Department of Mathematics

We will define the cohomology with values in a Picard category, and we will see its application to symmetric monoidal bi-categories followed by a use of biextensions to study the cohomology groups.More

We will discuss about smooth actions on manifolds by higher rank groups, such as lattices in SL(n,R) with n ≥ 3 or Z^k with k ≥ 2.



The higher rank property of the acting group sugge... More

In this expository lecture, we will recall the definition of modular forms, go over some properties of L-functions (e.g., analytic continuation), define Hecke operators, and use them in some construct... More

Cohomology groups of moduli spaces of algebraic curves consist of characteristic classes for surface bundles and appear in many disparate areas of mathematics, from low-dimensional topology to algebra... More

A surface is said to be of finite type if its fundamental group is finitely generated; otherwise we say it is of infinite type. Infinite type surfaces are, in a sense, much more mysterious than finite... More

A standard method for studying a singular variety is to resolve it by a smooth variety and to then relate invariants of the singular variety to invariants of the smooth one. Motivic integration provid... More

Anosov diffeomorphisms are a class of dynamical systems that exhibit strong chaotic behavior. Every known Anosov diffeomorphism is topologically conjugate to a linear Anosov diffeomorphism, called an ... More

In this talk we will give non-commutative generalizations of some classical facts about cohomological operations on the derived category of a hypersurface: finite generations of Ext groups, nilpotent ... More

We'll look at the double quotient B\G/H, where G is a semisimple Lie group, B a minimal parabolic subgroup, and H any closed subgroup which makes the double quotient finite. Then consider certain disc... More

This is the second and last part of a series with the following abstract: Let A be an optimal semistable elliptic curve over the rational numbers and r be a prime such that A has good reduction at r. ... More

A generic homogeneous polynomial defines a smooth hypersurface in projective space. Generalizing results of Tommasi and others, we establish homological stability for the space of such smooth polynomi... More

We will give some background on modular forms and elliptic curves that is relevant to Matthew Winters's talks. In particular, we will talk about modular curves, their Jacobians, etc., and time permitt... More

A divisible convex set is a convex, bounded, and open subset of an affine chart of the real projective space, on which acts cocompactly a discrete group of projective transformations. These objects ha... More

Let A be an optimal semistable elliptic curve over the rational numbers and r be a prime such that A has good reduction at r. In a previous talk, we indicated how to show that if r is greater than thr... More

For $n \geq 2$, the $n$-th curvature set $K_n(X)$ of a metric space $X$ is the set of all $n$-by-$n$ distance matrices of points sampled from $X$. We study the topological and geometric structures of ... More

Manin’s conjecture is about the asymptotics of the number of points on a (Fano) variety of a bounded height over a global field. In this we talk we focus on (a) when the variety is a (complete, simp... More

We will discuss the cubical Joyal model structures on the categories of cubical sets both with and without connections, by which cubical sets model the theory of (infinity,1)-categories, and the proof... More

Given six points A, B, C, D, E, F on a nonsingular conic in the complex projective plane, Pascal’s theorem says that the three intersection points AE∩BF, BD∩CE, AD∩CF are collinear. The line c... More

We will discuss the cubical Joyal model structures on the categories of cubical sets both with and without connections, by which cubical sets model the theory of (infinity,1)-categories, and the proof... More

Determinant functors map a category with a notion of exact

sequence to the 1-type of its K-Theory, in such a way that the exact

sequence in the former corresponds to the tensor p... More

Vector valued information can be transmitted by describing the coordinates of vectors in a fixed basis, however this method of transmission is not very robust since corruption of a single coordinate c... More

Determinant functors map a category with a notion of exact

sequence to the 1-type of its K-Theory, in such a way that the exact

sequence in the former corresponds to the tensor p... More

Vector valued information can be transmitted by describing the coordinates of vectors in a fixed basis, however this method of transmission is not very robust since corruption of a single coordinate c... More

This is an expository talk (mostly). It has been known for a long time that K-Theory of a ring has trace maps towards its Hochschild homology. In the last decade or so of the last century these maps w... More

I will discuss current work that extends the notion of determinant functor from triangulated categories to tensor triangulated categories. Specifically, I will discuss some background, motivation and,... More

I will discuss a categorification and generalization of the determinant. Specifically, we will start with the determinant of a matrix from undergraduate linear algebra and discuss how it became a func... More

The talk is based on a paper by W. G. Dwyer and J. Spalinski. In this talk, I define Model categories, see some examples, then I define homotopy category of Model categories in a couple of ways and se... More

Let A be a semistable elliptic curve over the rational numbers which is optimal, and r be a prime greater than three such that A has good reduction at r. We show that if A[r] is reducible, then A has ... More

Let A be a semistable elliptic curve over the rational numbers which is optimal, and r be a prime greater than three such that A has good reduction at r. We show that if A[r] is reducible, then A has ... More

A discussion on the relationship between standard and determinantal Cremona transformations and their Segre Zeta function. We will show that the homogenization of the numerator of the Segre Zeta funct... More

We will show why the quotient of the product of n upper half planes by a torsion-free discrete subgroup of SL(2,R)^n is a manifold and why the space of forms on the quotient manifold is isomorphic to ... More

here are a number of classical formulas for the generating functions of sequences defined with Legendre polynomials, but one case had eluded a closed form formula, namely for the sequence $\binom{2n}{... More

I describe some work in progress in collaboration with Keri D’angelo (Cornell — CS) to formalize crossed modules of groups, hence groups, in Homotopy Type Theory using the Agda proof assistant. No... More

I describe some work in progress in collaboration with Keri D’angelo (Cornell — CS) to formalize crossed modules of groups, hence groups, in Homotopy Type Theory using the Agda proof assistant. No... More

Automata are useful in the study of computation and ubiquitous in computer science. A major question that arises in this field is whether an automaton can be reduced and act the same way as the origin... More

The Segre class is an invariant capturing essential intersection-theoretic

information concerning embeddings of e.g., subvarieties of an algebraic variety.

Every homogeneous ideal of a p... More

The Segre class is an invariant capturing essential intersection-theoretic

information concerning embeddings of e.g., subvarieties of an algebraic variety.

Every homogeneous ideal of a p... More

We will watch June Huh’s ICM talk, titled "Combinatorial applications of the

Hodge–Riemann relations”. The abstract of this talk is



"Why do natural and interesting sequence... More

We will show that the cohomology of Hecke congruence subgroups of a certain degree can be completely described by compactly supported cohomology. A key step in this process will be to extend a form th... More

The Hecke congruence subgroup is a subgroup of the Hilbert Modular group. We will first talk about the number theory needed to understand these groups. Then we will move on to examine the stabilizer o... More

Stark-Heegner points (also called Darmon points) are certain

points defined on elliptic curves using analytic constructions, but

are conjectured to have algebraic coordinates. They are a... More

This will be an organizational meeting.

The goal of this talk is to stabilize 2-Crossed modules by putting a symmetric monoidal 2-category structure on them. We will also see outputs of the K-theory of Waldhausen categories in terms of corr... More

The Adjusted Predegree Polynomial (APP) is an algebraic invariant of hypersurfaces in n-dimensional projective space that encodes information about the closure of their orbit under the action of PGL(n... More

In this thesis we develop a few algorithms that are useful for factoring linear recurrence operators. We approach the factorization problem from three directions. First, considering reduction modulo a... More

The topic of this talk will be non-commutative geometry. Specifically, the categorical approach taken to realize certain abelian categories as "spaces". As an instance, the category of commutative r... More

This talk is based on the paper: Motives of Melonic Graphs, by Aluffi, Marcolli and Qaisar.

The Stable Homotopy Hypothesis connects symmetric monoidal algebra and stable homotopy theory. In a classical special case, the tensor product of graded abelian groups leads to a model for the 1-dimen... More

This talk will start with the Riemann Existence Theorem. We will show how it implies that for any finite group G, there exist number fields K and F with Gal(K/F) = G. Next we will discuss the issues o... More

This is an expository talk on Hilbert’s irreducibility theorem and the inverse Galois problem following Chapter 8 of Lang, Diophantine Geometry.

Elliptic curves appear in many areas of mathematics, and abelian varieties can be viewed as higher dimensional analogues of elliptic curves. This talk discusses possible generalizations of results on ... More

In this talk I'll present a method to obtain irrationality results using recurrence relations, and use it to derive the Lindemann-Weierstrass theorem. The approach is motivated by Apery's proof of th... More

In this talk I'll present a method to obtain irrationality results using recurrence relations, and use it to derive the Lindemann-Weierstrass theorem. The approach is motivated by Apery's proof of th... More

When we look at the moduli space of abelian varieties, some 'linear spaces' naturally pop up. A classical example is the Serre-Tate coordinates, which says that the local deformation space of an ordin... More

Elliptic curves are defined as smooth projective algebraic curves of genus one with a specified base point. This talk will show the equivalence of this definition to a curve cut out by an explicit cub... More

PGL(n+1) acts on homogeneous polynomials on n+1 variables by a linear change of variables. In this talk, we will interpret this as an action of PGL(n+1) on subvarieties of P^n and we will construct an... More

This is the second part of the talk with the same title. We will briefly review what is needed from the first talk. The abstract for the series is: The orders of the torsion and arithmetic component ... More

The orders of the torsion and arithmetic component groups of elliptic curves show up in the second part of the Birch and Swinnerton-Dyer conjecture. We shall recall the objects and the conjecture men... More

Differential Cohomology is a cohomological theory which extends topological cohomology theories. These latter generalized cohomology theories are those which satisfies Eilenberg-Steenrod axioms, an ex... More

This will be part 2 of the talk with the same abstract as before (I will recall whatever is needed from the first talk): We shall discuss the cohomology groups of quotients

of products of upper... More

We shall discuss the cohomology groups of quotients

of products of upper half planes and upper half spaces under the action of certain types of groups (usually arithmetic in nature). When the ... More

A classical method to desingularize a difference operator is taking the left common least multiple (LCLM) with another suitably chosen operator. In this talk we present a new desingularization algorit... More

In this talk we will discuss simplifying the general matrix differential equation by means of gauge transformations. We will first review the differential Galois theory for matrix differential equatio... More

Using the change of variables $x = y − a/2$, the general quadratic $x^2 + ax + b$ simplifies to the polynomial $y^2 + c$ in one parameter $c = b − a^2/2$. Similarly, a linear change of variables s... More

This will be an organizational meeting. I will email the Zoom link to those in the department before the scheduled time.

Univalent foundations (a.k.a homotopy type theory) is a foundation of mathematics based on dependent type theory. It has interpretations in the classical model structure on simplicial sets and other m... More

In higher algebra, we study algebraic objects endowed with a multiplication that is associative only up to (coherent) homotopy, or commutative up to (coherent) homotopy. In this Brave new algebra, we ... More

In this talk I will describe joint work with Yuki Maehara in which we give a model-independent (i.e. a purely infinity-categorical) construction of the (non-symmetric) Gray monoidal structure on the i... More

With the rising significance of (infinity, n)-categories, it is important to have easy-to-handle models for those and understand them as much as possible. In this introductory talk we will discuss how... More

Mirror symmetry predicts unexpected relationships between arithmetic properties of distinct families of algebraic varieties. For example, Wan and others have shown that for some mirror pairs, the numb... More

We present results necessary for developing model categories of enriched functors suitable for doing functor calculus. As a first example, we show how the discrete functor calculus of Bauer, Johnson a... More

The classifying diagram was defined by Rezk and is a generalization of the nerve of a category; in contrast to the nerve, the classifying diagram of two categories is equivalent if and only if the cat... More

This is joint work with Kevin McGown and Jonathan Sands. We will provide numerical evidence for related conjectures of Stark, Rubin, Popescu, and Burns concerning L-functions for abelian extensions of... More

We describe a new model structure on the category of cubical sets with connections whose cofibrations are the monomorphisms and whose fibrant objects are defined by the right lifting property with res... More

In this talk we will show how we can use the formalism of presentable infinity-categories and the fact that they are tensored over spaces to present a new and formal method to compute higher THH of va... More

Every category can be equipped with the canonical Grothendieck topology, which can be explicitly described using colimits. However, colimits are not homotopy invariant. By instead using homotopy colim... More

Deninger suggested that a Weil cohomology theory in Arakelov geometry (not yet constructed) could be used to prove the Riemann Hypothesis. I will survey the relationship between the zeta function and ... More

This will be a continuation of the previous talk with the same title. We will go over the notion of modular forms and give more details of the proof of the result that we mentioned last time. We will ... More

Beke-Bronstein algorithm is an elegant one for factoring differential operators. We reformulate the theory part in terms of exterior modules. It turns out the reformulation is also theoretically valid... More

In the theory of characteristic classes of singular varieties, the Mather class of a variety X plays a central role. It is a longstanding problem to calculate this class for X a Schubert variety in a ... More

The classifying space of a category allows us to view categories through a topological lens. When a category carries extra structure, such as a symmetric monoidal tensor product, this structure gets p... More

Thurston conjectured that quasi-Fuchsian manifolds are determined by the induced hyperbolic metrics on the boundary of their convex core and Mess generalized those conjectures to the context of global... More

Let E be an elliptic curve defined over the rational numbers. Then, up to isogeny, one can view E as an abelian subvariety of a certain Jacobian J of a modular curve. We will show that under some hypo... More

Hyperbolic Dehn filling theorem proven by Thurston is a fundamental theorem of hyperbolic 3-manifold theory, but it is not true anymore in dimension > 3. Since hyperbolic geometry is a sub-geometry of... More

Homotopy Type Theory is a new field of mathematics which combines

Homotopy Theory and Type Theory, a branch of mathematical logic and

computer science, in an innovative way. Type Theory ... More

Homotopy Type Theory is a new field of mathematics which combines

Homotopy Theory and Type Theory, a branch of mathematical logic and

computer science, in an innovative way. Type Theory ... More

This talk is a continuation of last week's talk. To make the talk accessible for all graduate students, I’ll

focus on one case and make it explicit as possible.



Part I: How ... More

The Markov numbers have been studied in connection with quadratic forms and diophantine approximation since the 1880s. These numbers satisfy a simple diophantine equation and have a particularly nice ... More

Let $K$ be a number field, and let $E/K$ be an elliptic curve over $K$.

The Mordell--Weil theorem says that the $K$-rational points $E(K)$ of $E$

form a finitely generated abelian group.... More

Lets construct the natural numbers:

N = {0, 1, 2, 3, 4, ...}

The dots mean "and so on”.  But what does “and so on” mean? It means this:

* I gave some examples.* Constructin... More

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To each subvariety of projective space we associate an integer sequence that records the maximal cardinalities of its intersections with linear subvarieties. We call this the sequence of secant indice... More

I plan to discuss topics from the 1998 paper "Quasialgebra structure of the octonions" by H. Albuquerque, S. Majid.

L-functions are fundamental objects in number theory. At the central point s = 1/2, an L-function L(s) is expected to vanish only if there is some deep arithmetic reason for it to do so (such as in th... More

The modular number and congruence number are certain invariants associated to modular abelian varieties (e.g., elliptic curves), and in joint work with K. Ribet, we proved a relation between them. Mul... More

The modular number and congruence number are certain invariants associated to modular abelian varieties (e.g., elliptic curves), and in joint work with K. Ribet, we proved a relation between them. Mul... More

Having defined the Lê cycles in the previous talk, I will explain how they are related to the exceptional divisor of the blow-up of the jacobian ideal. This will enable us to see that the sum of the ... More

I will recall the definition of the Milnor fibration of a complex hypersurface, define the Lê cycles and Lê numbers of a hypersurface with arbitrary singularities, and give several examples. Then I ... More

Prof. David Massey will visit the department for two weeks, and give talks in our seminar. My talk will review several notions from algebraic geometry which will be assumed as background in Massey’s... More

The Mahler measure of a polynomial with integer coefficients, is the product, taken over all roots whose modulus is greater than 1 and the absolute value of its leading coefficient. We'll talk about t... More

Tight frames in Euclidean spaces are widely used convenient generalizations of orthonormal bases. A particularly nice class of such frames is generated as orbits under irreducible actions of finite gr... More

Let L be a differential operator in C(x)[d/dx]. We say that L descends to a simpler differential operator L1 if L is obtained from L1 under change of variables transformation (x maps to f , deg(f)>1)... More

One important step in Stoll, Katz, Rabinoff, and Zureick-Brown's recent results, regarding conditional uniform bounds on the number of rational points in curves of genus 2-or-greater, is a covering pr... More

A basic consequence of the Chinese remainder theorem is that for any m, any matrix in SL(2,Z/mZ) can be obtained by reducing the coefficients of a matrix in SL(2,Z) mod m. This is the simplest instanc... More

Considerations of the action of a non-associative algebra

motivates the non-associative modules of Serge Lang. Under suitable restrictions, one can construct a category of modules. This admits ... More

Considerations of the action of a non-associative algebra

motivates the non-associative modules of Serge Lang. Under suitable restrictions, one can construct a category of modules. This admits ... More

The essential skeleton is an invariant of a degeneration that appears in both Berkovich geometry and minimal model theory. I will show that for degenerations with a semistable model, the essential ske... More

2-stratifolds are a generalization of 2-manifolds in that there are disjoint simple closed branch curves. Since every 3-manifold has some 2-complex as a spine one may ask which 3-manifolds have 2-stra... More

In the study of factoring linear difference operators, we have found p-curvature a powerful tool. I will talk about algorithms for computing p-curvatures and the math behind them.

n 1956, Busemann and Petty posed a series of questions about symmetric convex bodies, of which only the first one has been solved. Their fifth problem asks the following. Let K be an origin symmetric ... More

It is well known that isomorphism classes of central extensions, A-->E-->G, correspond to homotopy classes of maps between classifying spaces [BG,B^2A]. It is natural then to ask if there is a corresp... More

Given n degree d hypersufaces in projective n-space with zero-dimensional intersection, the degree of the intersection is d^n by Bézout's theorem. Determining the maximal degree of a zero-dimensional... More

I will explain how we can use techniques from algebraic number theory to construct examples of hyperbolic 3-manifolds. Outside of some elementary Galois theory, no background in number theory will be ... More

Zome is a neat construction kit featured during Math Fun Day. It is based

on the icosahedral symmetry group A5.

This talk will show how material taught in graduate algebra can be

... More

CP1 structures are geometric structures modelled on the complex projective line, acted on by the projective group PSL(2,C). These structures are not as rigid as Riemmannian structures (like Euclidean,... More

We consider smooth manifolds endowed with two metrics and the resulting geometric structures.

A motivating example come from classical tests of general relativity, where an empirical Galilean m... More

In this talk the algorithm for bounding x-degrees of right-hand factors will be introduced. I will focus on the notion of generalized exponents, which plays a crucial role in the algorithm. Generalize... More

I'll talk about the problem of finding the algebraic integer of smallest Mahler measure in a given number field. (Where this integer is not a root of unity and is a primitive element.) I'll begin wit... More

This is an ATE exam.

One of Thurston’s many amazing ideas are his hyperbolic gluing equations. Roughly speaking, given an ideally triangulated 3-manifold M, one can construct a set of complex polynomial equations whose ... More

We describe a generalization of the Segre zeta function defined by Aluffi to subschemes of projective bundles. Given a rational map of projective bundles P(F) to P(E) induced by a surjection of the un... More

We will discuss some old and new conjectures about the colored Jones polynomial.

These include the volume conjecture, AJ conjecture, slope conjecture, and strong slope conjecture.

The vo... More

In Geometric and Combinatorial Group Theory, we study finitely generated groups by relating them with their geometric counterparts, Cayley Graphs, endowed with a metric, the so called the word metric.... More

Last semester, I gave several talks introducing the non-Archimedean analytic background necessary for Katz, Rabinoff, and Zureick-Brown's proof that there exists a uniform bound on the number of ratio... More

For a finite separable field extension K/k, all subfields can be obtained by intersecting so-called principal subfields of K/k. The talk will present a way to quickly compute these intersections. If t... More

2-Stratifolds are a generalization of closed surfaces in that they contain simple closed curves where several sheets meet. Currently, there is no general classification of these spaces or their funda... More

We will start reading the paper by Bourgain, Katz and Tao on the so-called "sum-product" estimates.

We will present a new approach to the computation of the Segre class of a subscheme of projective space, based on the construction of a suitable Newton-Okounkov body. The result may be viewed as a co... More

Gromov introduced some procedures to turn a given polyhedron into a new one endowed with a piecewise Euclidean metric of non-positive curvature, while preserving some of its original topological featu... More

The octonions provide an inner product on R^8. Left or right multiplication provides a map from the octonions to SO(8), thus is an operator on R^8. With an inner product and an operator we can ask wha... More

For certain number fields, Stark's conjecture gives an analytic

formula for the absolute value of certain algebraic numbers in field

extensions. The ultimate goal is to give an analytic ... More

We will begin by discussing some background needed for simplicial groups and simplicial groupoids. We will then show that there is a correspondence between central extensions of simplicial groups A-->... More

Geometric Group Theory is one of the most explored topic in Mathematics for last few decades. In this topic we mostly try to find connections between abstract algebraic properties of groups and geome... More

We will begin by discussing some background needed for simplicial groups and simplicial groupoids. We will then show that there is a correspondence between central extensions of simplicial groups A-->... More

I plan to talk about relatively recent work of E. Katz, J. Rabinoff, and D. Zureick-Brown. These three authors established a uniform bound on the number of rational points in smooth projective algebra... More

2-Stratifolds are a generalization of surfaces where there is a family of disjoint simple closed curves where several sheets meet. They are able to be represented as

bipartite graphs from which... More

A standard result in metric geometry is that every compact geodesic metric space can be approximated arbitrarily well by finite metric graphs in the Gromov-Hausdorff sense. It is well known that the f... More

I will introduce a definition of kissing numbers of a sphere packings in 3-dimensional real space.

We will talk about Greg Kuperberg and Oded Schramn’s Theorem showing that the average number... More

For a monic polynomial in Z[x], the Mahler measure is the product, taken over all roots r, of max{1,|r|}. In some sense, this measures how far a polynomial is from being cyclotomic. We know, from Kro... More

The classification of central extensions of a group $G$ by a

(necessarily abelian) group $A$ is very well understood from both the

Algebra and Geometry viewpoints. Indeed, one of the mos... More

I plan to talk about relatively recent work of E. Katz, J. Rabinoff, and D. Zureick-Brown. These three authors established a uniform bound on the number of rational points in smooth projective algebra... More

We will discuss the idea of the Segre zeta function which in the case of closed subschemes of projective spaces explains how Segre classes behave under the operation of taking cones. Following this, w... More

We begin our study of noncommutative geometry by roughly following the book, Basic Noncommutative Geometry, M. Khalkhali.

Quadratic laminations of the unit disk were introduced by Thurston as a vehicle for understanding the Julia sets of quadratic polynomials and the parameter space of quadratic polynomials. The Central ... More

I plan to talk about relatively recent work of E. Katz, J. Rabinoff, and D. Zureick-Brown. These three authors established a uniform bound on the number of rational points in smooth projective algebra... More

Quadratic laminations of the unit disk were introduced by Thurston as a vehicle for understanding the Julia sets of quadratic polynomials and the parameter space of quadratic polynomials. The Central ... More

We will make the first step towards understanding the recent article on Matrix Weighted Estimates for singular integral operator.





We will begin by presenting the classical sparc... More

We will discuss the idea of the Segre zeta function which in the case of closed subschemes of projective spaces explains how Segre classes behave under the operation of taking cones. Following this, w... More

Roughly speaking a geometric structure is a recipe for gluing together pieces of a geometric space using geometry preserving maps to build a manifold with interesting topology. Some classical examples... More

We prove a formula computing the Chern-Schwartz-MacPherson (CSM) class of an arbitrary subscheme of a nonsingular variety in terms of the Segre class of an associated scheme. This formula generalizes ... More

The octonions are noted for there exceptional symmetry. This can be expressed by examining the size of automorphism group relative to other algebras of a similar size. In particular, the automorphism ... More

We will choose a topic to discuss first.

In the complex projective plane, how many degree d rational curves pass through 3d - 1 given points in general position? The answers N_d to this question for d = 1,2,3,4 were known classically, but on... More

Any compact irreducible 3-manifold admits a decomposition (JSJ decomposition) into pieces which are either Seifert fibered or Atoroidal. Waldhausen studied and classified manifolds where the JSJ decom... More

In the complex projective plane, how many degree d rational curves pass through 3d - 1 given points in general position? The answers N_d to this question for d = 1,2,3,4 were known classically, but on... More

We will present a proposed proof of the Andrews-Curtis Conjecture.

In the complex projective plane, how many degree d rational curves pass through 3d - 1 given points in general position? The answers N_d to this question for d = 1,2,3,4 were known classically, but on... More

It is known that the category of Waldhausen categories is a closed symmetric multicategory and algebraic K-theory is a functor from the category of Waldhausen categories to the category of spectra. By... More

We will provide more details about the construction of Stark-Heegner/Darmon points on elliptic curves over arbitrary number fields that was discussed in our talk last Fall.

Let M, N be finite-dimensional manifolds with M compact. In this talk I

will look at the Riemnannian geometry on the space C^\infty(M,N) of

smooth maps equipped with the L^2-Riemannian m... More

K-theory is a very large, useful subject with applications in many areas of mathematics. Vaguely speaking, it is very useful for classifying invariants and it can also be seen as a generalization of l... More

K-theory is a very large, useful subject with applications in many areas of mathematics. Vaguely speaking, it is very useful for classifying invariants and it can also be seen as a generalization of l... More

We present a general method to compute a presentation for any cusped hyperbolic lattice Gamma, applying a classical result of Macbeath to a suitable Gamma-invariant horoball cover of the corresponding... More

K-theory is a very large, useful subject with applications in many areas of mathematics. Vaguely speaking, it is very useful for classifying invariants and it can also be seen as a generalization of... More

Various generalizations of the octonions will be presented, and difficulties that arise in higher dimension explored. In particular, any two octonions generate a quaternionic sub algebra. However, th... More

A subset W of a Manifold M is k-contractible if the inclusion factors homotopically through k, where k is a cw-complex of dimension lower then M. The k-category of a manifold is the smallest number o... More

Various generalizations of the octonions will be presented, and difficulties that arise in higher dimension explored. In particular, any two octonions generate a quaternionic sub algebra. However, th... More

In this talk I will introduce some the basic concepts in the study of the qualitative theory of



dynamical systems. After that I will move into the setting of discrete dynamical system ... More

In this talk I will give a brief summary of the relation between some interesting characteristic classes and geometry invariants for projective varieties.

We will review the octonion algebra, including how to use it to define a parallelization of S^7. We will then proceed to show that an octonion valued function as a point can be recovered by an integra... More

In this talk I will give a brief summary of the relation between some interesting characteristic classes and geometry invariants for projective varieties.

We derive new equations for the idealized motion of a flag on a pole in 3-space, viewed as an unstretchable piece of fabric with no friction and ignoring self-collisions. Geometrically the flag repres... More

One way to define a central extension of simplicial groups is by saying it a short exact sequence of simplicial groups so that level-wise it is a central extension of groups. From this definition we w... More

One way to define a central extension of simplicial groups is by saying it a short exact sequence of simplicial groups so that level-wise it is a central extension of groups. From this definition we w... More

In this talk I will describe certain algebraic structures related to Matroid Theory. After defining a matroid over a ring, I will focus on matroids over a field k

and matroids over the integers... More

We present a new axiomatic formulation of a 2D TQFT. Our formalism is based on the edge-contraction operations on graphs drawn on a Riemann surface (cellular graphs). We assign to each cellular graph... More

We would expect there to be singular elements in a complete family of curves. One can ask: what is the largest complete family of smooth curves in projective space? Surprisingly, there is very little ... More

The derived McKay correspondence is an equivalence of derived categories between two different resolutions of certain singular varieties. The simplest example is given by the quotient of the complex 2... More

The Chern-Schwartz-MacPherson class is a generalization of the total Chern class to singular varieties. When the variety admits a smooth group action of the general linear group, we can generalize the... More

We will discuss two methods for distributing nodes according to a

prescribed continuous density, in such a way that certain local

properties hold and the resulting configuration is well-... More

A survey of loop algebras, Moufang loops, (split)-octonions and there analysis and Malcev (i.e. Moufang Lie) algebras.

We will discuss entropy of real and complex dynamics for automorphisms of rational surface that are birationally equivalent to quadratic birational maps of the plane and give examples where the entrop... More

By assigning to any Waldhausen category the fundamental groupoid of the 1-type of its K-theory spectrum, we get a 1-functor from the category of Waldhausen categories to the category of Picard groupoi... More

A survey of the universal properties of Clifford algebras, the usefulness of the composition property and an outline of the proof of Hurwitz' theorem: There are precisely four positive definite compos... More

Stark-Heegner points (also called Darmon points) are certain

points defined on elliptic curves using analytic constructions, but

are conjectured to have algebraic coordinates. They are a... More

Given a bisimplicial abelian group A there is a natural isomorphism between the nth homology of C(diagA) and Tot(C(A)), where C is the unnormalized chain complex functor. We will go over the backgroun... More

A Lie algebroid, or Lie-Rinehart algebra, is an A-module, where A is a

commutative k-algebra, and at the same time it is a k-Lie Algebra. The

prototypical example is the module of deriva... More

This will be an introductory/survey talk

in this topic. These diffeomorphisms are currently an

extremely important area within dynamical systems. They

are diffeomorphisms so that ... More

Stark’s main conjecture has been refined by various authors especially in the abelian setting. Stark himself gave a more precise conjecture “over Z” for imprimitive L-functions having precisely ... More

We give a proof that there is no nonzero harmonic differential form on the upper half space that is invariant under SL_2(C) that is of degree other than zero or the top degree. We will define the obje... More

In this talk we will discuss a theorem which gives a sufficient condition for two embedded surfaces in an irreducible 3-manifold to be isotopic. We shall also discuss some generalizations. For exampl... More

Determinantal Varieties are quite interesting varieties. In this talk we give an algorithm to compute the Chern-Mather class, Chern-Schwartz-Macpherson class, and the local Euler obstruction. We also ... More

It is conjectured that the dual variety of any smooth subvariety of dimension >2N/3 in projective N-space is a hypersurface. This is weaker than Hartshorne’s complete intersection conjecture but is ... More

This will be an introduction to an ongoing project that I am working on in collaboration with S. Preston and M. Bauer. To give some background we're going to review some rigidity theory for surfaces i... More

The Appell F1 function is a bivariate A-hypergeometric function, and satisfies a system of differential equations of rank 3. It is known for which parameters this system is reducible. We will show for... More

Hirzebruch–Riemann–Roch theorem is a fascinating result in Complex Manifold, and I will talk about how to tell the story on singular spaces.

A simplicial abelian group is a contravariant functor from the ordinal number category to the category of abelian groups and a bisimplicial abelian group is a contravariant functor from the ordinal nu... More

Let $$f:X \to X$$ be a quadratic rational surface automorphism fixing a cuspidal cubic. Let $$X_R$$ be the closure of the $$\mathbb{RP}^2$$ inside $$X$$. When the multiplier at the invariant cuspidal ... More

There has been a lot of effort to find the graded Betti numbers of the Veronese ring. I am going to define the (pinched) Veronese modules and present a sample of literature results, like the work of E... More

Let M be a hyperbolic manifold. The SL2(C) character variety of M is essentially the set of all representations  pi_1(M) --> SL2(C) up to trace equivalence.

This algebraic set is connected to ... More

Muro and Tonks constructed an algebraic model for the stable 1-type of the K-Theory spectrum of any Waldhausen by ways of stable quadratic modules. Stable 1-types are classified by Picard groupoids, s... More

A Lie algebroid, or Lie-Rinehart algebra, is a module over a commutative k-algebra A

and at the same time it is a k-Lie Algebra. The prototypical example is the algebra of derivations of a k-al... More

This talk introduces a combinatorial algebraic framework to encode, compute, and analyze topological summaries of geometric data. The motivating problem from evolutionary biology involves statistics o... More

The goal of this talk is to find a transformation that reduces a complicated differential operator L (with numerous apparent singularities) to a simpler one. To find such transformation, we present a ... More

I will describe the SL2(C) character variety for a family of hyperbolic two-bridge knots. These character varieties have multiple components which intersect at points corresponding to non-integral ir... More

There are conditions that must be met in order to attain rigidity of circle packings on certain surfaces. Ma-Schlenker pairs of circle packings provide an example of how rigidity fails in the 2-sphere... More

I will talk about the great theorem by Serre on connecting Analytic spaces and Algebraic Schemes.

In his notes, Thurston described a procedure for producing quasi-Fuchsian hyperbolic 3-manifolds from Fuchsian surface groups. Subsequently, this procedure was greatly generalized by Johnson and Mills... More

I'll be discussing the A-polynomial and some of its properties.

Motivic measure can be defined by a similar construction as the scissor-group of the area of polygons in the plane. Any polygon can be cut into finitely many triangles then reassembled into a rectangl... More

Motivic integration is a notion in algebraic geometry that was introduced by Kontsevich in 1995 and was developed by Denef and Loeser. Motivic integration can be quite useful in various branches of a... More

Two pseudo-Anosov homeomorphisms are flow equivalent if there is a homeomorphism between their mapping tori identifying the pseudo-Anosov flows induced by the respective vertical directions of the map... More

The theory of the tritangent planes to a canonically embedded sextic in CP^3 is classical. The number of such planes was known to Klein and their structure worked out by Coble. In this talk we'll qui... More

The Borromean rings have been used as everything from a logo for Ballantine beer to a symbol of marriage. But how many people have considered their hyperbolic structure?



Let M be a hyp... More

How can we quantify the difference between two curves in a homogeneous space in a manner that is invariant under reparameterization and rigid motions? How can we define and compute the "average" of se... More

The theory of the tritangent planes to a canonically embedded sextic in CP^3 is classical. The number of such planes was known to Klein and their structure worked out by Coble. In this talk we'll qui... More

A subset W of a closed 3-manifold M is torus-contractible if the inclusion map from W to M

factors homotopically through a map to the torus. We obtain a list of closed prime 3-manifolds that c... More

The Fatou set of a holomorphic mapping f is the set where the iterates f^n are locally equicontinuous. The Julia set is defined as the complement of the Fatou set. The Julia set is where any chaotic b... More

The Fatou set of a holomorphic mapping f is the set where the iterates f^n are locally equicontinuous. The Julia set is defined as the complement of the Fatou set. The Julia set is where any chaotic b... More

I'll discuss an algorithm to produce representations of the fundamental group of a knot into SL(2,C).

We compute the Chern-Schwartz-MacPherson classes of Schubert varieties in flag manifolds. These classes are obtained by constructing a representation of the Weil group, by means of certain Demazure-Lu... More

We compute the Chern-Schwartz-MacPherson classes of Schubert varieties in flag manifolds. These classes are obtained by constructing a representation of the Weil group, by means of certain Demazure-Lu... More

A c-polyhedron is a generalization of circle packings on the sphere to circle patterns with specified inversive distances between adjacent circles where the underlying 1-skeleton need not be a triangu... More

The "Heisenberg Group" is a functor that to any pair of abelian groups $A$ and $B$, assigns a nilpotent central extension of $A\times B$ by $A\otimes B$. I will show that the universal cup product map... More

This is a very general introduction to the topic

of partially hyperbolic diffeomorphisms, particularly

in dimension 3.

This talk continues the last talk. I will first give a short review of the algebraic and topological K-theory. Then I will introduce the operator K-theory and the applications of it to topology. Hopef... More

A chain complex of R-modules is a family, {P_n}, of R-modules together with module morphisms d_n:P_n --> P_{n-1} with the relation d_n d_{n+1}=0 for all n. With this relation it makes it possible to t... More

In 1989, A. Libgober defined a polynomial invariant for curve complements using the representations of the braid group. In this talk, I will introduce the Krammer polynomial which is obtained using th... More

In this talk, we will present algorithms which find a simple closed form formula or a recurrence relation for a given symbolic summation problem whose summand is a hypergeometric term.

In the 1990's Adam Epstein defined a deformation space of rational maps

on the Riemann sphere and asked: is this deformation space connected? In this talk, I will

give the definitio... More

We will define a stable quadratic module directly from a given Waldhausen Category, whose homology groups are isomorphic to the homotopy groups of the 1-type of the K-theory spectrum of the Waldhausen... More

In this talk, we consider the homogeneous space M, which is a space with a transitive group action by a Lie group G. We use the Square root velocity function introduced by Dr. Srivastava and Dr. Klass... More

In abstract algebra classes, students learn certain methods, such as Eisenstein's Criterion, for proving that a polynomial is irreducible. However, in general it can be very difficult to determine whe... More

Zeta functions of number fields are generalizations of the classical Riemann zeta function. This talk is the fourth and last in a series of talks whose goal is to describe the first few terms of the L... More

Zeta functions of number fields are generalizations of the classical Riemann zeta function. This talk is the third in a series of talks whose goal is to describe the first few terms of the Laurent ser... More

Zeta functions of number fields are generalizations of the classical Riemann zeta function. This talk is the second in a series of talks whose goal is to describe the Laurent series expansion of the z... More

Zeta functions of number fields are generalizations of the classical Riemann zeta function. This talk is the first in a series of talks whose goal is to describe the Laurent series expansion of the ze... More

The first part of the talk will be to review ideas such as Murasugi sums, fibered 3-manifolds, Coxeter links, and the Thurston norm. The goal of the talk will be to outline a result that describes the... More

Properly convex structures are a flexible generalization of complete hyperbolic structures. In general it is quite difficult to understand the set of all properly convex structures on a fixed manifold... More

A Theorem of Matsushima-Shimura describes harmonic complex-valued differential forms on n copies of the upper half place that are invariant with respect to a discrete subgroup of SL(2,R)^n; these form... More

A short introduction to algebraic and topological K-Theory with applications and a statement of the Novikov conjecture.

We will illustrate the proof of Grushko's Theorem given last week with an explicit example and point out some consequences.

Finding Gauss hypergeometric solutions of second order linear differential equations with rational function coefficients is an active research area of computational differential algebra. In this talk ... More

Given two complex polynomials of degree d, say p and q, we can

form the mating of the polynomials S^2 to S^2 by gluing together the

Julia sets of p and q in a dynamically meaningful way.... More

We'll review the classical problems in intersection theory and enumerative geometry, and then we'll look at some of the modern theory of Fulton and McPherson. One of the main tools in the modern theo... More

I am going to introduce how group action and cohomology are combined via the help of classify G space, and talk about how the action effects the character class on a possible singular space.

The purpose of this talk is to present a theorem which characterizes the malnormality of peripheral subgroups of the fundamental group of compact orientable irreducible 3- manifolds with toral boundar... More

The purposes of this talk is to introduce tensor triangulated categories and explain how one might define a form of stable K-theory for them. The rationale for attempting such construction is that ten... More

Let M be a hyperbolic 3-manifold. If G is the fundamental group of M, the set of all homomorphisms of G to SL_2 (C) is an algebraic set. We will look at this set for a certain class of 2-bridge knots.... More

Kapranov has introduced a motivic zeta function which generalizes the Weil zeta function of varieties over finite fields. However, its behavior with respect to products of varieties is very mysterious... More

We will talk about Thurston's paper in which he gave an elementary construction to show the following

Theorem: Let (M,BdM) be a compact, oriented 3-manifold (where BdM may be empty). There is a... More

We will continue with Stalling's proof of his fibering theorems for 3-manifolds.

We will talk about John Stalling's paper, in which he proves:

Theorem 1. If E is a compact 3-manifold and its fundamental group contains a finitely generated normal subgroup G, whose quotient g... More

In this talk we will review some basic ideas about mapping class groups of surfaces and fibered 3-manifolds. The Murasugi sum is an operation that lets us build new fibered links from old ones. A fibe... More

The classical Kronecker's first limit formula gives the

constant term in the Laurent expansion of a certain two variable

Eisenstein series, which in turn gives the constant term in theMore

For a manifold M in a large class of 3-manifolds, M admits a collection of algebraic numbers arising from mapping classes of essential surfaces in M connecting the geometry of M with the dynamics of t... More

We will continue with the proof of the loop theorem

Based on Hatcher's "Notes on Basic 3-Manifold Topology" we will talk about the loop theorem: Let M be a 3-manifold and let D be a 2-disk. If there is a map f: (D,∂D)→(M,∂M) with ... More

Let f:X->Y be a non-constant analytic map where X and Y are compact Riemann surfaces. The Hurwitz formula gives a relation between the ramification indices of f and the Euler characteristics of X and ... More

This talk will present the concept of local isomorphism in the context of a single example-the discrete hyperbolic plane H. I will explore the local isomorphism class (H) of this hyperbolic complex, ... More

The set of all PSL_2(C) representations of a hyperbolic knot complement is an algebraic set. The set of characters of these representations is an algebraic set as well, which is related to the topolo... More

We will talk about the prime decomposition, following Hatcher's notes on $3$-manifolds