During the academic year, the AGANT group runs three weekly seminars,
in Algebra (Mon 3:30-4:30),
in Algebraic Geometry (Wed 2:30-3:30), and
in Arithmetic Geometry/Number Theory (Wed 3:45-5:15).

We further link the three fields of AGANT with the development of our Oberseminar, which started in the fall of 2011. The Oberseminar takes place twice a semester, in general at the time of one of the current Wednesday seminars, with the following goals:

(1) A place where an incoming postdoc or an incoming tenure-track faculty gives a talk in their first semester at UGA, addressing the whole AGANT group, and thus giving a talk that would be less specialized that if given in one of the separate seminars that we hold. It is important that incoming people meet and interact with the current members of the group as soon as possible after they arrive at UGA. To give such a talk gives an incoming member the opportunity to explain their research interests to a wider audience. Such a talk, for a postdoc, could also be construed as putting in place the foundation of the job talk that he or she will need for their next job application.

(2) The Oberseminar is a place where big research ideas are discussed rather than a focus on technical details. A place where connections between the areas of AGANT are sought, or explained. A place where the latest major achievements in one of the AGANT areas are explained with a twist: how can these results be used, interpreted, viewed, in the other AGANT areas. A place where fertilization and cross-pollination can easily occur.

### 2018-2019

Oberseminar: Wednesday August 29, 2018, 2:30 – 4:30, Boyd, Room 304

David Wen, 2:30 : Towards Minimal Models of Elliptic Fibrations

The Minimal Model Program is a research program in Algebraic Geometry with the goal of constructing "simple" birational models to be used towards a birational classification of varieties. In this talk, I will give a not so technical overview of the Minimal Model Program and present some results on minimal models of certain elliptic fibrations.

Tea and Social: 3:00-3:30

Brandon Hanson, 3:30: Problems about squares that I like

I want to introduce the kind of problems that I am interested in, and I plan to do that by way of three examples, spending just a few minutes on each. I will (very) briefly introduce each problem, mention some work that I contributed toward a solution/some methods of attack, and if time permits, talk about where we could go next. The first problem is about the distribution of sums of squares in short intervals. It is a very old problem to estimate how big the space between consecutive integers which are the sums of two squares. The estimate which is the current state of the art is really just an exercise, and is over 70 years old. But, by replacing x^2+y^2 with nearby quadratic forms, the estimates get a lot better. The second problem is about something called the Large Sieve, which is a tool for understanding sequences of integers which are controlled by different congruence constraints (i.e. by forbidding certain residue classes modulo different primes). The Large Sieve is really quite powerful, but it could do a lot better, unless the integer sequence is very special (the squares!). The final problem has to do with quadratic residues (the squares mod p). These squares form a multiplicative subgroup, and seem to behave quite erratically if we perturb them by applying a translation. But what we believe, and what we can prove are quite different. So, I will finish by introducing Sarkozy’s problem, which is to refute that the set of quadratic residues arises as the collection of sums from a pair of subsets.

Kevin Dilks, 4:00: The q=-1 Phenomenon and the Cyclic Sieving Phenomenon

Often time in algebraic combinatorics, one is not just interested in enumerating the number of objects in a set, but also understanding the symmetries of those objects with respect to natural involutions and rotations. In this talk, we will look at classical examples coming from subsets and from Catalan objects, where the symmetries with respect to involution and rotation can be extracted (non-obviously) by evaluating an associated generating function at complex values, in what are respectively called the 'q=-1' phenomenon and the cyclic sieving phenomenon.

Oberseminar welcoming new faculty, Raju Krishnamoorthy, Philip Engel, and Robert Schneider, Wed August 22, 2018, 2:30 – 4:30, Boyd, Room 304

Raju Krishnamoorthy, 2:30: Correspondences without a Core

This talk will be an introduction to my NT seminar talk. Modular/Shimura curves are basic examples of compact Riemann surfaces. A remarkable theorem of Margulis implies that Shimura curves are characterized by having certain special correspondences called Hecke correspondences. We will sketch this background and explain the elementary notion of a "correspondence without a core", which generalizes Hecke correspondences. We will end with two theorems on bounded orbits, which imply e.g. the following:

1. Any two supersingular elliptic curves over \bar{F_p} are related by an l-primary isogeny for any l\neq p.

2. A Hecke correspondence of compactified modular curves is always ramified at at least one cusp.

3. There is no canonical lift of supersingular points on a (projective) Shimura curve.

Tea and Social: 3:00-3:30

Philip Engel, 3:30: Counting triangulations in two ways

We say that a triangulated surface has positive curvature if every vertex has six or fewer edges emanating from it. Weighting each triangulation appropriately, the number c_n of positive curvature triangulations of the sphere with 2n triangles can be expressed very simply as a multiple of the ninth divisor power sum of n. This is because the generating function sum c_n q^n is a special type of function called amodular form. We will outline two approaches to proving modularity. The first approach uses number theory. The positive curvature triangulated spheres are lattice points in a moduli space of flat metrics on the sphere, which is an arithmetic quotient of nine-dimensional complex hyperbolic space. The second approach uses representation theory. Every triangulation with 2n triangles admits a branched covering of degree 6n over the 2-sphere branched over 0, 1, and infinity. Generating functions of such covers are modular using representation theory of the symmetric group, and ideas from mathematical physics.

Robert Schneider, 4:00: Multiplicative theory of (additive) partitions

Much like the positive integers $\mathbb Z^+$, the set $\mathcal P$ of integer partitions ripples with interesting patterns and relations. Now, the prime decompositions of integers are in bijective correspondence with the set of partitions into prime parts, if we associate 1 to the empty partition. One wonders: might some number-theoretic theorems arise as images in $\mathbb Z^+$ (i.e. in prime partitions) of greater algebraic and set-theoretic structures in $\mathcal P$?

We show that many well-known objects from elementary and analytic number theory are in fact special cases of phenomena in partition theory: a multiplicative arithmetic of partitions that specializes to classical cases; a class of partition zeta functions'' containing the Riemann zeta function and other Dirichlet series (as well as exotic non-classical cases); deep connections to an operator from statistical physics, the $q$-bracket of Bloch-Okounkov; and other phenomena at the intersection of the additive and multiplicative branches of number theory.

### 2017-2018

Oberseminar: Ben Bakker, Thursday April 19, 2018, 4:00 – 5:00, Boyd, Room 304

Title: An invitation to o-minimal geometry

A notion originating in model theory, an o-minimal structure specifies a collection of tame subsets of R^n which obeys an important finiteness property—-one should for example think of the collection of real semi-algebraic sets.  Geometric objects locally modeled on these subsets behave very much like algebraic varieties, but the additional functions permitted by the definition allow for much greater flexibility in constructions.  Beginning with the pioneering work of Pila and Zannier, these techniques have recently led to a number of important breakthroughs in arithmetic and algebraic geometry.  In this talk we’ll introduce the basic notions and sample some applications, including results in functional transcendence theory, the Manin-Mumford conjecture, the Andre-Oort conjecture, and recent joint work with Klingler and Tsimerman in Hodge theory.

Oberseminar: Pete L. Clark, Thursday April 5, 2018, 3:30 – 4:30, Boyd, Room 304

Title: More honored in the breach: the Hasse Principle for algebraic curves

Let C be a (nice) algebraic curve defined over the rational numbers.  One says that "C satisfies the Hasse Principle" if the following implication holds: if C has real points and has a point modulo n for all integers n, then it has a rational point.  The desire for the Hasse Principle to hold is one of the main organizing ideas in arithmetic geometry...which is strange, because it holds for genus zero curves (a weak form of a 1785 result of Legendre), but for curves of genus at least one, it seems to be violated most of the time!  In this talk we will discuss various ways of making precise the idea that the Hasse Principle is "more honored in the breach": both old and new, and both proven and conjectural.  In particular we will discuss the case of hyperelliptic curves, including joint work with L.D. Watson.

Oberseminar: Dino Lorenzini, Wed Nov 1, 2017, 3:45 – 4:45, Boyd, Room 304

Title: The Direct Summand Conjecture

The Direct Summand Conjecture, a 1973 conjecture of Hochster, was recently proved by Andre. This conjecture is one of the many related Homological Conjectures, and this talk will be a brief survey on this topic for non-experts.

Oberseminar welcoming new postdocs Chun-Ju Lai, Nikon Kurnosov, and Alex Stathis, Wed August 30, 2017, 2:30 – 4:30, Boyd, Room 304

Chun-Ju Lai, 2:30: From Schur-Weyl duality to quantum symmetric pairs

The famous Schur-Weyl duality is a double centralizer property between the symmetric groups and the general linear groups/Lie algebras bypassing Schur algebras. It plays a fundamental role in early development of representation theory around a century ago, and there are still novel mathematical ideas that can be drawn from a Schur-type duality. Around 1985, Jimbo introduced a quantized duality between the Hecke algebras and the quantum groups via the q-Schur algebras. In contrast, it is made possible to construct quantum groups from the family of q-Schur algebras by Beilinson, Lusztig and MacPherson(BLM). In a seemingly unrelated direction, Letzter and Kolb developed a theory quantizing the symmetric pairs consisting of a Lie algebra and its fixed-point subalgebra. The objects constructed are called the quantum symmetric pairs, including examples arising from the reflection equations, the Onsager algebras from Ising model, the (twisted) Yangians. In this talk I will provide examples of certain Schur-type dualities beyond type A, and exhibit a new family of quantum symmetric pairs in terms of the algebras constructed a la BLM. I will summarize with applications in representation theory.

Nikon Kurnosov, 3:00: Hypekahler manifolds: restrictions and subvarieties

A triple of complex Kahler structures gives us a hyperkahler manifold. And a number of questions arise naturally - what are examples of hyperkahler manifolds and their "good" subvarieties? I will introduce my interest in this questions.

Tea and Social: 3:30-4:00

Alex Stathis, 4:00: The Hilbert Scheme of Points in the Projective Plane and its Intersection Theory

I will introduce the Hilbert scheme of points, give the background and motivation for the work conducted in my thesis, and finish by stating my results. This talk is intended to be accessible to nonalgebraic geometers.

Oberseminar welcoming new postdocs Ben Lund, Scott Mullane, and Kei Yuen Chan, Wed, August 23, 2017, 2:30 – 4:30, Boyd, Room 304

Ben Lund, 2:30: Flats determined by points

Start with a set of n points in the real plane, and draw a line through each pair. How many lines have you drawn? In 1948, de Bruijn and Erdos showed that this number is either 1, or at least n. In 1983, Beck showed that either nearly all of the points lie on a single line, or the number of lines is a constant fraction of n^2. I will discuss these results, along with their generalization to higher dimensions.

Scott Mullane, 3:00: Flat geometry, the strata of abelian differentials and the birational geometry of M_g,n.

An abelian differential defines a flat metric with singularities at its zeros and poles, such that the underlying Riemann surface can be realized as a polygon whose edges are identified pairwise via translation. A number of questions about geometry and dynamics on Riemann surfaces reduce to studying the strata of abelian differentials with prescribed number and multiplicities of zeros and poles. After introducing abelian differentials or flat surfaces, we'll discuss how flat surfaces degenerate and my interest in how flat geometry informs the birational geometry of the underlying moduli spaces of Riemann surfaces.

Tea and Social: 3:30-4:00

Kei Yuen Chan, 4:00: Dirac cohomology versus homological properties for graded affine Hecke algebras

Dirac operator has its origin in the study of quantum mechanics. It has been applied in the representation theory of reductive groups to realize discrete series by the work of Parthasarathy and Schmid. The notion of  Dirac cohomology was introduced by Vogan along with a deep conjecture relating to the infinitesimal character of Harish-Chandra modules. The conjecture has been later proved by Huang-Pandzic.

Graded affine Hecke algebras have been a useful tool in the study of the representation theory of p-adic groups. Motivated from analogies between real groups and p-adic groups, Barbasch-Ciubotaru-Trapa generalized the notion of Dirac cohomology to the setting of graded affine Hecke algebras. In this talk, I shall explore connections between the Dirac cohomology and homological properties for the modules of graded Hecke algebras, centraling around some of my results.

### 2016 - 2017

Oberseminar: Paul Pollack, Tu April 25, 2017, 3:30 – 4:30, Boyd, Room 23

Title: Arithmetic functions: old and new

I will survey some of what is known (and still unknown) about the value distribution of classical arithmetic functions. The problems discussed have in common that they owe their origin, in one way or another, to the fascination of the ancients with sums of divisors.

Oberseminar welcoming new postdocs Asilata Bapat, Anand Deopurkar, Andrew Niles, and Michael Schuster, Wed, August 24, 2016, 2:30 – 5:00, Boyd, Room 304

Anand Deopurkar, 2:30: The algebra of canonical curves and the geometry of their moduli space

Every non-hyperelliptic curve of genus g canonically embeds in the projective space of dimension (g-1). There are fascinating connections between the algebra of the corresponding homogeneous ideal and the geometry of the curve. Going further, it seems that understanding the algebra of homogeneous ideals will shed light on the birational geometry of the moduli space of all curves. I will discuss an ongoing project to understand this connection (partly joint with Fedorchuk and Swinarski).

Asilata Bapat, 3:00: Calogero-Moser space and GIT

The Calogero-Moser space is a symplectic algebraic variety that deforms the Hilbert scheme of points on a plane. It can be interpreted in many ways, for example as the parameter space of irreducible representations of a Cherednik algebra, or as a Nakajima quiver variety. It has a partial compactification that can be described combinatorially using Schubert cells in a Grassmannian. The aim of my talk is to introduce the Calogero-Moser space, and some work in progress towards constructing another partial compactification using Geometric Invariant Theory (GIT).

Tea and Social, 3:30-4:00

Andrew Niles, 4:00: The Picard Groups of Certain Moduli Problems

The Picard group of the stack M_{1,1} of elliptic curves, over an algebraically closed field of characteristic coprime to 6, was computed in 1965 by Mumford. However, the Picard group of M_{1,1} over more general base schemes (such as over the integers) was not known until it was computed in 2010 by Fulton and Olsson; their result holds over an arbitrary reduced base scheme or an arbitrary base scheme on which 2 is invertible. We present a partial generalization of the result of Fulton-Olsson, computing the Picard groups of the stacks Y_0(2) and Y_0(3) over any base scheme on which 6 is invertible.

Michael Schuster, 4:30: The multiplicative eigenvalue polytope

The multiplicative eigenvalue problem asks the following: for which sets of eigenvalues do there exist special unitary matrices A_1,...,A_n having those eigenvalues, that when multiplied A_1*A_2*...*A_n give you the identity? The set of such eigenvalues forms a convex polytope called the multiplicative eigenvalue polytope, which is connected to a number of important objects and spaces in representation theory and algebraic geometry. In this talk I will discuss the multiplicative polytope and its connections with quantum cohomology, conformal blocks, moduli spaces of parabolic bundles, and moduli spaces of curves, time permitting.

### 2015-2016

Oberseminar: Dan Nakano, Wed May 4, 2016, 2:30 – 3:30, Boyd, Room 304

Title: Irreducibility of Weyl modules over fields of arbitrary characteristic

In the representation theory of split reductive algebraic groups, the following is a well-known fact: for every minuscule weight, the Weyl module with that highest weight is irreducible over every field. The adjoint representation of E_8 is also irreducible over every field.  Recently, Benedict Gross conjectured that these two examples should be the only cases where the Weyl modules are irreducible over arbitrary fields. In this talk I will present our proof of Gross' suggested converse to these statements, i.e., that if a Weyl module is irreducible over every field, it must be either one of these, or trivially constructed from one of these. My coauthors will be revealed during my talk.

Oberseminar: Paul Pollack, Wed March 30, 2016, 2:30 – 3:30, Boyd, Room 304

Title: A survey of recent work on gaps between primes

I will present an overview of the spectacular progress from the past few years towards the (in)famous twin prime conjecture. At the conclusion of the talk, I will discuss a very recently discovered (just this month!) "repulsion phenomenon" for consecutive primes in residue classes.

Oberseminar: Eric Katz (Waterloo), Wed December 2, 2015, 3:45 – 4:45, Boyd, Room 304

Title: Hodge Theory on Matroids

The chromatic polynomial of a graph counts its proper colorings.  This polynomial's coefficients were conjectured to form a unimodal sequence by Read in 1968.  This conjecture was extended by Rota in his 1970 address to assert the log-concavity of the characteristic polynomial of matroids which are the common generalizations of graphs and linear subspaces.  We discuss the resolution of this conjecture which is joint work with Karim Adiprasito and June Huh.  The solution draws on ideas from the theory of algebraic varieties, specifically Hodge theory, showing how a question about graph theory leads to a solution involving Grothendieck's standard conjectures.

Oberseminar: Valery Alexeev and Elham Izadi, Wed November 4, 2015, 3:30 – 4:30, Boyd, Room 304

Title:  What is an abelian 6-fold?

Abelian varieties, higher-dimensional generalizations of elliptic curves, are basic objects in algebraic geometry, arithmetic geometry, and number theory. Over the complex numbers, they are quotients of vector spaces by lattices. Classically, (principally polarized) abelian varieties of low dimension g have a very special description: for g up to 3 they are Jacobians of curves, and for g up to 5 they are Pryms associated to curves with involution. This implies that moduli spaces of abelian varieties for g up to 5 are unirational: they can be rationally parameterized by g(g+1)/2 parameters. On the other hand, Harris-Mumford proved that for g \ge 7 the moduli spaces are of general type, which is on the opposite side of the spectrum. The situation for g=6 has been open since the 1980s. In this work, joint with Donagi, Farkas, Ortega, we prove a beautiful conjecture of Kanev, describing a general abelian 6-fold as a "Prym-Tyurin" variety for a 27:1 cover of curves with the same symmetry as the 27 lines on a cubic surface in P3. We also make a big advance towards determining the birational type of the moduli of abelian 6-folds.

Oberseminar welcoming new postdocs Julian Rosen, Reza Seyyedali, and Paul Sobaje, Wed, August 26, 2015, 2:30 – 4:30, Boyd, Room 304

Paul Sobaje, 2:30: Modular representation of algebraic groups

Let G be a linear algebraic group over a field of positive characteristic.  We'll look at questions and methods which arise from studying the representation theory of G by restriction to its various finite subgroups (and subgroup schemes), in particular focusing on the theory of support varieties for modules.

Julian Rosen, 3:00: Periods and multiple zeta values

A period is complex number that, roughly speaking, arises as an integral of a rationally defined function over a rationally defined region. Although periods are often transcendental, they have lots of algebraic structure, including a (largely conjectural) Galois theory. The multiple zeta values are a particular class of periods that arise in many areas of pure an applied math. These periods can also be described be infinite series, and finite truncations of these series are rational numbers with interesting arithmetic properties. This talk will be an introduction to periods, multiple zeta values, and their finite truncations.

Tea and Social, 3:30-4:00

Reza Seyyedali, 4:00:  Chow stability of ruled manifolds

In 2001, Donaldson proved that the existence of cscK metrics on a polarized manifold (X,L) with discrete automorphism group implies that (X,L^k) is Chow stable for k large enough. We show that if E is a simple stable bundle over a polarized manifold (X,L), (X,L) admits cscK metric and have discrete automorphism group, then (PE^*, \O(d) \otimes L^k)  is Chow stable for k large enough.

### 2014-2015

Oberseminar: Amber Russell, Mon, April 27, 2015, 3:30 – 4:30, Room 302, Boyd

Title: Perverse Sheaves:  Powerful results and related constructions

The focus of this talk will be on results whose proofs rely on perverse sheaves.  For example, the Fundamental Lemma and Deligne's proof of the Weil Conjecture.  I will also discuss the properties of perverse sheaves which have made them so useful, and if time permits, I will discuss briefly perverse coherent sheaves and parity sheaves, focusing on their relation to perverse sheaves and the new results associated to them.

Oberseminar: Dino Lorenzini, Wed, Feb 18, 2015, 3:45 – 4:45, Room 304, Boyd

Title: The Mathematics of Alexander Grothendieck (1928-2014)

Alexander Grothendieck (1928-2014) died last November. He wrote a thesis on topological vector spaces under Laurent Schwartz (Fields medal 1950) in which he completely solved a series of 14 problems published in a paper by Dieudonné and Schwartz in 1949. Sixty years ago in 1955, Grothendieck completely changed his field of research and began a revolution in algebra, algebraic geometry and number theory. I plan to discuss Grothendieck's mathematics starting with a letter of his dated February 18, 1955.

Oberseminar welcoming new faculty Noah Giansiracusa, Wed, November 19, 2014, 3:45pm – 4:45pm, Boyd, Room 304

Title: Berkovich analytification and the universal tropicalization

I'll discuss joint work with my brother in which we use our theory of tropical schemes to develop a purely algebraic (based on semirings) view of non-archimedean analytification. Given an integral scheme X over a non-archimedean valued field k, there is a universal closed embedding of X into a k-scheme equipped with a model over the field with one element (a generalization of a toric variety). An embedding into such an ambient space determines a tropicalization of X, and I'll explain how the tropicalization of X with respect to this universal embedding set-theoretically is the Berkovich analytification of X. The scheme-theoretic tropicalization of this embedding can be thought of as an intrinsic, universal tropicalization: it maps to all other tropicalizations and satisfies a scheme-theoretic enrichment of a limit result of Payne. In addition, this universal tropicalization turns out to be the moduli space of non-archimedean valuations on X. In particular, this construction yields a universal valuation on any integral k-algebra. ​​

Oberseminar welcoming new postdocs Abbey Bourdon, and Patricio Gallardo, Wed, September 3, 2014, 2:30pm – 3:30pm, Boyd 304

Abbey Bourdon: A Uniform Version of a Finiteness Conjecture for CM Elliptic Curves

One approach to studying the absolute Galois group is to examine its action on other objects, such as the algebraic fundamental group of $\mathbb{P}^1_{\bar{\mathbb{Q}}} \setminus \{0,1,\infty\}$. In the 1980s, Ihara studied a pro-$\ell$ version of this representation for a prime $\ell$ and asked whether a natural field associated to this action is maximal. Torsion point fields of abelian varieties provide concrete examples of fields with similar properties, and they are poised to give either explicit subfields of Ihara's abstract field or to provide a negative answer to his question. Unfortunately, appropriate abelian varieties to use as test cases for Ihara's question are quite rare. A 2008 conjecture made by Rasmussen and Tamagawa addresses the scarcity of such abelian varieties, and states that for a fixed dimension and field of definition there exists such an abelian variety for only a finite number of primes. In this talk, I will present a uniform version of the Rasmussen-Tamagawa conjecture in the case of elliptic curves with complex multiplication. The method, which relies on the connection between CM elliptic curves and class field theory, has the advantage of providing explicit bounds on the primes $\ell$ in many cases.

Patricio Gallardo: On the parameter space of elliptic quartics in the projective space. How can we compactify the open set that parametrizes smooth elliptic quartics in the projective space?

In this talk, we discuss several answers to this question, open problems, and partial new results. This is an ongoing project with C. Lozano-Huertas and B. Schmidt.

### 2013-2014

Oberseminar welcoming new postdoc Anna Kazanova, Wed, October 2, 2013, 3:45pm – 4:45pm, Room 303, Boyd

Title: Degenerations of surfaces of general type and vector bundles.

We will describe a relation between some boundary components of the moduli space of stable surfaces of general type and certain vector bundles.

Oberseminar welcoming new postdoc Joseph Vandehey, Wed, September 4, 2013, 2:30pm – 3:30pm

Title: Digit patterns in the number of prime divisors function

The function omega(n) counts the number of distinct prime factors of n. It is well-known that omega(n) acts similar to a random variable with mean and variance log log n; or, roughly speaking, given a random n, we can guess the first half of the digits of n with high probability of being correct. But what about the rest of the digits? What can we say about them, if anything? The answer will take us from ergodic theory, through analytic and elementary number theory, asymptotic and Fourier analysis.

### 2012-2013

Oberseminar: Robert Varley, Wed, April 3, 2013, 3:30pm – 4:30pm, Room 304, Boyd

Title: Operad actions on configurations and cohomology

The operad concept basically took off in 1963 with J. Stasheff's criterion for a topological space to have the homotopy type of a loop space. His analysis of associativity up to homotopy was expressed in terms of an A-infinity space in topology and an A-infinity algebra on the cohomology level. Then Boardman and Vogt used E-infinity (for "homotopy everything") to characterize infinite loop spaces, and in 1972 J.P. May formalized the definition of operad exactly as it stands now, at least in the topological category. Following an introduction I will discuss some self-contained and interesting known examples of operad actions (or compositional structures) related to (1) moduli of Riemann surfaces with marked points, and (2) cohomology operations. I am not planning to give the general definition of an operad in a symmetric monoidal category; the best short introduction is probably Stasheff's piece "What is ... an operad?" in the 2004 Notices.

Oberseminar: Dino Lorenzini, Wed, February 6, 2013, 2:30pm – 3:30pm, Room 304 in Boyd

Title: On Kim's approach to Faltings' theorem and other diophantine problems using fundamental groups

Abstract: A general talk accessible to graduate students in algebra, number theory, and algebraic geometry.

Oberseminar welcoming new postdoc Jie Wang, Wed, November 7, 2012, 3:45pm – 4:45pm, Room 302 in Boyd

Title: Generic vanishing results on certain Koszul cohomology groups

Abstract: A central problem in curve theory is to describe algebraic curves in a given projective space with fixed genus and degree. One wants to know the extrinsic geometry of the curve, i.e., information on the equations defining the curve. Koszul cohomology groups in some sense carry 'everything one wants to know' about the extrinsic geometry of curves in projective space: the number of equations of each degree needed to define the curve, the relations between the equations, etc. In this talk, I will present a new method using deformation theory to study Koszul cohomology of general curves. Using this method, I will describe a way to determine number of defining equations of a general curve in some special degree range (but for any genus).

Oberseminar welcoming new postdoc Amber Russell, Wed, October 3, 2012, 3:45pm – 4:45pm, Room 302 in Boyd

Title: Perverse Sheaves and the Springer Correspondence

Abstract: Perverse sheaves were first defined in the early 1980's, and they arise largely out of the theory of intersection homology. They have been instrumental in results in multiple areas of mathematics, but particularly in representation theory. In this talk, we will begin by discussing briefly the usefulness of these objects, and then focus on Borho and MacPherson's particular application to the Springer Correspondence, a Lie theoretic result relating representations of a Lie algebra's Weyl group to its nilpotent orbits.

Oberseminar welcoming new postdoc Lola Thompson, Wed, September 5, 2012, 3:45pm – 5:15pm, Room 302 in Boyd

Title: Products of distinct cyclotomic polynomials

Abstract: A polynomial is a product of distinct cyclotomic polynomials if and only if it is a divisor over Z[x] of x^n-1 for some positive integer n. In this talk, we will examine two natural questions concerning the divisors of x^n-1: "For a given n, how large can the coefficients of divisors of x^n-1 be?" and "How often does x^n-1 have a divisor of every degree between 1 and n?" We will consider the latter question when x^n-1 is factored in both Z[x] and F_p[x].

### 2011-2012

Oberseminar: Valery Alexeev, Wed, February 15, 2012, 2:30pm – 4:00pm, Boyd 328

Title: Moonshine

Abstract: "All you ever wanted to know about... Moonshine", or: "Mathieu groups, K3 surfaces, and moonshine" This is going to be a general-audience talk about some fascinating, mysterious and largely unexplained connections between algebra (sporadic simple groups and their representations), number theory (elliptic curves and modular curves), algebraic geometry (elliptic curves, K3 surfaces), and physics (conformal field theory). The "Monstrous moonshine" was a 1979 conjecture of Conway and Norton concerning a totally unexpected connection between the monster group (the largest simple group, of order about 10^54) and modular functions. It was proved by Borcherds. Mathieu group M24 is another sporadic simple group, and has order about 10^8. It is a subgroup of S24 preserving the binary Golay code (an error-correcting code used in digital communications) and the Witt design, otherwise known as the Steiner system S(5,8,24). Its definition is intimately connected with the 24 24-dimensional unimodular Niemeier lattices, which include the fabulous Leech lattice. Subgroups of M24 stabilizing 1, resp. 2 points are called Mathieu groups M23 and M22. The connection with algebraic geometry was discovered by Mukai in a famous 1988 paper whose main result is that G is a finite group of symplectic automorphisms of a K3 surface iff G is a subgroup of M23 with at least 5 orbits. Recently, physicists found the "Mathieu moonshine", a mysterious connection between M24, elliptic genus of a K3 surface, and mock theta functions (and black holes, of course). Currently, no explanation for this moonshine exists.

Oberseminar:  Brian Boe and Angela Gibney, Wed, November 30, 2011, 2:30pm – 4:00pm, Boyd 303

Title: Conformal blocks

Oberseminar:  Pete Clark, Daniel Nakano, and Dino Lorenzini, Wed, October 26, 2011, 2:30pm – 4:00pm

Title: On characteristic p problems.

Oberseminar:  Danny Krashen and William Graham, Wed, September 21, 2011, 2:30pm – 3:30pm, Boyd 323

Title: On Langlands Program

Abstract: The Langlands program relates geometry, number theory and representation theory. The relation arises because given an algebraic variety, one can define certain functions ($L$-functions) which are related to number theory (to the number of solutions to the equations defining the variety with coefficients in a finite field). These functions turn out to be related to functions which are automorphic", i.e., functions which are (almost) invariant under some group action; more generally, automorphic representations" of the group appear. In this talk we will attempt to explain some of the motivation and ideas involved in this relationship. The talk should be accessible to graduate students in algebra, number theory and algebraic geometry.