss a compactification of the moduli space of degree d ≥ 5 surfaces in P3. In other words, we'll find a parameter space whose interior points correspond to (equivalence classes of) smooth hypersurfaces and whose boundary points correspond to their degenerations. To achieve this, we will consider a surface D in P3 as a pair (P3, D) and study an enlarged class of these pairs, including singular degenerations of both D and the ambient space. The moduli space of the enlarged class of pairs will be the desired compactification and, as long as the degree d is odd, we can give a rough classification of the objects on the boundary of the moduli space., Password: 993752
2/19/2021Joaquín Moraga (Princeton University)Maximal log Fano manifolds are generalized Bott towersA log Fano manifold is a pair (X,D) so that (X,D) is log smooth, D is reduced, and -(KX+D) is ample. We show that D has at most dim(X) components. Furthermore, we prove that if D has dim(X) components, then X is a generalized Bott tower. If time permits, we will discuss how this result allows to study degenerations of Fano manifolds., Password: 993752
2/12/2021Yuanqi Wang (University of Kansas)Atiyah classes and the essential obstructions in deforming a singular G2−instantonWhen the rank of the bundle is ≥ 2, in a certain sense, we found an essential obstruction for the gluing construction of G2−instantons with 1−dimensional singularities. It involves the Atiyah classes generated by contracting a vector in C3 with the curvature. Intuitively speaking, the gluing does not work if the tangent connection at a component of the 1−dimensional singular locus is not the twisted Fubini-Study connection on a twisted tangent bundle of P2. Particularly, it fails if the rank of the bundle is ≥ 3., Password: 993752

Fall 2020
11/20/2020Jingchen Niu (University of Arizona)Towards a smooth compactification of the space of curves in projective spacesThe moduli spaces of stable maps to projective spaces play a prominent role in algebraic and symplectic geometry. Following the fundamental work [VakilZinger], desingularizations of such spaces for genus 1 and 2 have been achieved from different perspectives: [RanganathanSantos-ParkerWise] and [BattistellaCarocci] via log geometry, and [HuLi] and [HuLiNiu] via constructive blowing-ups. Towards possible generalizations to higher genera, we provide certain interpretations of [HL] and [HLN] in [HN1] and [HN2] using "twisted fields". In this talk, I will explain some motivation and examples for [HN1,2]., Password: 996661
11/6/2020Patricio Gallardo Candela (UC Riverside)Explicit compactifications of moduli spaces of surfaces of general type, Password: 996661
10/30/2020Calum Spicer (King's College London)Birational geometry and holomorphic dynamicsA foliation on an algebraic variety is a partition of the variety into “parallel" disjoint immersed complex submanifolds. This turns out to be a very useful notion and holomorphic foliations have played a central role in several recent developments in the study of the geometry of projective varieties. I will explain some recent work building towards the birational classification of holomorphic foliations on projective varieties in the spirit of the Kodaira-Enriques classification of algebraic surfaces, as well as indicating some applications of these ideas to the study of the dynamics and geometry of foliations and foliation singularities. Features joint work with P. Cascini and R. Svaldi., Password: 996661
10/16/2020Roberto Nunez (University of Missouri, Coumbia)Existence of limits on schemes with full-dimensional non-reduced locus, Password: 996661
10/9/2020Andrea Ricolfi (Scuola Internazionale Superiore di Studi Avanzati)Higher rank K-theoretic Donaldson-Thomas theory of pointsRecently Okounkov proved Nekrasov’s conjecture expressing the partition function of K-theoretic DT invariants of the Hilbert scheme of points Hilb(C^3,points) on affine 3-space as an explicit plethystic exponential. The higher rank analogue of Nekrasov’s formula is a conjecture in String Theory by Awata-Kanno. We state this conjecture and sketch how to prove it mathematically via Quot schemes. Specialising from K-theoretic to cohomological invariants, we obtain the statement of a conjecture of Szabo. This is joint work with Nadir Fasola and Sergej Monavari., Password: 996661
10/2/2020Gavin Ball (Université du Québec à Montréal)Closed G2-structures inducing a conformally flat metric, Password: 996661
9/25/2020Sien Gong (University of Kansas)The Collapsing Limits of Kähler Ricci-flat Metrics, Password: 996661
9/18/2020Giovanni Inchiostro (University of Washington)Wall crossing morphisms for moduli of stable pairsConsider a quasi-compact moduli space M of pairs (X,D) consisting of a variety X and a divisor D on X. If M is not proper, it is reasonable to find a compactification of it. Assume furthermore that there are two rational numbers 0 < b < a < 1 such that, for every pair (X,D) corresponding to a point in M, the pairs (X,aD) and (X,bD) are klt, and the Q-divisors KX+aD and KX+bD are ample. Using Kollár's formalism of stable pairs, one can construct two different compactifications of M (Ma and Mb), corresponding to a and b. The goal of this project is to relate these two compactifications. The main result is that, up to replacing Ma and Mb with their normalizations, there are birational morphisms Ma → Mb. This project is inspired by Hassett's work on weighted stable curves, and is joint with Kenny Ascher, Dori Bejleri and Zsolt Patakfalvi., Password: 996661
9/11/2020Justin Lacini (University of Kansas)Coverings with log canonical centersWe continue the study of pluricanonical maps of varieties of general type by introducing the key technical tools. Starting from ideas of McKernan, we find a new type of covering with log canonical centers, which arises naturally from the geometry of the variety under consideration., Password: 996661
9/4/2020Justin Lacini (University of Kansas)On pluricanonical maps of varieties of general typeHacon and McKernan have proved that there exist integers rn such that if X is a smooth variety of general type and dimension n, then the pluricanonical maps |rKX| are birational for r ≥ rn. These values are typically very large: for example r3 ≥ 27 and r4 ≥ 94. In this talk we will show that the r-th canonical maps of smooth threefolds and fourfolds of general type have birationally bounded fibers for r ≥ 2 and r ≥ 4 respectively. Furthermore, we will generalize these results to higher dimensions in terms of the constants rn and we will discuss recent progress on a conjecture of Chen and Jiang., Password: 996661

Spring 2020
5/29/2020Sujoy Chakraborty (Tata Institute of Fundamental Research)Chow group of 1-cycles of moduli of Parabolic bundles over a smooth projective curveChow groups are interesting and important objects to study for various reasons. Unfortunately, not much is known about the Chow groups for various moduli spaces, for example the moduli space of semistable vector bundles of a fixed rank and degree over a curve. We will study the Chow group of 1-cycles for the moduli of semistable Parabolic bundles of fixed rank, Parabolic degree and weight. We will first define the various notions related to Parabolic bundles, and study the effect on Chow group of 1-cycles as we vary the generic weight. As a consequence, we will find explicitly the Chow group of 1-cycles for the case of rank 2 Parabolic bundles, using an earlier result of I. Choe and J. Huang.
5/22/2020Akash Sengupta (Columbia University)Geometric invariants and geometric consistency of Manin's conjectureLet X be a Fano variety with an associated height function defined over a number field. Manin's conjecture predicts that, after removing a thin set, the growth of the number of rational points of bounded height on X is controlled by certain geometric invariants (e.g. the Fujita invariant of X). I will talk about how to use birational geometric methods to study the behaviour of these invariants and propose a geometric description of the thin set in Manin's conjecture. Part of this is joint work with Brian Lehmann and Sho Tanimoto., Password: 344283
5/15/2020Joseph DimosTorelli Small Generating Sets and Darboux Coordinate SystemsIn [Putnam, 2011]], the small generating set for the Torelli group was explained. To this, we grant the connection with the abelianization of Torelli groups for genus g ≥ 3 on bounding pair maps that is due to [Johnson]. I will show the generation of symplectic structure for moduli that coincides with a Darboux coordinate system. The relationship is with C. Leininger and D. Margalit for any pair of elements of the Torelli group that commute or generate a free group. Using the bounding class maps of Leininger-Margalit, the bounding pairs of Torelli subgroups are products of Dehn twists. The subsequent Dehn and Nielsen relating self-homomorphisms of a surface and automorphisms of fundamental groups of a surface is a consistent feature. The extension is upon symplectic bases that permit a Prym matrix for generative homological coordinates on T^*M_g., Password: 927539
5/8/2020Sien Gong (University of Kansas)Asymptotically Cylindrical Calabi-Yau ManifoldsExpository talk on the paper "Asymptotically Cylindrical Calabi-Yau Manifolds" by Mark Haskins, Hans-Joachim Hein and Johannes Nordström., Password: 959418
5/1/2020Jayan Mukherjee (University of Kansas)Deformations of Quadruple Galois Canonical CoversWe show that for any quadruple Galois canonical cover X (with at worst canonical singularities) of a smooth surface Y of minimal degree, there exists a smooth affine algebraic curve along which the canonical morphism of X deforms to a two-to-one canonical morphism unless X is a product of two genus 2 curves, or Y is a linear P2. We also give lower bounds of the dimension of the subspace of the deformation space of the canonical morphism along which it deforms to a two-to-one morphism. This result is in sharp contrast with earlier results of Gallego-González-Purnaprajna that show that the general deformation of a canonical cover of degree two or three onto a projective bundle over P1, embedded by a complete linear series, will again have the same degree. This is a joint work with P. Bangere, F.J. Gallego, M. González and D. Raychaudhury., Password: 792219
2/28/2020Promit Kundu (University of Kansas)Intrinsic Normal Cone by Kai BehrendExpository talk.
2/21/2020Yuanqi Wang (University of Kansas)The spectrum of an operator associated with G2 - instantons with 1-dimensional singularitiesIn the model setting of G2 - instantons with 1 - dimensional conic singularities, the linearized operator yields a self-adjoint first order elliptc operator P on a bundle over S5. Using the "Quaternion structure" in the Sasakian geometry of S5, we describe the spectrum of P, and give an example. Particularly, we relate certain eigenspaces of P to sheaf cohomologies on P2.
02/14/2020Yunfeng Jiang (University of Kansas)A proof of Vafa-Witten's S-duality conjecture for projective plane and K3 surfacesFor a real four manifold, the S-duality conjecture of Vafa-Witten (1994) predicts that the S-transformation sends the gauge group SU(r)-invariants counting instantons to the Langlands dual gauge group SU(r)/Zr-invariants; and both of the invariants satisfy modularity properties. The SU(r) - Vafa-Witten invariants have been constructed by Tanaka-Thomas using the moduli space of semistable Higgs bundle or sheaves on a smooth projective surface. In this talk I will present the idea of using moduli space of twisted sheaves and twisted Higgs sheaves on a projective surface to define the Langlands dual gauge group SU(r)/Zr - Vafa-Witten invariants, and prove the S-duality conjecture of Vafa-Witten for projective plane in rank two and K3 surfaces in prime ranks.

Fall 2019
11/15/2019Jayan MukherjeeDeformations of Galois canonical covers of varieties of minimal degree IIThis is the continuation of the previous talk. We study the deformations of higher dimensional, bi-double Galois canonical covers of smooth varieties of minimal degree and we produce examples of smooth, regular, bi-double, Galois canonical covers of higher dimensional smooth varieties of minimal degree whose general deformations will again be four-to-one.
11/08/2019Debaditya RaychaudhuryDeformations of Galois canonical covers of varieties of minimal degreeWe show that for any smooth, regular, bi-double, Galois canonical cover X of a smooth surface of minimal degree, there exists a curve along which the canonical morphism of X deforms to a two-to-one canonical morphism. This result is in sharp contrast with an earlier result of Gallego-González-Purnaprajna that shows that the general deformation of a canonical cover of degree two or three onto a projective bundle over P1, embedded by a complete linear series, will again have the same degree.
10/25/2019Promit KunduDeligne Faltings object on schemesExpository talk.
10/18/2019Sien GongModuli space of Elliptic Curves IIExpository talk.
10/11/2019Debjit BasuModuli space of Elliptic CurvesExpository talk.
09/27/2019Yuanqi WangG2 - instantons with 1 - dimensional singularities I: spectral theoryWe consider G2-instantons with 1 - dimensional conic singularities. The linearized operator of such an instanton, under the model data, yields a self-adjoint first order elliptc operator P on a bundle over S5. Using the "Quaternion structure" in the Sasakian geometry of S5, we describe the spectrum of P, and give an example.