Wednesdays at 2:00-3:00 PM in H705

Here is a link to the algebra seminars in previous semesters

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 (Rutgers) TBA 18 Oct no seminar (Tiep-60 conference at Princeton U) 25 Oct Zeyu Shen (RU) " 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

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

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.

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 CPSpring 2021 classes begin Tuesday January 19 and end Monday May 3.^{5}" 12 May Avery Wilson (N. Carolina) "Compactifications of moduli of G-bundles and conformal blocks"

9 Sept Lev Borisov Rutgers "A journey from the octonionicFall 2020 classes begin Tuesday September 1 and end Thursday December 10P^{2}to a fakeP^{2}" 16 Sept Stefano Filipazzi (UCLA) "On the boundedness of n-folds of Kodaira dimension n-1" 23 Sept Yifeng Huang (U.Michigan) "Betti numbers of unordered configuration spaces of a punctured torus" 30 Sept Andrea Ricolfi (SISSA, Italy) "Moduli of semiorthogonal decompositions" 7 Oct Giacomo Mezzedimi (Hannover) "The Kodaira dimension of some moduli spaces of elliptic K3 surfaces" 14 Oct Katrina Honigs (Oregon) "An obstruction to weak approximation on some Calabi-Yau threefolds" 21 Oct Alex Wertheim (UCLA) "Degree One Milnor K-Invariants of Groups of Multiplicative Type" 28 Oct Inna Zakharevich (Cornell)3:30PM Colloquium"The Dehn complex: scissors congruence, K-theory, and regulators" 4 Nov Michael Wemyss (Glasgow) "Tits cone intersections and Applications" 11 Nov Pieter Belmans (Univ. Bonn)12:00 noon"Graph potentials as mirrors to moduli of vector bundles on curves" 18 Nov David Hemminger (UCLA) "Lannes' T-functor and Chow rings of classifying spaces" 25 Nov --- no seminar --- Thanksgiving is Nov. 26; Friday class schedule 2 Dec Clover May (UCLA) "Classifying perfect complexes of Mackey functors" 9 Dec Be'eri Greenfeld (UCSD) "Combinatorics of words, symbolic dynamics and growth of algebras"

22 Jan Shira Gilat (Bar-Ilan U.) "Higher norm principles for norm varieties" 12 Feb Chuck Weibel (Rutgers) "The K'-theory of monoid sets" 19 Feb Linhui Shen (Michigan State) "Quantum geometry of moduli spaces of local system" 26 Feb Lev Borisov Rutgers "Six explicit pairs of fake projective planes" 4 Mar Saurabh Gosavi (Rutgers) "Generalized Brauer dimension" 18 Mar no seminar ------------------- Spring Break ---------- 25 Mar CANCELLED -- Future seminars moved on-line 1 Apr Joaquin Moraga (Princeton) "On the Jordan property for local fundamental groups" 8 Apr Ian Coley (Rutgers) "Higher K-theory via generators and relations" 15 Apr Franco Rota (Rutgers) "Moduli spaces on the Kuznetsov component of Fano threefolds of index 2" 22 Apr James Cameron (UCLA) "Group cohomology rings via equivariant cohomology" 29 Apr Luca Schaffler (U. Mass) "Compactifications of moduli of points and lines in the projective plane" 6 May Angela Gibney (Rutgers) CANCELLEDSpring 2020 classes begin Tuesday January 22 and end Monday May 4

18 Sep Sándor Kovács (U.Washington) "Rational singularities and their cousins in arbitrary characteristics" 20 Sep Jacob Lurie (IAS) "On Makkai's Strong Conceptual Completeness Theorem" 25 Sep Saurabh Gosavi (Rutgers) "Generalized Brauer dimension and other arithmetic invariants of semi-global fields" 2 Oct Danny Krashen (Rutgers) "The arithmetic of semiglobal fields via combinatorial topology" 9 Oct Ian Coley (Rutgers) "What is a derivator?" 16 Oct Sumit Chandra Mishra (Emory U.) "Local-global principle for norm over semi-global fields" 23 Oct Chengxi Wang (Rutgers) "strong exceptional collections of line bundles" 30 Oct Angela Gibney (Rutgers) "Vertex algebras of CohFT-type" 6 Nov Robert Laugwitz (Nottingham, UK) "dg categories and their actions" 13 Nov Volodia Retakh (Rutgers) "Noncommutative Laurent Phenomenon: two examples" 20 Nov Carl Lian (Columbia) "Enumerating pencils with moving ramification on curves" 27 Nov --- no seminar --- Thanksgiving is Nov. 28; Friday class schedule 11 Dec Diane Mclagan (U.Warwick) "Tropical scheme theory"Fall 2019 classes begin Tuesday September 3 and end Wednesday Dec.11. Finals are December 14-21, 2019

Note: First 3 seminars were in H-005; 4th in H705; all others in SERC-206 23 Jan Patrick Brosnan U.Maryland "Palindromicity and the local invariant cycle theorem" 30 Jan Khashayar Sartipi UIUC "Paschke Categories, K-homology and the Riemann-Roch Transformation" 6 Feb Chuck Weibel Rutgers "The Real graded Brauer group" 13 Feb Volodia Retakh Rutgers "An analogue of mapping class groups and noncommutative triangulated surfaces" 20 Feb Dawei Chen Boston College&IAS "Volumes and intersection theory on moduli spaces of abelian differentials" 6 Mar no seminar 13 Mar Jeanne Duflot Colorado State U. "A Degree Formula for Equivariant Cohomology" 20 Mar no seminar ------------------- Spring Break ---------- 27 Mar Louis Rowen Bar-Ilan Univ "The algebraic theory of systems" 3 Apr Iulia Gheorghita Boston College "Effective divisors in the Hodge bundle" 10 Apr Gabriel Navarro U.Valencia "Character Tables and Sylow Subgroups of Finite Groups" 17 Apr John Sheridan Stony Brook "Continuous families of divisors on symmetric powers of curves" 24 Apr Yaim Cooper IAS "Severi degrees via representation theory" 1 May Dave Anderson Ohio State "Schubert calculus and the Satake correspondence"Classes end Monday May 6; Finals are May 9-15, 2019

19 Sep Nicola Tarasca Rutgers "Geometry and Combinatorics of moduli spaces of curves" 26 Sep Angela Gibney Rutgers "Basepoint free loci on $M_{0,n}$-bar from Gromov-Witten theory of smooth homogeneous varieties" 5 Oct(FRI) Michael Larsen Indiana U "Irrationality of Motivic Zeta Functions" *** Friday at 10:00 AM in Hill 005 *** 10 Oct Yotam Hendel Weizmann Inst. "On singularity properties of convolutions of algebraic morphisms" 17 Oct Qixiao Ma Columbia Univ. "Brauer class over the Picard scheme of curves" 24 Oct Sandra Di Rocco KTH-Sweden "Generalized Polar Geometry" 31 Oct Igor Rapinchuk Michigan State "Algebraic groups with good reduction and unramified cohomology" 7 Nov Isabel Vogt MIT "Low degree points on curves" 14 Nov Bob Guralnick USC "Low Degree Cohomology" 21 Nov --- no seminar --- Thanksgiving is Nov. 22; Friday class schedule 28 Nov Julie Bergner U.Virginia "2-Segal spaces and algebraic K-theory" 5 Dec Chengxi Wang Rutgers "Quantum Cohomology of Grassmannians" 12 Dec Patrick Brosnan U.Maryland POSTPONEDClasses end Wednesday Dec. 12; Finals begin Dec. 15, 2018

Here is a link to the algebra seminars in previous semesters

**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, October 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.

**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.

**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], including

**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.

**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.

**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 M_{g,n}(X,d) of n-pointed stable maps to X
of genus g and degree d, to the product M_{g,n} × X^{n}.
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

**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 CP ^{5}
(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 CP

**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.

**Stability of pencils of plane curves
(Aline Zanardini, February 3, 2021):**

I will discuss some recent results on the problem of classifying pencils of
plane curves via geometric invariant theory. We will see how the stability of
a pencil is related to the stability of its generators, to the log canonical
threshold of its members, and to the multiplicities of its base points.

**What is Tensor Triangulated Geometry?
(Ian Coley, January 27, 2021):**

Based on work of Thomason, Balmer defined a way to think about varieties from
a purely category-theoretic point of view. By considering not only the
triangulated structure of the derived category but also the tensor product,
one can (nearly) do geometry within the category D^{b}(X) itself.
I will discuss
the construction of the 'Balmer spectrum' and give some pertinent examples.

**Combinatorics of words, symbolic dynamics and growth of algebras
(Be'eri Greenfeld, December 9, 2020):**

The most important invariant of a finite dimensional algebra is its
dimension. Let A be a finitely generated, infinite dimensional associative
or Lie algebra over some base field F. A useful way to 'measure its
infinitude' is to study its growth rate, namely, the asymptotic behavior of
the dimensions of the spaces spanned by (at most n)-fold products of some
fixed generators. Up to a natural asymptotic equivalence relation, this
function becomes a well-defined invariant of the algebra itself, independent
of the specification of generators.

**Classifying perfect complexes of Mackey functors
(Clover May, December 2, 2020):**

Mackey functors were introduced by Dress and Green to encode operations that
behave like restriction and induction in representation theory. They play a
central role in equivariant homotopy theory, where homotopy groups are
replaced by homotopy Mackey functors. In this talk I will discuss joint work
with Dan Dugger and Christy Hazel classifying perfect chain complexes of
Mackey functors for G=Z/2. Our classification leads to a computation of the
Balmer spectrum of the derived category. It has topological consequences as
well, classifying all modules over
the equivariant Eilenberg--MacLane spectrum HZ/2.

**Lannes' T-functor and Chow rings of classifying spaces
(David Hemminger, November 18, 2020):**

Equivariant Chow rings, including Chow rings of classifying spaces of
algebraic groups, appear often in nature but are difficult to compute. Like
singular cohomology in topology, these Chow rings modulo a prime p carry the
additional structure of unstable modules over the Steenrod algebra. We utilize
this extra structure to refine estimates of equivariant Chow rings mod p. As a
special case, we prove an analog of Quillen's stratification theorem,
generalizing and recovering prior results of Yagita and Totaro.

**Graph potentials as mirrors to moduli of vector bundles on curves
(Pieter Belmans, November 11, 2020):**

In a joint work with Sergey Galkin and Swarnava Mukhopadhyay we have
introduced a class of Laurent polynomials associated to decorated trivalent
graphs which we called graph potentials. These Laurent polynomials satisfy
interesting symmetry and compatibility properties. Under mirror symmetry they
are related to moduli spaces of rank 2 bundles (with fixed determinant of odd
degree) on a curve of genus $g\geq 2$, which is a class of Fano varieties of
dimension $3g-3$.

I will discuss (parts of) the (enumerative / homological)
mirror symmetry picture for Fano varieties, and then explain what we
understand for this class of varieties and what we can say about the
(conjectural) semiorthogonal decomposition of the derived category.

**Tits cone intersections and Applications
(Michael Wemyss, November 4, 2020):**

In the first half of the talk, I will give an overview of Tits cone
intersections, which are structures that can be obtained from (possibly
affine) ADE Dynkin diagrams, together with a choice of nodes. This is quite
elementary, but visually very beautiful, and it has some really remarkable
features and applications.

In the second half of the talk I will highlight
some of the applications to algebraic geometry, mainly to 3-fold flopping
contractions, through mutation and stability conditions. This should be
viewed as a categorification of the first half of my talk. Parts are joint
work with Yuki Hirano, parts with Osamu Iyama.

**The Dehn complex: scissors congruence, K-theory, and regulators
(Inn Zakharevich, October 28, 2020):**

Hilbert's third problem asks: do there exist two polyhedra with the same
volume which are not scissors congruent? In other words, if P and Qare
polyhedra with the same volume, is it always possible to write P as the union
of P_i, and Q as the union of Q_i, such that the P's and
Q's intersect only on the boundaries and such that P_i is congruent to Q_i?

In 1901 Dehn answered this question in the negative by constructing a second scissors
congruence invariant now called the "Dehn invariant," and showing that a cube
and a regular tetrahedron never have equal Dehn invariants, regardless of
their volumes. We can then restate Hilbert's third problem: do the volume and
Dehn invariant separate the scissors congruence classes? In 1965 Sydler showed
that the answer is yes; in 1968 Jessen showed that this result extends to
dimension 4, and in 1982 Dupont and Sah constructed analogs of such results in
spherical and hyperbolic geometries. However, the problem remains open past
dimension 4. By iterating Dehn invariants Goncharov constructed a chain
complex, and conjectured that the homology of this chain complex is related to
certain graded portions of the algebraic K-theory of the complex numbers, with
the volume appearing as a regulator.

**Degree One Milnor K-Invariants of Groups of Multiplicative Type
(Alex Wertheim, October 21, 2020):**

Many important algebraic objects can be viewed as G-torsors over a field F,
where G is an algebraic group over F. For example, there is a natural
bijection between F-isomorphism classes of central simple F-algebras of degree
n and PGL_n(F)-torsors over Spec(F). Much as one may study principal bundles
on a manifold via characteristic classes, one may likewise study G-torsors
over a field via certain associated Galois cohomology classes. This principle
is made precise by the notion of a cohomological invariant, which was first
introduced by Serre.

**An obstruction to weak approximation on some Calabi-Yau threefolds
(Katrina Honigs, October 14, 2020):**

The study of Q-rational points on algebraic varieties is fundamental to
arithmetic geometry. One of the few methods available to show that a variety
does not have any Q-points is to give a Brauer-Manin obstruction. Hosono and
Takagi have constructed a class of Calabi-Yau threefolds that occur as a
linear section of a double quintic symmetroid and given a detailed analysis of
them as complex varieties in the context of mirror symmetry. This construction
can be used to produce varieties over Q as well, and these threefolds come
tantalizingly equipped with a natural Brauer class. In work with Hashimoto,
Lamarche and Vogt, we analyze these threefolds and their Brauer class over Q
and give a condition under which the Brauer class obstructs weak
approximation, though it cannot obstruct the existence of Q-rational points.

**The Kodaira dimension of some moduli spaces of elliptic K3 surfaces
(Giacomo Mezzedimi, October 7, 2020):**

Let $\mathcal{M}_{2k}$ denote the moduli space of $U\oplus \langle
-2k\rangle$-polarized K3 surfaces. Geometrically, the K3 surfaces in
$\mathcal{M}_{2k}$ are elliptic and contain an extra curve class,
depending on $k\ge 1$. I will report on a joint work with M. Fortuna
and M. Hoff, in which we compute the Kodaira dimension of
$\mathcal{M}_{2k}$ for almost all $k$: more precisely, we show that it
is of general type if $k\ge 220$ and unirational if $k\le 50$,
$k\not\in \{11,35,42,48\}$. After introducing the general problem, I
will compare the strategies used to obtain both results. If time
permits, I will show some examples arising from explicit geometric
constructions.

**Moduli of semiorthogonal decompositions
(Andrea Ricolfi, September 30, 2020):**

We discuss the existence of a moduli space parametrising semiorthogonal
decompositions on the fibres of a smooth projective morphism X/U.
More precisely, we define a functor on (Sch/U) sending V/U to the set of
semiorthogonal decompositions on Perf(X_V). We show this functor defines an
etale algebraic space over U. As an application, we prove that if the generic
fibre of X/U is indecomposable, then so are all fibres. We discuss some
examples and applications.
Joint work with Pieter Belmans and Shinnosuke Okawa.

**Betti numbers of unordered configuration spaces of a punctured torus
(Yifeng Huang, September 23, 2020):**

Let X be a elliptic curve over C with one point removed, and consider the
unordered configuration spaces
Conf^n(X)={(x_1,...,x_n): x_i\ne x_j for i\ne j} / S_n.
We present a rational function in two variables from whose
coefficients we can read off the i-th Betti numbers of Conf^n(X) for
all i and n. The key of the proof is a property called "purity", which was
known to Kim for (ordered or unordered) configuration spaces of the complex
plane with r >= 0 points removed. We show that the unordered configuration
spaces of X also have purity (but with different weights). This is a joint
work with G. Cheong.

**On the boundedness of n-folds of Kodaira dimension n-1
(Stefano Filipazzi, September 16, 2020):**

One of the main topics in the classification of algebraic varieties is
boundedness. Loosely speaking, a set of varieties is called bounded if it can
be parametrized by a scheme of finite type. In the literature, there is
extensive work regarding the boundedness of varieties belonging to the three
building blocks of the birational classificaiton of varieties: varieties of
Fano type, Calabi--Yau type, and general type. Recently, work of
Di Cerbo--Svaldi and Birkar introduced ideas to deduce
boundedness statements for fibrations from boundedness results concerning
these three classes of varieties. Following this philosophy, in this talk I
will discuss some natural conditions for a set of n-folds of Kodaira dimension
n-1 to be bounded.

Part of this talk is based on joint work with Roberto Svaldi.

**A journey from the octonionic P^{2} to a fake P^{2}
(Lev Borisov, September 9, 2020):**

This is joint work with Anders Buch and Enrico Fatighenti. We discover a family of surfaces of general type with

**Compactifications of moduli of points and lines in the projective plane
(Luca Schaffler, April 29, 2020):**

Projective duality identifies the moduli space B_{n} parametrizing
configurations of n general points in projective plane with *X(3,n)*,
parametrizing configurations of n general lines in the dual plane. When
considering degenerations of such objects, it is interesting to compare
different compactifications of the above moduli spaces.

In this work, we consider Gerritzen-Piwek's compactification
B_{n}
and Kapranov's Chow quotient compactification
X(3,n),
and we show they have isomorphic normalizations.

We prove that
B_{n}
does not admit a
modular interpretation claimed by Gerritzen and Piwek, namely a family of
n-pointed central fibers of Mustafin joins associated to one-parameter
degenerations of n points in the plane. We construct the correct
compactification of B_{n} which admits such a family, and we describe
it for n=5,6. This is joint work in progress with Jenia Tevelev.

**Group cohomology rings via equivariant cohomology
(James Cameron, April 22, 2020):**

The cohomology rings of finite groups are typically very complicated, but
their geometric properties are often tractable and retain representation
theoretic information. These geometric properties become more clear once one
considers group cohomology rings in the context of equivariant cohomology. In
this talk I will discuss how to use techniques involving flag varieties dating
back to Quillen and a filtration of equivariant cohomology rings due to Duflot
to study the associated primes and local cohomology modules of group
cohomology rings.
*This talk will be online, using webex*

**Moduli spaces on the Kuznetsov component of Fano threefolds of index 2
(Franco Rota, April 15, 2020):**

The derived category of a Fano threefold Y of Picard rank 1 and index 2
admits a semiorthogonal decomposition. This defines a non-trivial subcategory
Ku(Y) called the *Kuznetsov component*, which encodes most of the
geometry of Y.

I will present a joint work with M. Altavilla and M. Petkovic, in which we
describe certain moduli spaces of Bridgeland-stable objects in Ku(Y), via the
stability conditions constructed by Bayer, Macri, Lahoz and
Stellari. Furthermore, in our work we study the behavior of the Abel-Jacobi
map on these moduli space. As an application in the case of
degree *d=2*, we prove a strengthening of a categorical
Torelli Theorem by Bernardara and Tabuada.
*This talk will be online, using webex*

**Higher K-theory via generators and relations
(Ian Coley, April 8, 2020):**

K_{0} (the Grothendieck group) of an exact category has a nice
description in terms of generators and relations. Nenashev
(after Quillen and Gillet-Grayson) proved that
K_{1} can also be described in terms of generators and relations, and
Grayson extended that argument to all higher K-groups. I will sketch
Grayson's argument and (ideally) show some advantages of the
generators and relations approach.
*This talk will be online, using webex*

**On the Jordan property for local fundamental groups
(Joaquin Moraga, April 1, 2020):**

We discuss the Jordan property for the local fundamental group of
*klt* singularities. We also show how the existence of a large Abelian
subgroup of such a group reflects on the geometry of the singularity. Finally,
we show a characterization theorem for klt 3-fold singularities with large
local fundamental group.
*This talk will be online, using webex*

**Six explicit pairs of fake projective planes
(Lev Borisov, February 26, 2020):**

I will briefly review the history of fake projective planes and will
talk about my latest work on the subject, joint with Enrico Fatighenti.

**Quantum geometry of moduli spaces of local system
(Linhui Shen, February 19, 2020):**

Let G be a split semi-simple algebraic group over Q. We introduce a natural
cluster structure on moduli spaces of G-local systems over surfaces with
marked points. As a consequence, the moduli spaces of G-local systems admit
natural Poisson structures, and can be further quantized. We will study the
principal series representations of such quantum spaces. It will recover
many classical topics, such as the q-deformed Toda systems, quantum groups,
and the modular functor conjecture for such representations. This talk will
mainly be based on joint work with A.B. Goncharov.

**The K'-theory of monoid sets (Chuck Weibel, February 5, 2020):**

There are three flavors of K-theory for a pointed abelian monoid A;
they depend on the A-sets one allows. This talk considers the
well-behaved family of *partially cancellative* (pc) A-sets,
and its K-theory. For example, if A is the natural numbers, then
pc A-sets are just rooted trees.

**Higher norm principles for norm varieties
(Shira Gilat, January 22, 2020):**

The norm principle for a division algebra states that the image of the
reduced norm is an invariant of its Brauer-equivalence class. This can
be generalized to symbols in the Milnor K-group K^{M}_{n}(F).
We prove a generalized norm principle for symbols in
K^{M}_{n}(F) for a prime-to-p closed field F
of characteristic zero (for some prime p).
We also give a new proof for the norm principle for division algebras,
using the decomposition theorem for (noncommutative)
polynomials over the algebra.

**Tropical scheme theory (Diane Mclagan, December 11, 2019):**

Tropical geometry can be viewed as algebraic geometry over
the tropical semiring (R union infinity, with operations min and +).
This perspective has proved surprisingly effective over the last
decade, but has so far has mostly been restricted to the study of
varieties and cycles. I will discuss a program to construct a scheme
theory for tropical geometry. This builds on schemes over semirings,
but also introduces concepts from matroid theory. This is joint work
with Felipe Rincon, involving also work of Jeff and Noah Giansiracusa
and others.

**Enumerating pencils with moving ramification on curves,
(Carl Lian, November 20, 2019):**

We consider the general problem of enumerating branched covers of the
projective line from a fixed general curve subject to ramification conditions
at possibly moving points. Our main computations are in genus 1; the theory of
limit linear series allows one to reduce to this case. We first obtain a
simple formula for a weighted count of pencils on a fixed elliptic curve E,
where base-points are allowed. We then deduce, using an inclusion-exclusion
procedure, formulas for the numbers of maps E→ P^{1}
with moving ramification
conditions. A striking consequence is the invariance of these counts under a
certain involution. Our results generalize work of Harris, Logan, Osserman,
and Farkas-Moschetti-Naranjo-Pirola.

**Noncommutative Laurent Phenomena: two examples
(Volodia Retakh, November 13, 2019):**

We discuss two examples when iterations of the noncommutative rational map
are given by noncommutative Laurent polynomials. The first example is related
to noncommutative triangulation of surfaces. The second example, which leads
to a noncommutative version of the Catalan numbers, is related to solutions of
determinant-like equations. The talk is based on joint papers with
A. Berenstein from U. of Oregon.

**Vertex algebras of CohFT-type
(Angela Gibney, October 30, 2019):**

Finitely generated admissible modules over "vertex algebras of
CohFT-type" can be used to construct vector bundles of coinvariants
and conformal blocks on moduli spaces of stable curves. In this talk
I will say what vertex algebras of CohFT-type are, and explain how
such bundles define semisimple cohomological field theories.

As an application, one can give an expression for their total Chern
character in terms of the fusion rules. I'll give some examples.

**Strong exceptional collections of line bundles
(Chengxi Wang, October 23, 2019):**

We study strong exceptional collections of line bundles on
Fano toric Deligne-Mumford stacks with rank of Picard group at most
two. We prove that any strong exceptional collection of line bundles
generates the derived category of the stack, as long as the number of
elements in the collection equals the rank of the (Grothendieck)
K-theory group of the stack.

The problem reduces to an interesting combinatorial problem and
is solved by combinatorial means.

**Local-global principle for norm over semi-global fields,
Sumit Chandra Mishra, Oct. 16, 2019):**

Let K be a complete discretely valued field with
residue field κ. Let F be a function field in
one variable over K and **X** a regular proper model
of F with reduced special fibre X a union of regular curves
with normal crossings. Suppose that the graph associated to
**X** is a tree (e.g. F = K(t). Let L/F be a Galois extension
of degree n with Galois group Gand n coprime to char(κ).
Suppose that κ is algebraically closed field or
a finite field containing a primitive n^{th} root of unity.
Then we show that an element in F* is a norm
from the extension L/F if it is a norm from the
extensions L⊗_{F}F_{ν}
(i.e., $L\otimes_F F_\nu/F_\nu$)
for all discrete valuations ν of F.

**What is a derivator? (Ian Coley, October 9, 2019):**

Derivators were introduced in the 90s by Grothendieck, Heller, and
Franke (independently) to generalize triangulated categories and
answer questions in homotopy theory and algebraic geometry using a
more abstract framework. Since then, applications to modular
representation theory, tensor triangulated geometry, tilting theory,
K-theory, equivariant homotopy theory, and more have been developed by
scores of mathematicians.

This talk will give the basic definition of a derivator, motivated by
the initial question of enhancing a triangulated category, describe
some of these useful applications to the "real world" away from
category theory. We assume a priori the listener's interest in
triangulated category theory and one or more of the above
disciplines. In particular, no knowledge of infinity/quasicategories
is required!

**Generalized Brauer dimension and other arithmetic invariants of semi-global fields
(Saurabh Gosavi, October 2, 19):**

Given a finite set of Brauer classes *B* of a fixed period ℓ, we
define *ind(B)* to be the *gcd* of degrees of field extensions L/F
such that α⊗_{F}L=0 for every α in *B*. We
provide upper-bounds for *ind(B)* which depends upon arithmetic
invariants of fields of lower arithmetic complexity. As a simple
application of our result, we will obtain upper-bounds for the
splitting index of quadratic forms and finiteness of symbol length for
function fields of curves over higher-local fields.

**
Rational singularities and their cousins in arbitrary characteristics
(Sándor Kovacs, Sept. 18, 2019):**

I will discuss several results about rational and closely related
singularities in arbitrary characteristics. The results concern
various properties of these singularities including their behavior
with respect to deformations and degenerations, and applications to
moduli theory.

**Severi degrees via representation theory
(Dave Anderson, May 1, 2019):**

As a vector space, the cohomology of the Grassmannian Gr(k,n) is
isomorphic to the k-th exterior power of C^n. The geometric Satake
correspondence explains how to naturally upgrade this isomorphism to
one of $gl_n$-representations. Inspired by work of Golyshev and Manivel
from 2011, we use these ideas to find new proofs of Giambelli formulas
for ordinary and orthogonal Grassmannians, as well as rim-hook rules
for quantum cohomology. This is joint with Antonio Nigro.

**Severi degrees via representation theory
(Yaim Cooper, April 24, 2019):**

The Severi degrees of $P^1$ x $P^1$ can be computed in terms of
an explicit operator on the Fock space $F[P^1]$. We will discuss this and
variations on this theme. We will explain how to use this approach to
compute the relative Gromov-Witten theory of other surfaces, such as
Hirzebruch surfaces and Ex$P^1$. We will also discuss operators for
calculating descendants. Joint with R. Pandharipande.

**Continuous families of divisors on symmetric powers of curves
(John Sheridan, April 17, 2019):**

For X a smooth projective variety, we consider its set of
effective divisors in a fixed cohomology class. This set naturally
forms a projective scheme and if X is a curve, this scheme is a
smooth, irreducible variety (fibered in linear systems over the Picard
variety). However, when X is of higher dimension, this scheme can be
singular and reducible. We study its structure explicitly when X is a
symmetric power of a curve.

**Character Tables and Sylow Subgroups of Finite Groups
(Gabriel Navarro, April 10, 2019):**

Brauer's Problem 12 asks which properties of Sylow subgroups can be
detected in the character table of a finite group. We will talk about recent
progress on this problem.

**Effective divisors in the Hodge bundle
(Iulia Gheorghita, April 3, 2019):**

Computing effective divisor classes can reveal important information
about the geometry of the underlying space. For example, in 1982 Harris and
Mumford computed the Brill-Noether divisor class and used it to determine the
Kodaira dimension of the moduli space of curves.

In this talk I will explain how to compute the divisor class of the
locus of canonical divisors in the projectivized Hodge bundle over the
moduli space of curves which have a zero at a Weierstrass point.
I will also explain the extremality of the divisor
class arising from the stratum of canonical divisors with a double zero.

**
The algebraic theory of systems
(Louis Rowen, March 27, 2019):**

The notion of ``system'' is introduced to unify classical algebra with
tropical mathematics, hyperfields, and other related areas for which we can
embed a partial algebraic structure into a fuller structure from which we can
extract more information. The main ideas are a generalized negation map since
our structures lack classical negatives, and a ``surpassing relation'' to
replace equality.

We discuss this theory with emphasis on the main applications, which will be
described from the beginning:

1. Classical algebra

2. Supertropical mathematics (used for valuations and tropicalization)

3. Symmetrized systems (used for embedding additively idempotent semi
structures into systems)

4. Hyperfields

**A Degree Formula for Equivariant Cohomology
(Jeanne Duflot, March 13, 2019):**

I will talk about a generalization of a result of Lynn on the "degree"
of an equivariant cohomology ring $H^*_G(X)$. The degree of a graded module
is a certain coefficient of its Poincaré series, expanded as a
Laurent series about t=1. The main theorem,
which is joint with Mark Blumstein,
is an additivity formula
for degree: $$\deg(H^*_G(X)) = \sum_{[A,c] \in \mathcal{Q'}_{max}(G,X)}\frac{1}{|W_G(A,c)|} \deg(H^*_{C_G(A,c)}(c)).$$

**Volumes and intersection theory on moduli spaces of
abelian differentials (Dawei Chen, February 20, 2019):**

Computing volumes of moduli spaces has significance in many
fields. For instance, the celebrated Witten's conjecture regarding
intersection numbers on the Deligne-Mumford moduli space of stable
curves has a fascinating connection to the Weil-Petersson volume,
which motivated Mirzakhani to give a proof via Teichmueller theory,
hyperbolic geometry, and symplectic geometry. The initial two other
proofs of Witten's conjecture by Kontsevich and by
Okounkov-Pandharipande also used various ideas in ribbon graphs,
Gromov-Witten theory, and Hurwitz theory.

In this talk I will introduce an analogous formula of intersection numbers
on the moduli spaces of abelian differentials that computes the Masur-Veech
volumes. This is joint work with Moeller, Sauvaget, and Zagier
(arXiv:1901.01785).

**The Real graded Brauer group (Chuck Weibel, February 6, 2019):**

We introduce a version of the Brauer--Wall group for Real vector
bundles of algebras (in the sense of Atiyah),
and compare it to the topological analogue of the Witt group.
For varieties over the reals, these invariants capture
the topological parts of the Brauer--Wall and Witt groups.

**Paschke Categories, K-homology and the Riemann-Roch Transformation
(Khashayar Sartipi, January 30, 2019):**

For a separable C*-algebra A, we introduce an exact C*-category called the
Paschke Category of A, which is completely functorial in A, and show that its
K-theory groups are isomorphic to the topological K-homology groups of the
C*-algebra A. Then we use the Dolbeault complex and ideas from the classical
methods in Kasparov K-theory to construct an acyclic chain complex in this
category, which in turn, induces a Riemann-Roch transformation in the homotopy
category of spectra, from the algebraic K-theory spectrum of a complex
manifold X, to its topological K-homology spectrum.

**Palindromicity and the local invariant cycle theorem
(Patrick Brosnan, January 23, 2019):**

In its most basic form, the local invariant cycle theorem of
Beilinson, Bernstein and Deligne (BBD) gives a surjection from the
cohomology of the special fiber of a proper morphism of smooth
varieties to the monodromy invariants of the general fiber. This
result, which is one of the last theorems stated in the book by BBD,
is a relatively easy consequence of their famous decomposition
theorem.

In joint work with Tim Chow on a combinatorial problem, we
needed a simple condition ensuring that the above surjection is
actually an isomorphism. Our theorem is that this happens if and only
if the special fiber has palindromic cohomology. I will explain the
proof of this theorem and a generalization proved using the (now
known) Kashiwara conjecture. I will also say a little bit about the
combinatorial problem (the Shareshian-Wachs conjecture on Hessenberg
varieties) which motivated our work.

**2-Segal spaces and algebraic K-theory
(Julie Bergner, November 28, 2018):**

The notion of a 2-Segal space was defined by Dyckerhoff and Kapranov
and independently by Galvez-Carrillo, Kock, and Tonks under the name
of decomposition space. Although these two sets of authors had
different motivations for their work, they both saw that a key example
is obtained by applying Waldhausen's S-construction to an exact
category, showing that 2-Segal spaces are deeply connected to
algebraic K-theory.

In joint work with Osorno, Ozornova, Rovelli, and
Scheimbauer, we show that any 2-Segal space arises from a suitable
generalization of this construction. Furthermore, our generalized
input has a close relationship to the CGW categories of Campbell and
Zakharevich. In this talk, I'll introduce 2-Segal structures and
discuss what we know and would like to know about the role they play
in algebraic K-theory.

**Low Degree Cohomology
(Bob Guralnick November 14, 2018):**

Let G be a finite group with V an absolutely
irreducible kG-module with k a field of positive characteristic.
We are interested in bounds on the dimension of the first
and second degree cohomology groups of G with coefficients
in V. We will discuss some old and new bounds, conjectures
and applications.

**Low degree points on curves
(Isabel Vogt, November 7, 2018):**

We will discuss an arithmetic analogue of the gonality of a curve over
a number field: the smallest positive integer $e$ such that the points
of residue degree bounded by $e$ are infinite. By work of Faltings,
Harris-Silverman and Abramovich-Harris, it is understood when this
invariant is 1, 2, or 3; by work of Debarre-Fahlaoui these criteria
do not generalize to $e$ at least 4. We will focus on scenarios under
which we can guarantee that this invariant is actually equal to the
gonality using the auxiliary geometry of a surface containing the
curve. This is joint work with Geoffrey Smith.

**Algebraic groups with good reduction and unramified cohomology
(Igor Rapinchuk, October 31, 2018):**

Let $G$ be an absolutely almost simple algebraic group over a field K,
which we assume to be equipped with a natural set V of discrete
valuations. In this talk, our focus will be on the K-forms of $G$ that
have good reduction at all v in V . When K is the fraction field of a
Dedekind domain, a similar question was considered by G. Harder; the
case where $K=\mathbb{Q}$ and V is the set of all p-adic places was analyzed in
detail by B.H. Gross and B. Conrad. I will discuss several emerging
results in the higher-dimensional situation, where K is the function
field $k(C)$ of a smooth geometrically irreducible curve $C$ over a number
field k, or even an arbitrary finitely generated field.

**Generalized Polar Geometry
(Sandra Di Rocco, October 24, 2018):**

Polar classes are very classical objects in Algebraic Geometry.
A brief introduction to the subject will be presented and ideas
and preliminarily results towards generalizations will be explained.
These ideas can be applied towards variety sampling and
relevant applications in Kinematics and Biochemistry.

**Brauer class over the Picard scheme of curves
(Qixiao Ma, October 24, 2018):**

We study the Brauer class rising from the obstruction to the existence
of a tautological line bundle on the Picard scheme of curves. If we
consider the universal totally degenerate curve with a fixed dual
graph, then, using symmetries of the graph, we give bounds on the
period and index of the Brauer classes. As a result, we provide some
division algebra of prime degree, serving as candidates for the
cyclicity problem.

**On singularity properties of convolutions of algebraic morphisms
(Yotam Hendel, October 10, 2018):**

In analysis, the convolution of two functions results in a smoother,
better behaved function. It is interesting to ask whether an analogue
of this phenomenon exists in the setting of algebraic geometry.
Let $f$ and $g$ be two morphisms from algebraic varieties X and Y to an
algebraic group $G$. We define their convolution to be a morphism $f*g$
from $X\times Y$ to $G$ by first applying each morphism and then multiplying
using the group structure of $G$.

**Irrationality of Motivic Zeta Functions
(Michael Larsen, October 5, 2018):**

It is a remarkable fact that the Riemann zeta function extends to a
meromorphic function on the whole complex plane. A conjecture of Weil,
proved by Dwork, asserts that the zeta function of any variety over a
finite field is likewise meromorphic, from which it follows that it can
be expressed as a rational function. In the case of curves, Kapranov
observed that this is true in a very strong sense, which continues to
hold even in characteristic zero. He asked whether this remains true
for higher dimensional varieties. Valery Lunts and I disproved his
conjecture fifteen years ago, and recently disproved a weaker conjecture
due to Denef and Loeser. This explains, in some sense, why Weil's
conjecture was so much easier in dimension 1 than in higher dimension.

Charles Weibel / weibel @ math.rutgers.edu / November 1, 2021