5 Feb 12 Feb 19 Feb Nariel Monterro (UC Santa Cruz) TBA 26 Feb 5 Mar 12 Mar 19 Mar no seminar ------------------- Spring Break ---------- 26 Mar 2 Apr
Spring 2025 classes begin Tuesday January 21 and end Monday May 5.
Fall 2024 Seminars (Wednesdays at 2:00 PM in H705) 18 Sept Robert Boltje (UC Santa Cruz) On a question of Feit 2 Oct Teddy Gonzales (RU) Matroids and tori varieties 9 Oct Joseph Fluegemann (IAS) Smooth Points on Positroid Varieties 16 Oct Igor Pak (UCLA and IAS) Spectral radii of Cayley graphs of fin. gen. groups 23 Oct Zengrui Han (RU) Pfaffian double mirrors, stringy Hodge numbers and Homological Projective Duality 30 Oct Marco Castronovo (Columbia) Cluster gluings and open-string Schubert calculus 6 Nov no seminar 13 Nov Anders Buch (RU) The Picard group of a cominuscule Richardson variety 20 Nov Joy Hamlin (RU) Hook Multiplication in the Quantum K-Theory Ring of the Grassmannian 27 Nov --- no seminar --- Thanksgiving is Nov. 28 4 Dec Jay Taylor (U. Manchester) Modular Reduction of Nilpotent Orbits 11 Dec Lev Borisov (RU) Modular curves X1(n) as moduli of point arrangementsFall 2024 classes begin Tuesday September 3 and end Wednesday Dec. 11.
Spring 2024 Seminars (Wednesdays at 2:00 AM in H705) 7 Feb no seminar 14 Feb Gabriel Navarro (U.Valencia) Recent advances in the representation theory of finite groups 21 Feb Eilidh McKemmie (RU) Monodromy groups of covers of genus 1 Riemann surfaces 28 Feb Teddy Gonzales (RU) Hodge theory of matroids 6 Mar Lev Borisov (RU) Phantom categories 13 Mar no seminar ------------------- Spring Break ---------- 20 Mar Jurij Volčič (Drexel) Matrix evaluations of noncommutative rational functions and Waring problems 27 Mar Anders Buch (RU) Classes of Dynkin quiver orbits 3 Apr Shira Gilat (U.Penn.) The graph of totally isotropic subspaces 10 Apr V. Retakh (RU) Noncommutative surfaces and their symmetries 17 Apr Zengrui Han (RU) Central charges in local mirror symmetry 24 Apr Dennis Hou (RU) Formal Colombeau theory 1 May Reading Day (no classes)Spring 2024 classes begin Tuesday September 16 and end April 29.
Fall 2023 Seminars (Wednesdays at 2:00 AM in H705) 20 Sept Itamar Vigdorovich (Weizmann Inst) Spectral gap and character limits in arithmetic groups 27 Sept Chuck Weibel (RU) The K-theory of polynomial-like rings 4 Oct Emily Riehl (Johns Hopkins) Colloquium (3:30 PM) Do we need a new foundation for higher structures? 11 Oct Eilidh McKemmie (RU) Group theoretic problems from cryptography 18 Oct no seminar (Tiep-60 conference at Princeton U) 25 Oct Zeyu Shen (RU) Computing the G-theory of two-dimensional toric varieties 8 Nov Robert Guralnick (USC) 3/2 generation of finite groups and spread 15 Nov Tae Young Lee (RU) Monodromy groups of hypergeometric sheaves 22 Nov --- no seminar --- Thanksgiving is Nov. 23 29 Nov Deepam Patel (Purdue) Motivic Properties of Generalized Alexander Modules 13 Dec Ben Steinberg (CUNY) Topological methods in monoid representation theoryFall 2022 classes begin Tuesday September 5 and end Wednesday Dec. 13.
Spring 2023 Seminars (Wednesdays at 2:00 AM in H705) 8 Feb Abi Ali (Rutgers) "Strong integrality of inversion subgroups of Kac-Moody groups" 15 Feb Zengrui Han (RU) "Duality of better-behaved GKZ systems" 1 Mar Danny Krashen (U.Penn) "Perspectives on local-global principles for the Brauer group" 8 Mar Connor Cassady (U.Penn) "Quadratic forms, local-global principles, and field invariants" 15 Mar no seminar ------------------- Spring Break ---------- 28 Mar Yael Davidov (RU) Tuesday at 1PM, in CORE 431 "Admissibility of Finite Groups over Semi-Global Fields in the Bad Characteristic Case" 5 Apr Chuck Weibel (RU) "Computing shift equivalence via Algebra" 12 Apr Mihail Tarigradschi "Classifying cominuscule Schubert varieties up to isomorphism" and Zeyu Shen (RU) "A brief introduction to toric varieties" 19 Apr Richard Lyons (RU) "Remarks on CFSG and CGLSS" 26 Apr Ishaan Shah (RU) 4 May Classes end Monday May 1 Spring 2021 classes begin Tuesday January 17 and end Monday May 1; Finals are May 4-10.
Fall 2022 Seminars (Wednesdays at 2:00 AM in H705) 14 Sept Laurent Vera (RU) "Super-equivalences and odd categorification of sl2" 21 Sept Tamar Blanks (RU) "Trace forms and the Witt invariants of finite groups" 28 Sept Lauren Heller (Berkeley) "Characterizing multigraded regularity on products of projective spaces" 12 Oct Yael Davidov (RU) "Admissibility of Groups over Semi-Global Fields in the 'Bad Characteristic' Case" 19 Oct Mandi Schaeffer Fry (Metropolitan State U) TBA (group representation theory) 26 Oct Chuck Weibel (Rutgers) "Grothendieck-Witt groups of singular schemes" 2 Nov Marco Zaninelli (U.Antwerp) “The Pythagoras number of a function field in one variable” 9 Nov Anders Buch (Rutgers) "Pieri rules for quantum K-theory of cominuscule Grassmannians" 16 Nov Francesca Tombari (KTH Sweden) "Realisations of posets and tameness" 23 Nov --- no seminar --- Thanksgiving is Nov. 24 30 Nov --- no seminar --- 7 Dec Eilidh McKemmie (Rutgers) "Galois groups of random additive polynomials" 14 Dec --- no seminar ---Fall 2022 classes begin Tuesday September 6 and end Wednesday Dec. 14. Next semester: Andrzej Zuk (Univ Paris VII) TBA
Spring 2022 Seminars (Wednesdays at 2:00 PM in H705) no seminar January 19 as we start in remote mode 26 Jan Weihong Xu (Rutgers) "Quantum $K$-theory of Incidence Varieties" (remote) 2 Feb Rudradip Biswas (Manchester) "Cofibrant objects in representation theory" (remote) 9 Feb Chuck Weibel (RU) "An introduction to monoid schemes" 16 Feb Ian Coley (RU) "Hochster's description of Spec(R)" 23 Feb no seminar -------------------------------------------- 2 Mar Alexei Entin (Tel Aviv) "The minimal ramification problem in inverse Galois theory" (remote) 9 Mar Pham Tiep (RU) "Representations and tensor product growth" 16 Mar no seminar ------------------- Spring Break ---------- 30 Mar Tim Burness (Bristol, UK) "Fixed point ratios for primitive groups and applications" 6 Apr Eugen Rogozinnikov (Strasbourg) "Hermitian Lie groups of tube type as symplectic groups over noncommutative algebras" 13 Apr Eilidh McKemmie (RU) "A survey of various random generation problems for finite groups" 20 Apr Yael Davidov (RU) "Exploring the admissibility of Groups and an Application of Field Patching" 27 Apr Daniel Douglas (Yale) "Skein algebras and quantum trace maps" 4 MaySpring 2022 classes begin Tuesday January 18 and end Monday May 2.
Spring break is March 12-20, 2022
Fall 2021 Seminars (Wednesdays at 11:00 AM in H525) 15 Sept Yom Kippur 29 Sept Yoav Segev "A characterization of the quaternions using commutators" 6 Oct Lev Borisov (RU) "Explicit equations for fake projective planes" 13 Oct Ian Coley (RU) "Introduction to topoi" 20 OctAnders Buch (RU) POSTPONED to 12/8 ("Tevelev Degrees") 27 Oct Eilidh McKemmie (RU) "The probability of generating invariably a finite simple group" 3 Nov Ian Coley and Chuck Weibel "Localization, and the K-theory of monoid schemes" 10 Nov Max Peroux (Penn) "Equivariant variations of topological Hochschild homology" 17 Nov Shira Gilat (RU) "The infinite Brauer group" 24 Nov --- no seminar --- Thanksgiving is Nov. 25 29 Nov (Monday) Wednesday class schedule 1 Dec --- no seminar --- 8 Dec Anders Buch (RU) "Tevelev Degrees"Fall 2021 classes begin Tuesday September 1 and end Monday Dec. 13.
Spring 2021 Seminars (Wednesdays at 2:00 PM, on-line) 27 Jan Ian Coley (Rutgers) "Tensor Triangulated Geometry?" 3 Feb Aline Zanardini (U.Penn) "Stability of pencils of plane curves" 10 Feb Svetlana Makarova (U.Penn) "Moduli spaces of stable sheaves over quasipolarized K3 surfaces, and Strange Duality" 17 Feb no seminar 24 Feb Patrick McFaddin (Fordham) "Separable algebras and rationality of arithmetic toric varieties" 3 Mar Christian Klevdal (U.Utah) "Integrality of G-local systems" 10 Mar Justin Lacini (U.Kansas) "On log del Pezzo surfaces in positive characteristic" 17 Mar no seminar ------------------- Spring Break ---------- 24 Mar John Kopper (Penn State) "Ample stable vector bundles on rational surfaces" 14 Apr Allechar Serrano Lopez (U.Utah) "Counting elliptic curves with prescribed torsion over imaginary quadratic fields" 21 Apr Franco Rota (Rutgers) "Motivic semiorthogonal decompositions for abelian varieties" 28 Apr Ben Wormleighton (Washington U./St.Louis) "Geometry of mutations: mirrors and McKay" 5 May Morgan Opie (Harvard) "Complex rank 3 vector bundles on CP5" 12 May Avery Wilson (N. Carolina) "Compactifications of moduli of G-bundles and conformal blocks"Spring 2021 classes begin Tuesday January 19 and end Monday May 3.
Here is a link to the algebra seminars in previous semesters
Abstracts of seminar talks
Spring 2025
Nariel Monterro, Feb. 19, 2025
Fall 2024
Modular curves $X_1(n)$ as moduli of point arrangements (Lev Borisov, Dec. 1, 2024
For a complex elliptic curve $E$ and a point $p$ of order $n$ on it, the images of the points $p_k=kp$ under the Weierstrass embedding of $E$ into $CP^2$ are collinear if and only if the sum of indices is divisible by $n$. We prove that for $n$ at least $10$ a collection of $n$ points in $P^2$ with these properties comes (generically) from a point of order $n$ on an elliptic curve. In the process, we discover amusing identities between logarithmic derivatives of the theta function at rational points. I will also discuss potential applications of these results to bounds on the numbers of Hecke eigenforms for $\Gamma_1(n)$ of positive analytic rank, although this is rather speculative. This is joint work with Xavier Roulleau
Modular Reduction of Nilpotent Orbits (Jay Taylor, Dec. 4, 2024)
The general linear group G=GLn(k) over a field k acts on the spacegl= gln of (nxn)-matrices by conjugation. The set N(g) of nilpotent matrices is preserved by this action and G acts with finitely many orbits. The Jordan normal form gives a representative of each orbit that is contained in gln(Z). Importantly, the structure of the centralizer of this nilpotent matrix is independent of k.
It is natural to ask to what extent this statement extends to a connected reductive algebraic group G acting on its Lie algebrag or the dual space g* via the (co-)adjoint representation. In general, the structure of centralisers of nilpotent elements will depend on k but one can hope that the centraliser dimension remains the same. In this talk I will propose two variants of this idea, generalising the classical situation above, and report on on-going joint work with Adam Thomas (Warwick) to establish the existence of elements satisfying the proposed properties.
Hook Multiplication in the Quantum K-Theory Ring of the Grassmannian (Joy Hamlin, Nov. 20, 2024)
Quantum K-theory is a formal deformation of K-theory that encodes Gromov-Witten invariants of the moduli space of curves meeting Schubert varieties in the Grassmannian. There are two Pieri rules, dual to each other, for multiplication in the quantum K-theory of a Grassmannian, but no positive formula for more complicated products is known.
I will generalize both Pieri rules to find a complete description of the product when an arbitrary class is multiplied by a "hook class", and show a reduction that relates elements in quantum K-theory rings of different Grassmannians.
The Picard group of a cominuscule Richardson variety (Anders Buch, Nov. 13, 2024)
A Richardson variety R in a cominuscule Grassmannian is defined by a skew diagram of boxes. If this diagram has several connected components, then R is a product of smaller Richardson varieties given by the components. I will show that the Picard group of R is a free abelian group of rank equal to the number of components in its skew diagram. In addition, the integration pairing identifies the Picard group with the dual of the first Chow group of R. The proof is based on a resolution of singularities that is interesting in itself.
Cluster gluings and open-string Schubert calculus (Marco Castronovo, Oct.30, 2024)
When is a gluing of affine schemes affine? I will state a complete but conjectural answer when the pieces are tori, and the gluing maps are cluster mutations in the sense of Fomin-Zelevinsky. The combinatorics of Demazure weaves allows to prove the conjecture in interesting cases, notably those in which there are finitely many tori, as well as the simplest ones with infinitely many. This has applications to open-string Schubert calculus, i.e. computations of the Fukaya category of G/P. Joint work with M. Gorsky, J. Simental, D. Speyer.
Pfaffian double mirrors, stringy Hodge numbers and Homological Projective Duality (Zengrui Han, Oct.23, 2024)
The Pfaffian double mirror construction is historically the first example of derived equivalent but not birationally equivalent Calabi-Yau 3-folds.
I will define them and talk about results on their stringy Hodge numbers.
I will then explain how these results provide numerical predictions for the shape of Lefschetz decompositions of the categorical crepant resolutions of Pfaffian varieties.
Spectral radii of Cayley graphs of fin. gen. groups (Igor Pak, Oct.16, 2024)
Spectral radius is a classical parameter of a graph and in the context of Cayley graphs of finitely generated groups it measures "how far" is the graph from being amenable. Unfortunately, computing the spectral radius is rather challenging and has been done in only a handful of examples.
We prove that the set of all possible radii has continuum cardinality. Time permitting, I will mention what's known about other monotone parameters of Cayley graphs: asymptotic entropy, critical percolation, Cheeger constant, rate of exponential growth, and connective constant. This is joint work with Martin Kassabov.
Smooth Points on Positroid Varieties (Joseph Fluegemann, Oct.9, 2024)
Positroid varieties are subvarieties in the Grassmannian defined by cyclic rank conditions and which are related to Schubert varieties. They are also connected to N=4 supersymmetric Yang-Mills theory. In this talk, we will discuss recent work by the author studying whether torus-fixed points on positroid varieties are smooth. This will involve looking at combinatorial diagrams called "affine pipe dreams."
Matroids and Toric varieties (Teddy Gonzales, Oct.2, 2024)
I will discuss one "geometric model" of matroids which arises from viewing their representations as points on the Grassmannian and taking torus orbit closures due to Gelfand, Goresky, MacPherson, and Serganova. If time permits, I will describe how this relates to Berget, Eur, Spink, and Tseng's "tautological classes" of matroids which are certain equivariant classes on the permutohedral variety.
On a question of Feit (Robert Boltje, Sept.18, 2024)
In 1979 Feit asked the following question: Let χ be an irreducible character of a finite group G and let $n$ be the conductor of χ, i.e., the smallest positive integer $n$ such that all values of χ can be expressed as a rational linear combination of $n$-th roots of unity. Does this force G to have an element of order $n$?
Feit's question has been answered positively for solvable groups in 1986 by Ferguson and Turull, and independently by Amit and Chillag. Little progress has been made in the following decades. This talk reports on recent advances on this question in joint work with Alexander Kleshchev, Gabriel Navarro and Pham Huu Tiep.
Spring 2024
Formal Colombeau theory (Dennis Hou, April 24, 2024)
The vector space of doubly infinite formal Laurent series does not form a ring under the usual multiplication. In the theory of vertex operator algebras, a partially defined associative product is developed in terms of formal calculus. We consider a nonstandard approach to embedding this space into a differential algebra.
Central charges in local mirror symmetry (Zengrui Han, April 17, 2024)
In the framework of local mirror symmetry, a mirror of a toric Calabi-Yau variety (B-model) is given by a family of certain Laurent polynomials (A-model). In 2004, Hosono defined the central charges on A- and B-models using period integrals and hypergeometric series, and conjectured their equality.
In this talk, I will explain a proof of (a version of) Hosono's conjecture, which is a combination of the tropical method introduced by Abouzaid-Ganatra-Iritani-Sheridan and the hypergeometric duality established by Borisov and myself.
Noncommutative surfaces and their symmetries (Vladimir Retakh, April 10, 2024)
I will discuss some noncommutative invariants associated with triangulated surfaces and their relations to noncommutative Laurent phenomenon. The talk is based on joint work with Arkady Berenstein and Min Huang.
The graph of totally isotropic subspaces (Shira Gilat, April 3, 2024)
Given a qudratic form, a totally isotopic subspace is a linear subspace where the form vanishes. The collection of such subspaces gives rise to a polar geometry. We impose a graph structure on the set of maximal totally isotropic subspaces of a quadratic form and try to reconstruct algebraic properties from the geometric properties.
Motivated by the construction of polar spaces from isotropic vectors in a quadratic space, we define hyperbolicity of polar spaces which help us detect if our quadratic form is hyperbolic.
Classes of Dynkin quiver orbits (Anders Buch, March 27, 2024)
A result of Rimanyi shows that the Chern-Schwartz-MacPherson class of any orbit in the representation space of a Dynkin quiver can be computed as a product in the cohomological Hall algebra (COHA) of elementary factors that correspond to the indecomposable representations of the quiver.
I will speak about a new proof of this result based on an algebra homomorphism from Ringel's Hall algebra to the COHA. This algebra homomorphism can also be used to prove a simple explicit formula for the elementary factors that was conjectured by Rimanyi. This is joint work with Richard Rimanyi.
Matrix evaluations of noncommutative rational functions and Waring problems (Jurij Volčič, March 20, 2024)
Noncommutative polynomials and noncommutative rational functions are elements of the free associative algebra and free skew field, respectively. One may view them as multivariate functions in matrix arguments; this perspective is nowadays propelled by advances in free probability, noncommutative function theory, free real algebraic geometry, and noncommutative optimization.
This talk concerns the images of noncommutative polynomials and noncommutative rational functions on large matrices.
Firstly, every nonconstant noncommutative rational function attains values with pairwise distinct eigenvalues on sufficiently large matrix tuples.
Secondly, one can then apply this to noncommutative variants of the Waring problem. In particular, given a nonconstant noncommutative rational function, every large enough trace-zero matrix is a difference of its values, and every large enough nonscalar determinant-one matrix is a quotient of its values.
Phantom categories (Lev Borisov, March 6, 2024)
Phantom categories are certain hard-to-construct triangulated subcategories in categories of coherent sheaves on smooth algebraic varieties. I will define them and talk about their history and recent developments.
Hodge Theory of Matroids (Teddy Gonzales, February 28, 2024)
Adiprasito, Huh, and Katz have introduced the Chow ring of a matroid, which they have used to prove the log concavity of a matroid's characteristic polynomial. Following the presentation in Huh's "Tropical Geometry of Matroids" and Eur's "Essence of Independence,"
I will briefly introduce matroids, their characteristic polynomial, and their Chow ring and discuss some geometry behind these constructions by describing the cohomology classes on the permutohedral variety that arise from matroids.
Monodromy groups of covers of genus 1 Riemann surfaces (Eilidh McKemmie, February 21, 2024)
Consider a cover of the Riemann sphere by a compact connected Riemann surface. The monodromy group of the cover is an important invariant describing how badly the cover degenerates. It is natural to ask which groups can appear in such a context. We will discuss how to genus of the surface and the Aschbacher-Scott type of the group influences the answer to this question, and provide an answer for groups of type B in genus at most 1.
Recent advances in the representation theory of finite groups (Gabriel Navarro, February 14, 2024)
Some of the most important global/local conjectures in the representation theory of finite groups have been recently proven. We will survey these results and some of the still open conjectures.
Fall 2023
Topological methods in monoid representation theory (Ben Steinberg, December 13, 2023)
Over the past 15 years there has been a resurgence of interest in the representation theory of finite monoids motivated by applications to probability and algebraic combinatorics (e.g., the work of Brown, Diaconis, Chung, Graham, Björner, Reiner and others). The algebra of a finite monoid over the field of complex numbers is rarely semisimple. If one picks up a modern book on the representation theory of finite dimensional algebras, the first thing that is proved is that the module category is equivalent to the category of representations of a quiver with relations. But from a practical viewpoint, it is not so easy to find this quiver and relations. For example, the category of modules for the monoid of all order-preserving maps of $\{0,...,n\}$ is equivalent to the category of chain complexes of length $n$ and hence is presented by the oriented $A_{n+1}$-Dynkin quiver with relations that any composition of consecutive arrow is $0$. But the proof of this is essentially the Dold-Kan theorem, which is a nontrivial result.An important stepping stone to computing the quiver and relations is computing $Ext^1$ and $Ext^2$ between simple modules. In this talk, we consider the problem of computing $Ext^n$ between simple modules for the complex algebra of a finite monoid and also computing the global dimension. Our main result is that Ext between simple modules inflated from the group completion of $M$ and the group of units $G$ of $M$ can be computed by looking at the homology of a certain $G$-simplicial complex as a module over the group algebra. For a large class of monoids, all Ext computations for simple modules can be reduced to this case, and even when this is not a case computing the global dimension of the monoid surprisingly often boils down to this computing Ext between such modules.
Motivic Properties of Generalized Alexander Modules (Deepam Patel, November 29, 2023)
This will be a survey of some joint work with Madhav Nori on the theory of Gamma Motives. Classical Alexander modules are examples, and we will explain the analogs of the classical monodromy theorem and period isomorphisms in this context. If time permits, we will discuss some motivation coming from Beilinson’s conjectures on special values of L-functions.
Monodromy groups of hypergeometric sheaves (Tae Young Lee, November 15, 2023)
Let Gm be the multiplicative group over an algebraically closed field of characteristic p>0. The finite groups which are quotients of the fundamental group of Gm are precisely those generated by its Sylow p-subgroups together with at most one additional element.
Katz, Rojas Leon and Tiep used certain continuous representations of the fundamental group, called
hypergeometric sheaves, to construct explicit quotient maps for many such groups. In this talk, I will talk about the hypergeometric sheaves whose geometric monodromy group is finite almost quasisimple with the nonabelian factor PSLn(q).
3/2 generation of finite groups and spread (Bob Guralnick, November 8, 2023)
Let G be a finite group. We say G is 3/2 generated if given any nontrivial element g in G, there exists h in G with G generated by g and h. We will discuss the recent classification of such groups and the related notion of spread.
We will consider a variation of this question for almost simple groups related to a question of Lucchini about profinite groups.
Computing the G-theory of two-dimensional toric varieties (Zeyu Shen, October 25, 2023)
The G-theory of a Noetherian scheme X is the algebraic K-theory of the abelian category M(X) of coherent sheaves on X. i.e., G_n(X)=K_n(M(X)) for every non-negative integer n. We compute the G-theory of some two-dimensional toric varieties over an algebraically closed field of characteristic zero. I will explain how to compute the G-theory groups of a two-dimensional toric variety coming from a single cone. The computations of the G-theory groups of other two-dimensional toric varietiees will also be discussed. Using the semi-orthogonal decomposition of the bounded derived category of coherent sheaves, the G_0 of the weighted projective plane P(1,1,m) can be computed. If time permits, I will also discuss an example of computing the G-theory groups of a 3-dimensional affine simplicial toric variety.
Group theoretic problems from cryptography (Eilidh McKemmie, October 11, 2023)
We discuss three group theoretic problems that have applications to cryptography.
The K-theory of polynomial-like rings (Chuck Weibel, September 27, 2023)
We first prove that the K-theory of a polynomial ring k[x,y,...] has a previously unknown ray-like decomposition.
Then we show this when A is a polynomial-like ring, i.e., a normal subring of k[x,y,...] generated by monomials and k contains a field. The proof in characteristic 0 is different from the proof in characteristic p.
Spectral gap and character limits in arithmetic groups (Itamar Vigdorovich, September 20, 2023)
To any group G is associated the space of characters, often called the Thoma dual of G. This space is central for harmonic analysis on abstract groups. After defining this space properly, I will discuss its geometry in the case the group exhibits certain rigidity properties, most notably Kazhdan's property (T). Further restricting to a class of arithmetic groups, I will explain why any sequence of characters must converge to the Dirac character at the identity, and demonstrate this with certain examples and relations to character bounds of finite groups. Time permitting, I will discuss another result on the free group which is somewhat complimentary (and yet opposite) to the case above. The talk is based on a joint work with Levit and Slutsky.
Spring 2023
Remarks on CFSG and CGLSS (Richard Lyons, April 19, 2023)
The talk will consist of random remarks about the classification of the finite simple groups (CFSG)
and the strategy of the long-term project (GLS, more recently CGLSS, the initials of the participants)
to write a "second-generation" proof with explicit and limited foundation in the literature.
The "G" is for Daniel Gorenstein (1923-1992, Rutgers 1969-1992), who initiated and drove the project during his lifetime.
A brief introduction to toric varieties (Zeyu Shen, April 12, 2023)
Affine toric varieties are defined using strongly convex, rational, polyhedral cones. Smooth and simplicial cones are important special classes of such cones.
Normal toric varieties are constructed from fans. The bijective correspondence between properties of fans and geometric properties of the associated toric varieties will be mentioned.
Some examples of cones, fans and the associated toric varieties will be presented.
I will also demonstrate a calculation of the Weil divisor class group of projective space using torus-invariant divisors.
Computing shift equivalence via Algebra (Chuck Weibel, April 5, 2023
Shift equivalence of matrices over ℤ is an important invariant of discrete dynamical systems.
To compute it, we translate the problem into commutative ring theory, using localization,
Picard groups, conductor ideals, and basic linear algebra over ℤ[t].
On the Admissibility of Finite Groups over Semi-Global Fields in the Bad Characteristic Case (Yael Davidov, March 28, 2023)
In the last 20 years, a method of constructing various algebraic objects over semi-global fields (one-variable function fields over complete discretely valued fields) by patching together compatible objects constructed on a network of field extensions has been introduced and developed by Harbater, Hartmann, and Krashen. For example, field patching can be used to study central simple algebras and Galois extensions over these fields. This has been a powerful tool in considering the problem of admissibility over these fields.
Given a finite group G and a field K we say that G is admissible over K if there is a division algebra central over K with a maximal subfield that is a Galois extension of K with group G. Fixing a field K, we can ask, which groups are admissible over K?
I will present a recent result which completely solves the admissibility problem for a class of semi-global fields (equicharacteristic with algebraically closed residue fields) using field patching techniques.
Quadratic forms, local-global principles, and field invariants (Connor Cassady, March 8, 2023)
Given a quadratic form (homogeneous degree two polynomial) q over a field k, some basic questions one could ask are:
* Does q have a non-trivial zero (is q isotropic)?
* Which non-zero elements of k are represented by q?
* Does q represent all non-zero elements of k (is q universal)?
Over a global field F, the Hasse-Minkowski theorem, which is one of the first examples of a local-global principle, allows us to use answers to these questions over the completions of F to form answers to these questions over F itself. In this talk, we'll explore when the local-global principle for isotropy holds over more general fields k, as well as connections between this local-global principle and universal quadratic forms over k.
Perspectives on local-global principles for the Brauer group (Danny Krashen, March 1, 2023)
The Brauer group, which describes the collection of finite dimensional division algebras whose center is a given field, is an invariant capturing interesting and important aspects of field arithmetic. Understanding when a given Brauer class is trivial, which corresponds to understanding whether or not an algebra has zero-divisors, is often a surprisingly subtle problem. In this talk, I'll describe some new local-to-global principles arising from joint with with Max Lieblich and Minseon Shin, which use algebro-geometric tools, to reduce the context of such questions to simpler fields. I'll also describe applications of this result to the problem of finding rational points on genus 1 curves over function fields of complex varieties.
Duality of better-behaved GKZ systems (Zengui Han, Feb. 15, 2023)
GKZ hypergeometric systems of PDEs were introduced by Gelfand, Kapranov and Zelevinsky. They arise naturally in the moduli theory of toric Calabi-Yau varieties and play an important role in toric mirror symmetry.
In this talk I will discuss a better-behaved version of GKZ systems introduced by Borisov-Horja and my recent work on the duality of such systems. This is joint work with Borisov.
Strong integrality of inversion subgroups of Kac-Moody groups (Abid Ali, Feb. 8, 2023)
Let g be a symmetrizable Kac-Moody algebra over ℚ. Let V be an integrable highest weight g-module and let V_ℤ be a ℤ-form of V. Let G(ℚ) be an associated minimal representation-theoretic Kac-Moody group and let G(ℤ) be its integral subgroup. Let Γ(ℤ) be the Chevalley subgroup of G, that is, the subgroup that stabilizes the lattice V_ℤ in V.
It is a difficult question to determine if G(ℤ)=Γ(ℤ). We establish this equality for inversion subgroups U_w of G where, for an element w of the Weyl group, U_w is the group generated by positive real root groups that are flipped to negative roots by w^{-1}. This result extends to other subgroups of G, particularly when G has rank 2. This is joint work with Lisa Carbone, Dongwen Liu and Scott H. Murray.Fall 2022
Galois groups of random additive polynomials (Eilidh McKemmie, Dec. 7, 2022)
The Galois group of an additive polynomial over a finite field is contained in a finite general linear group. We will discuss three different probability distributions on these polynomials, and estimate the probability that a random additive polynomial has a "large" Galois group. Our computations use a trick that gives us characteristic polynomials of elements of the Galois group, so we may use our knowledge of the maximal subgroups of GL(n,q). This is joint work with Lior Bary-Soroker and Alexei Entin.
Realisations of posets and tameness (Francesca Tombari, Nov.16, 2022)
Persistent homology is commonly encoded by vector space-valued functors indexed by posets. These functors are called tame, or persistence modules, and capture the life-span of homological features in a dataset. Every poset can be used to index a persistence module, however some posets are particularly well suited.
We introduce a new construction called realisation, which transforms posets into posets. Intuitively, it associates a continuous structure to a locally discrete poset by filling in empty spaces. Realisations share several properties with upper semi-lattices. They behave similarly with respect to certain notions of dimension for posets that we introduce. Moreover, as indexing posets of persistence modules, they allow for good discretisations and effective computation of homological invariants via Koszul complexes.
Pieri rules for quantum K-theory of cominuscule Grassmannians (Anders Buch, Nov.9, 2022)
The quantum K-theory ring QK(X) of a flag variety X is constructed using the K-theoretic Gromov-Witten invariants of X, defined as arithmetic genera of Gromov-Witten varieties parametrizing curves meeting fixed subvarieties in X, and can be used to compute these invariants. A Pieri formula means a formula for multiplication with a set of generators of QK(X). Such a formula makes it possible to compute efficiently in this ring.
I will speak about a Pieri formula for QK(X) when X is a cominuscule Grassmannian, that is, an ordinary Grassmannian, a maximal orthogonal Grassmannian, or a Lagrangian Grassmannian. This formula is expressed combinatorially in terms of counting diagrams of boxes labeled by positive integers, also known as tableaux. This is joint work with P.-E. Chaput, L. Mihalcea, and N. Perrin.
The Pythagoras number of a function field in one variable (Marco Zaninelli, Nov.2, 2022)
The Pythagoras number of a field K is the minimum number n such that any sum of squares in K can be written as a sum of n squares in K.
Despite its elementary definition, the computation of the Pythagoras number of a field can be a very complicated task, to the point that for many families of fields we are not even able to produce an upper bound for it. When we are, it is usually thanks to local-global principles for quadratic forms and to modern techniques from algebraic geometry.
In this seminar we will focus on the Pythagoras number of function fields in one variable, and more precisely we will show how to obtain the upper bound 5 for the Pythagoras number of a large family of such fields by exploiting a recent local-global principle due to V. Mehmeti.
Grothendieck-Witt groups of singular schemes (Chuck Weibel, Oct.26, 2022)
We establish some new structural results for the Witt and Grothendieck–Witt groups of schemes over Z[1/2], includinghomotopy invariance of Witt groups Witt groups of X[t,1/t] Witt groups of punctured affine spaces.
Admissibility of Groups over Semi-Global Fields in the “Bad Characteristic” Case (Yael Davidov, Oct.12, 2022)
We say a finite group, G, is admissible over a field, F, if there exists a division algebra with center F and a maximal subfield K such that K/F is Galois with group G. The question of which groups are admissible over a given field is generally difficult to answer but has been solved in the case that F is a transcendence degree 1 extension of a complete discretely valued field with algebraically closed residue field, so long as the characteristic of the residue field does not divide the order of the group. This result was obtained in a paper by Harbater, Hartmann and Krashen using field patching techniques in 2009.
In this talk we will be discussing progress towards generalizing this result and trying to answer the question, what happens when the characteristic of the residue field does divide the order of G? We will restrict our attention to a special case to make the discussion accessible.
Trace forms and the Witt invariants of finite groups (Tamar Blanks, Sept. 21, 2022):
A Witt invariant of an algebraic group G over a field k is a natural transformation from G-torsors to the Witt ring, that is, a rule that assigns quadratic forms to algebraic objects in a way that respects field extensions over k. An important example is the invariant sending each etale algebra to its trace form. Serre showed that the ring of Witt invariants of the symmetric group is generated by the trace form invariant and its exterior powers.
In this talk we will discuss work towards generalizing Serre's result to other Weyl groups, and more generally to other finite groups. We will also describe the connection between Witt invariants and cohomological invariants via the Milnor conjecture.
Super-equivalences and odd categorification of sl2 (Laurent Vera, Sept. 14, 2022):
In their seminal work on categorifications of quantum groups, Chuang and Rouquier showed that an action of sl2 on a category gives rise to derived equivalences. These equivalences can be used to prove Broué’s abelian defect group conjecture for symmetric groups.
In this talk, I will present a “super version” of these results. I will introduce the odd 2-category associated with sl2 and describe the properties of its 2-representation theory. I will then describe the super analogues of the Chuang-Rouquier complexes and explain how they give rise to derived equivalences on 2-representations. These derived equivalences lead to a proof of the abelian defect group conjecture for spin symmetric groups. This is joint work with Mark Ebert and Aaron Lauda.
Spring 2022
Skein algebras and quantum trace maps (Daniel Douglas, April 27, 2022):
Skein algebras are certain noncommutative algebras associated to surfaces, appearing at the interface of low-dimensional topology, representation theory, and combinatorics. They occur as quantum deformations of character varieties with respect to their natural Poisson structure, and in particular possess fascinating connections to quantum groups. In this talk, I will discuss the problem of embedding skein algebras into quantum tori, the latter of which have a relatively simple algebraic structure. One such embedding, called the quantum trace map, has been used to shed light on the representation theory of skein algebras, and is related to Fock and Goncharov’s quantum higher Teichmüller theory.
Exploring the admissibility of Groups and an Application of Field Patching (Yael Davidov, April 20, 2022):
Similarly to the inverse Galois problem, one can ask if a group G is admissible over a given field F. This is answered in the affirmative if there exists a division algebra with F as its center that contains a maximal subfield that is a Galois extension of F, with Galois group G.
We will review admissibility results over the rationals that have been proven by Schacher and Sonn. We will also give some idea as to how one might try to construct division algebras that prove the admissibility of a particular group. Finally, we will briefly outline how Harbater, Hartmann and Krashen were able to obtain admissibility criteria for groups over a particular class of fields using field patching techniques.
Hermitian Lie groups of tube type as symplectic groups over noncommutative algebras (Eugen Rogozinnikov, Aprli 6, 2022):
We introduce the symplectic group Sp_2(A,σ) over a noncommutative algebra A with an anti-involution σ. We realize several classical Lie groups as Sp2 over various noncommutative algebras, which provides new insights into their structure theory. We construct several geometric spaces, on which the groups Sp2(A,σ) act. We introduce the space of isotropic A-lines, which generalizes the projective line. We describe the action of Sp2(A,σ) on isotropic A-lines, generalize the Kashiwara-Maslov index of triples and the cross ratio of quadruples of isotropic A-lines as invariants of this action.
When the algebra A is Hermitian or the complexification of a Hermitian algebra, we introduce the symmetric space XSp2(A,σ), and construct different models of this space. Applying this to classical Hermitian Lie groups of tube type (realized as Sp2(A,σ)) and their complexifications, we obtain different models of the symmetric space as noncommutative generalizations of models of the hyperbolic plane and of the three-dimensional hyperbolic space.
Fixed point ratios for primitive groups and applications (Tim Burness, March 30, 2022):
Let G be a finite permutation group and recall that the fixed point ratio of an element x, denoted fpr(x), is the proportion of points fixed by x. Fixed point ratios for finite primitive groups have been studied for many decades, finding a wide range of applications.
In this talk, I will present some of the main results and applications, focussing on recent joint work with Bob Guralnick where we determine the triples (G,x,r) such that G is primitive, x has prime order r and fpr(x) > 1/(r+1). The latter result allows us to prove new results on the minimal degree and minimal index of primitive groups, and we have used it in joint work with Moreto and Navarro to study the commuting probability of p-elements in finite groups.
Representations and tensor product growth (Pham Tiep, March 9, 2022):
The deep theory of approximate subgroups establishes 3-step product growth for subsets of finite simple groups G of Lie type of bounded rank. We will discuss 2-step growth results for representations of such groups G (including those of unbounded rank), where products of subsets are replaced by tensor products of representations. This is joint work with M. Larsen and A. Shalev.
The minimal ramification problem in inverse Galois theory (Alexei Entin, March 2, 2022):
For a number field K and a finite group G the Boston-Markin Conjecture predicts the minimal number of ramified places (of K) in a Galois extension L/K with Galois group G. The conjecture is wide open even for the symmetric and alternating groups S_n, A_n over the field of rational numbers Q. We formulate a function field version of this conjecture, settle it for the rational function field K=F_q(T) and G=S_n with a mild restriction on q,n and make significant progress towards the G=A_n case. We also discuss some other groups and the connection between the minimal ramification problem and the Abhyankar conjectures on the etale fundamental group of the affine line in positive characteristic.
An introduction to monoid schemes (Chuck Weibel, Feb. 9, 2022):
If A is a pointed abelian monoid, its prime ideals make sense and form a topological space, analogous to Spec of a ring; the notion of a monoid scheme is analogous to the notion of a scheme in Algebraic Geometry. The monoid ring construction k[A] gives a link to geometry. In this talk I will give an introduction to the basic ideas, including toric monoid schemes, which model toric varieties.
Cofibrant objects in representation theory (Rudradip Biswas, Feb. 2, 2022):
Cofibrant modules, as defined by Benson, play important roles in many cohomology questions of infinite discrete groups. In this talk, I will (a) talk about my new work on the relation between the class of these modules and Gorenstein projectives where I'll build on Dembegioti-Talelli's work, and (b) highlight new results from one of my older papers on the behaviour of an invariant closely related to these modules. If time permits, I'll show a possible generalization of many of these results to certain classes of topological groups.
Quantum $K$-theory of Incidence Varieties (Weihong Xu, Jan. 26, 2022):
Buch, Mihalcea, Chaput, and Perrin proved that for cominuscule flag varieties, (T-equivariant) K-theoretic (3-pointed, genus 0) Gromov-Witten invariants can be computed in the (equivariant) ordinary K-theory ring. Buch and Mihalcea have a related conjecture for all type A flag varieties.
In this talk, I will discuss work that proves this conjecture in the first non-cominuscule case--the incidence variety Fl(1,n-1;n). The proof is based on showing that Gromov-Witten varieties of stable maps to Fl(1,n-1;n) with markings sent to a Schubert variety, a Schubert divisor, and a point are rationally connected. As applications, I will also discuss positive formulas that determine the equivariant quantum K-theory ring of Fl(1,n-1;n). The talk is based on the arxiv preprint at https://arxiv.org/abs/2112.13036.
Fall 2021
Tevelev degrees (Anders Buch, December 8, 2021):
Let X be a non-singular complex projective variety. The virtual Tevelev degree of X associated to (g,d,n) is the (virtual) degree of the forgetful map from theKontsevich moduli space Mg,n(X,d) of n-pointed stable maps to X of genus g and degree d, to the product Mg,n × Xn. Recent work of Lian and Pandharipande shows that this invariant is enumerative in many cases, that is, it is the number of degree-d maps from a fixed genus-g curve to X, that send n fixed points in the curve to n fixed points in X.
I will speak about a simple formula for this degree in terms of the (small) quantum cohomology ring of X. If X is a Grassmann variety (or more generally, a cominuscule flag variety) then the virtual Tevelev degrees of X can be expressed in terms of the (real) eigenvalues of a symmetric endomorphism of the quantum cohomology ring. If X is a complete intersection of low degree compared to its dimension, then the virtual Tevelev degrees of X are given by an explicit product formula. I will do my best to keep this talk student-friendly, so the most of it will be about explaining the ingredients of this abstract. The results are joint work with Rahul Pandharipande.
Equivariant variations of topological Hochschild homology (Maximilien Peroux, November 10, 2021):
Topological Hochschild homology (THH) is an important variant for rings and ring spectra. It is built as a geometric realization of a cyclic bar construction. It is endowed with an action of the circle, because it is a geometric realization of a cyclic object. The simplex category factors through Connes' category Λ. Similarly, real topological Hochschild homology (THR) for rings (and ring spectra) with anti-involution is endowed with a O(2)-action. Here instead of the cyclic category Λ, we use the dihedral category Ξ.
From work in progress with Gabe Angelini-Knoll and Mona Merling, I present a generalization of Λ and Ξ called crossed simplicial groups, introduced by Fiedorwicz and Loday. To each crossed simplical group G, I define THG, an equivariant analogue of THH. Its input is a ring spectrum with a twisted group action. THG is an algebraic invariant endowed with different action and cyclotomic structure, and generalizes THH and THR.
Localization, and the K-theory of monoid schemes (Ian Coley and Chuck Weibel, November 3, 2021):
We develop the K-theory of sets with an action of a pointed monoid (or monoid scheme), analogous to the $K$-theory of modules over a ring (or scheme).
In order to form localization sequences, we construct the quotient category of a nice regular category by a Serre subcategory. A special case is the localization of an abelian category by a Serre subcategory.
The probability of generating invariably a finite simple group (Eilidh McKemmie, October 27, 2021):
We say a group is invariably generated by a subset if every tuple in the product of conjugacy classes of elements in that subset is a generating tuple. We discuss the history of computational Galois theory and probabilistic generation problems to motivate some results about the probability of generating invariably a finite simple group, joint work with Daniele Garzoni. We also highlight some methods for studying probabilistic invariable generation.
Introduction to topoi (Ian Coley, October 13, 2021):
The theory of sheaves on a topological space or scheme admits a generalization to sheaves on a category equipped with a topology, which we call a site. This level of generality allows us access to interesting cohomology theories on schemes that don't make sense at the point-set level. We'll give the basic definitions, warm up by categorifying the notion of sheaves on a topological space, then get into these new topologies and their associated sheaf cohomologies.
Explicit equations for fake projective planes (Lev Borisov, October 6, 2021):
There are 50 complex conjugate pairs of fake projective planes, realized as quotients of the complex 2-ball. However, in most cases there are no known explicit embeddings into a projective space. In this talk I will describe my work over the past several years (with multiple co-authors) which resulted in explicit equations for 9 out of the 50 pairs. It is a wild ride in the field of computer assisted AG computations.
A characterization of the quaternions using commutators (Yoav Segev, September 29, 2021):
Let D be a quaternion division algebra over a field F. Thus D=F +F i +F j+ F k, with i^2, j^2 in F and k=ij=-ji. A pure quaternion is an element p in D such that p is in F i+F j+F k.
It is easy to check that p^2 is in F, for a pure quaternion p, and that given x,y in D, the commutator (x,y)=xy-yx is a pure quaternion.
We show that this characterizes quaternion division algebras, namely, any associative ring R with 1, such that the commutator (x,y) is not a zero divisor and satisfies (x,y)^2 is in the center of R, for all nonzero x,y in R, is a quaternion division algebra. The proof is elementary and self contained.
This is joint work with Erwin Kleinfeld
Spring 2021
Compactifications of moduli of G-bundles and conformal blocks (Avery Wilson, May 12, 2021):
I will talk about Schmitt and Munoz-Castaneda's compactification of the moduli space of G-bundles on a curve and its relation to conformal blocks. I use this compactification to prove finite generation of the conformal blocks algebra over the stack of stable curves of genus >1, which Belkale-Gibney had previously proven for G=SL(r). This yields a nice compactification for the relative moduli space of G-bundles.
Complex rank 3 vector bundles on CP5 (Morgan Opie, May 5,2021:
Given the ubiquity of vector bundles, it is perhaps surprising that there are so many open questions about them -- even on projective spaces. In this talk, I will outline my ongoing work on complex rank 3 topological vector bundles on CP5. I will describe a classification of such bundles, involving a connection to topological modular forms. I will also discuss a topological, rank-preserving additive structure which allows for the construction of new rank 3 bundles on CP^5 from "simple" ones. This construction is an analogue to an algebraic construction of Horrocks. As time allows, I will discuss future algebro-geometric directions related to this project.
Geometry of mutations: mirrors and McKay (Ben Wormleighton, April 28, 2021):
There are several notions of mutation that arise in different parts of algebra, geometry, and combinatorics. I will discuss some of these appearances in mirror symmetry and in the McKay correspondence with a view towards approaching classification problems for Fano varieties and for crepant resolutions of orbifold singularities.
Motivic semiorthogonal decompositions for abelian varieties (Fanco Rota, April 21, 2021):
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.
Counting elliptic curves with prescribed torsion over imaginary quadratic fields (Allechar Serrano Lopez, April 14, 2021):
A generalization of Mazur's theorem states that there are 26 possibilities for the torsion subgroup of an elliptic curve over a quadratic extension of Q. If G is one of these groups, we count the number of elliptic curves of bounded naive height whose torsion subgroup is isomorphic to G in the case of imaginary quadratic fields.
Ample stable vector bundles on rational surfaces (John Kopper, March 24, 2021):
Ample vector bundles are among the most important "positive" vector bundles in algebraic geometry, but have resisted attempts at classification, especially in dimensions two and higher. In this talk, I will discuss a moduli-theoretic approach to this problem that dates to Le Potier and is particularly powerful on rational surfaces: study Chern characters for which the general stable bundle is ample.
After reviewing the ideas of stability and ampleness for vector bundles, I will discuss some new results in this direction for minimal rational surfaces. First, I will give a complete classification of Chern characters on these surfaces for which the general stable bundle is both ample and globally generated. Second, I will explain how this classification also holds in an asymptotic sense without the assumption of global generation. This is joint work with Jack Huizenga.
On log del Pezzo surfaces in positive characteristic (Justin Lacini, March 10, 2021):
A log del Pezzo surface is a normal surface with only Kawamata log terminal singularities and anti-ample canonical class. Over the complex numbers, Keel and McKernan have classified all but a bounded family of log del Pezzo surfaces of Picard number one.
In this talk we will extend their classification to positive characteristic. In particular, we will prove that for p>3 every log del Pezzo surface of Picard number one admits a log resolution that lifts to characteristic zero over a smooth base. As a consequence, we will see that Kawamata-Viehweg vanishing holds in this setting. Finally, we will conclude with some counterexamples in characteristic two, three and five.
Integrality of G-local systems (Christian Klevdal, March 3,2021):
Simpson conjectured that for a reductive group G, rigid G-local systems on a smooth projective complex variety are integral. I will discuss a proof of integrality for cohomologically rigid G-local systems. This generalizes and is inspired by work of Esnault and Groechenig for GL_n. Surprisingly, the main tools used in the proof (for general G and GL_n) are the work of L. Lafforgue on the Langlands program for curves over function fields, and work of Drinfeld on companions of \ell-adic sheaves. The major differences between general G and GL_n are first to make sense of companions for G-local systems, and second to show that the monodromy group of a rigid G-local system is semisimple.
All work is joint with Stefan Patrikis.
Separable algebras and rationality of arithmetic toric varieties (Patrick McFaddin, February 24, 2021):
The class of toric varieties defined over the complex numbers gives a robust testing ground for computing various invariants, e.g., algebraic K-theory and derived categories. To obtain a broader sense of the capabilities of these invariants, we look to the arithmetic setting and twisted forms of toric varieties. In this talk, I will discuss work on distinguishing forms of toric varieties using separable algebras and how this sheds light on the connection between derived categories and rationality questions. This is joint work with M. Ballard, A. Duncan, and A. Lamarche.
Moduli spaces of stable sheaves over quasipolarized K3 surfaces, and Strange Duality (Svetlana Makarova, February 10, 2021):
I will talk about a construction of relative moduli spaces of stable sheaves over the stack of quasipolarized surfaces. For this, I first retrace some of the classical results in the theory of moduli spaces of sheaves on surfaces to make them work over the nonample locus. Then I will recall the theory of good moduli spaces, whose study was initiated by Alper and concerns an intrinsic (stacky) reformulation of the notion of good quotients from GIT. Finally, I use a criterion by Alper-Heinloth-Halpern-Leistner, coupled with some categorical arguments, to prove existence of the good moduli space.