AdIMOM

Adelaide Institute of Mediocre and Outstanding Mathematics

About us | Past Seminars | Conference

Homemade Seminars

If you wish to be added to the mailing list, please send an email to homemadeseminar[at]gmail[dot]com. Here is a full list of the talks:

2017

2016

2015

Abstracts

Speaker: Justin Martel
Date: September 30, 2017
Title: How to make a group to act on a space with not that big dimension
Abstract: We describe homological-duality on manifolds-with-corners $(X, \del X)$, where $X$ is a contractible space supporting a free $G$-action, with $G$ a finitely-generated discrete matrix group.
Effectively computing $G$-equivariant (co)homology on $X$ is expensive process, which is burdened by the fact that (!): the apparent space dimension $dim[X]$ is typically much larger than the algebraic symmetry dimension $vcd[G]$ of the symmetry group $G$. This observation originates with Borel-Serre from ~1980s in their paper, and leads to interesting ideas.
Our goal is to describe explicit $G$-equivariant subvarieties $Z$ of $X$, whose topological dimension coincides with the symmetry dimension. We display the subvarieties $Z$ as ``Kantorovich Singularities" of an optimal semicoupling measure, with respect to a (gated)-electron-cost. The Kantorovich duality principle underyling mass transport methods is, we claim, the fundamental fact responsible for poincare duality. Attendees can expect details!

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Speaker: Fedya Kogan
Date: September 16, 2017
Title: Combinatorial species
Abstract: Tired of doing enumerative combinatorics by using shady operations on generating series? Try a fresh conceptual method for counting graphs and much more! This technique was developed by a king of categories Andre Joyal; the idea is quite abstract, but essentially very accessible. This talk will be challenging, yet simple, and most importantly fun!

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Speaker: Derya Ciray
Date: June 27, 2017
Title: Estimating the density of rational points on subsets of the reals
Abstract: This talk is about a method used to estimate the density of rational points (of bounded height) on subsets of the reals. A set has a mild parametrization, if it can be covered by a finite number of smooth functions with certain bounds on their derivatives (mild functions). I will explain how mild parametrization is used to achieve 'good bounds' on the density of rational points for certain sets.

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Joshua Wen:
Date: June 04, 2017
Title: Uniformization of G-bundles on curves
Abstract: To construct and classify G-bundles on the projective line, one can cut the line into two disks and use an element of the loop group to glue trivial bundles on each disk. By accounting for automorphisms of the bundles on each disk, we get a presentation of moduli of G-bundles on the line in terms of double cosets of the loop group. Surprisingly, a similar double coset construction exists for moduli of G-bundles on any compact Riemann surface---this is the content of the uniformization theorems of Beauville-Laszlo and Laszlo-Sorger. These presentations allow one to relate certain 'strange' induced representations of the loop group to vector bundles on the moduli spaces, which leads to the study of conformal blocks.

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Speaker: Özgür Esentepe
Date: March 18, 2017
Title: Cohomology annihilators and chicken mcnuggets
Abstract: A very nice theorem due to Serre states that a ring is regular (read smooth) if and only if it there is a finite number d such that for all n bigger than d and for all modules M and N, we have Ext^n(M,N)=0. This is equivalent to saying that any element inside the ring annihilates all large extension groups between any two modules. When the ring is not regular, the elements which annihilate all large extension groups form a nontrivial ideal. I will talk about this ideal, what happens in dimension one and how the answer was motivated by the coin problem (aka chicken mcnugget problem).

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Speaker: Jeffrey Carlson
Date: March 11, 2017
Title: Cohomogeneity-one actions and a little-remarked structure on the Mayer-Vietoris sequence
Abstract: The action of a Lie group G on a manifold M is said to be of cohomogeneity one if the orbit space M/G is a 1-manifold; such actions are arguably the simplest after the transitive, and accordingly have been long discussed and also classified in low dimensions. In this talk, we compute the equivariant K-theory and Borel equivariant cohomology rings $K^*_G M$ and $H^*_G(M;\mathbb Q)$ of such actions. The proof follows readily from a result of potentially much wider interest, namely the existence of an additional algebraic structure on the Mayer-Vietoris sequence for any multiplicative cohomology theory which, though easily stated and simply demonstrated, has-as best we can tell-nevertheless escaped mention in introductory topology texts. This work is joint with Oliver Goertsches, Chen He, and Augustin-Liviu Mare.

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Speaker: Ananth Shankar
Date: March 4, 2017
Title: The p-curvature Conjecture and Monodromy About Simple Closed Loops
Abstract: We will discuss vector bundles with flat connections, and talk about algebraic criteria for the existence of a full set of flat sections.

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Speaker: Lucia Mocz
Date: February 10, 2017
Title: Variation of Faltings Heights of CM Abelian Varieties
Abstract: We discuss a proof of a new Northcott property for Faltings' heights of CM abelian varieties. In particular, we show that there are finitely many CM abelian varieties of a fixed dimension of bounded Faltings height. We will focus this talk on understanding the variation of Faltings heights for CM abelian varieties within isogeny classes, with a full discussion on the explicit local (intersection) computations.

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Speaker: Sasha Shapiro
Date: February 4, 2017
Title: Poisson Geometry and Representation Theory
Abstract: I will outline relations between Poisson geometry and representation theory. In particular, I will explain how the universal enveloping algebras of simple Lie algebras on one hand and the quantum groups on the other arise in the same Poisson geometric framework, and why differential and q-difference operators are so important in their representation theory.

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Speaker: Max Klambauer
Date: January 28, 2017
Title: Modular Forms and Adeles
Abstract: Modular forms are the prototypical example of an automorphic form, though you wouldn't know that from the definitions. By the end of the talk we shall see how a modular form, defined in a straightforward way on the complex upper half plane, can be viewed as an automorphic form, which lives on the more complicated adeles. We will begin by talking about modular forms (just for the sake of it), then move on to talk about Tate's groundbreaking thesis to get a hang for the adeles. After this we will march on to our goal.

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Speaker: Aaron Fenyes
Date: January 21, 2017
Title: Feynman diagrams from inner products
Abstract: Last Saturday, the audience complained when the speaker recalled a little undergraduate linear algebra, so this Saturday I'll give a talk consisting entirely of undergraduate linear algebra. I'll show you how fancy-sounding gadgets from quantum field theory, like Feynman diagrams and Wick's theorem, arise naturally from the linear algebra of polynomials on an inner product space.
The main part of the talk will follow the attached notes, though I'll skip most of the background and cut right to the chase. With luck, I'll also manage to prepare a few interesting examples.

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Speaker: Ian Le
Date: January 14, 2017
Title: Hives and the Saturation Conjecture
Abstract: I will start by introducing some of the basics of the finite-dimensional representation theory of GL_n. I will then define Littllewood-Richardson coefficients, and talk about the problem of computing them. The saturation conjecture is a very interesting conjecture about Littlewood-Richardson coefficients that tells us that in some ways they behave regularly. I will explain how hives were used to solve the saturation conjecture, and also explain some relations of this problem to other areas of math.

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Speaker: Ethan Yale Jaffe
Date: January 7, 2017
Title: The Hodge Theorem
Abstract: The Hodge Theorem is a beautiful theorem connecting the topology of a compact Riemannian manifold to its geometry. In this talk I will sketch a proof of the Hodge theorem using pseudodifferential operators and the techniques of (global) microlocal analysis.

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Speaker: Jason van Zelm
Date: December 10, 2016
Title: The Moduli Space of Stable Curves and Its Intersection Theory
Abstract: In this talk I will introduce the moduli space of stable curves and its Chow ring. I will define a particularly nice subring of the Chow ring called the tautological ring and give a formula for the intersection of two classes in the tautological ring. Finally, I will discuss some recent developments about the limitations of this tautological subring.

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Speaker: Eduard Duryev
Date: November 25, 2016
Title: Dynamics on Moduli Spaces
Abstract: - Every Riemann surface of genus 2 admits a non-constant holomorphic map to a Riemann sphere. But does it admit one to a given elliptic curve? Any elliptic curve?
- Take a bunch of unit squares in a complex plane and glue them along their opposite sides so that they form a closed surface. You will obtain a Riemann surface, a point in the moduli space of complex structures. Can you obtain any complex structure via such construction? What if you slightly generalize and start with a bunch of polygons instead of squares?
- Take your favorite matrix from SL(2,Z) and apply it to that bunch of unit squares in the plane remembering the identification you chose. What kind of surface do you get? Do you ever get the same thing? (spoiler: very often you do!)
These questions come from the area called 'dynamics on the moduli space’ or 'translation surfaces'. It allows another view on the moduli space of curves as polygons in the plane. There is an action of GL(2,R) on that space and there is a rich study of the orbits of that action as they provide subvarieties of M_g.

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Speaker: Fima Abrikosov
Date: November 19, 2016
Title: Cluster Algebras and Varieties and Their Applications
Abstract: Cluster algebras were introduced by Fomin & Zelevinskiy in early 2000’s. Since then this subject became a striving area of research and many applications were found ‘in nature'. Brightest of them include theory of canonical bases for Lie algebras and categorical framework for the theory of Donaldson-Thomas invariants. I will give a gentle introduction to the theory of cluster algebras motivated by some geometric examples. I’ll try to describe a general framework for understanding cluster varieties. If time permits I’ll outline how Teichmüller spaces and spaces of laminations fit into this picture.

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Speaker: Peter Crooks
Date: November 12, 2016
Title: Some compactifications in Lie theory‎
Abstract: I will discuss some techniques for compactifying varieties in Lie-theoretic contexts, with an emphasis on how such compactifications arise naturally from algebraic group actions.

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Speaker: Payman Eskandari
Date: November 5, 2016
Title: Gross' proof of the Chowla-Selberg formula
Abstract: A classical formula of Chowla and Selberg expresses periods of an elliptic curve with complex multiplication up to an algebraic factor in terms of products of special values of the gamma function. Gross gave an algebro-geometric proof of this result in his 1978 Inventiones paper "On the Periods of Abelian Integrals and a Formula of Chowla and Selberg". I plan to sketch some ideas of Gross' proof.

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Speaker: Bernd Schober
Date: October 22, 2016
Title: Resolution of singularities
Abstract:A starting point of algebraic geometry is the study of varieties, zero sets of polynomial equations. In general, a variety $ X $ may have singular points where information on the geometry of $ X $ is hidden. In order to understand this situation better one tries to find a model $ Y $ of $ X $ which shares many properties with $ X $, but which is easier to handle. One approach is to resolve the singularities. In 1964, Hironaka has proven the existence of such a model for varieties over fields of characteristic zero. The aim of this talk is to give a gentle introduction to resolution of singularities over the complex numbers. By discussing some examples, we will explore together the ideas and techniques developed from Hironaka's famous proof. If time permits, I will finally explain some obstruction which appear when we consider fields of positive characteristic.

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Speaker: Fulgencio Lopez
Date: October 8, 2016
Title: Math of Musical Scale
Abstract:We will study the different frequencies of musical scales and how their history.

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Speaker: Timothy Magee
Date: October 4, 2016
Title: Toric varieties, log Calabi-Yau's, and Combinatorial Representation Theory
Abstract: Roughly speaking, a log Calabi-Yau is a space that comes equipped with a volume form in a natural way. Let X be an affine log CY with volume form Ω. We can partially compactify X by adding divisors along which Ω has a pole. The set of these divisors says a lot about X's geometry-- for a torus (the simplest example of a log CY) this set is just the cocharacter lattice. We can actually give this set a geometrically motivated multiplication rule too, which I hope not to get into. But this multiplication rule allows us to construct an algebra A, defined purely in terms of the geometry of X, and conjecturally A is the algebra of regular functions on the mirror to X. Viewed as a vector space, A naturally comes with a basis-- the divisors we used to define it. So we get a canonical basis for the space of regular functions on X's mirror. All of the technology involved is a souped-up version of something from the world of toric varieties. I'll describe the picture for toric varieties first, then say how this fits into the broader log CY setting. Finally, many objects of interest to representation theorists (semi-simple groups, flag varieties, Grassmannians...) are nice partial compactifications of log CY's. This gives us a chance to use the machinery of log CY mirror symmetry to get results in rep theory. Maybe the most obvious application given what I've said so far is finding a canonical basis for irreducible representations of a group. I'll discuss this, as well as how this machinery reproduces cones that make combinatorial rep theorists feel warm and fuzzy inside (like the Gelfand-Tsetlin cone and the Knutson-Tao hive cone).

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Speaker: Benjamin Briggs
Date: September 17, 2016
Title: Complex reflection groups, Duality groups, such and such
Abstract: I'll try to say something about the beautiful theory of complex reflection groups. We'll cover some classical things which, maybe, everyone should see. Then we'll talk about some of these reflection groups which satisfy a curious duality property, and what you can get out of it. Maybe we will talk about other things, like flag varieties. This is a matter of invariant theory, representation theory, and some nice geometry, but it should be very accessible to anyone who wants to come.

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Speaker: Mykola Matviichuk
Date: August 21, 2016
Title: Deformation theory of Dirac structures via L infinity algebras
Abstract: A Dirac structure is a lagrangian subalgebroid in a Lie bialgebroid. One should think of Lie bialgebroids as a generalization of Lie bialgebras, which play a crucial role in the theory of Poisson-Lie groups. To any Dirac structure we associate a natural L infinity algebra (aka strong homotopy Lie algebra) governing deformation theory of the Dirac. The plan of the talk is to start with defining the relevant notions of Dirac geometry and motivate the problem. Then I will give the necessary background on L infinity algebras. Then I will try to explain our construction. Based on the audience demand, the plan can be modified as we go, or abandoned altogether. No background in Dirac geometry or L infinity algebras will be assumed.

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Speaker: Elliot Cheung
Date: July 25, 2016
Title: Geometric Invariant Theory
Abstract: GIT is a way to study the equivariant geometry of varieties or schemes (especially in the projective case). A popular application of GIT is in the construction of coarse moduli spaces. In this context, GIT is a good tool for providing compactifications of such spaces. Unfortunately, it is often the case that the boundary of such a compactification has a "weaker modular meaning". One may partially resolve the singularities of a GIT quotient and provide an alternative compactification of these moduli spaces with better modular properties. In the GIT language: it's a resolution that turns all semi-stable points into either properly stable or unstable.

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Speaker: Iordan Ganev
Date: April 11, 2016
Title: The wonderful compactification for quantum groups
Abstract: The wonderful compactification of a semisimple group links the geometry of the group to the geometry of its partial flag varieties, encodes the asymptotics of matrix coefficients for the group, and captures the rational degenerations of the group. It plays a crucial role in several areas of geometric representation theory, and can be realized as a quotient of the Vinberg semigroup. In this talk, we will review several constructions of the wonderful compactification and its relevant properties. We then introduce quantum group versions of the Vinberg semigroup, the wonderful compactification, and the latter's stratification by G x G orbits. The talk will include an overview of necessary background from the representation theory of reductive groups, and a discussion of noncommutative projective geometry.

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Speaker: Peter Budrin
Date: April 2, 2016
Title: Laurence Sterne and Soviet Literary Scholars of the 1930s
Abstract: We will explore the reception of Laurence Sterne's novels in Stalin's Russia in the 1930s. In that tragic time of executions, arrests and the suppression of free thought an interesting and paradoxical discussion about the personality and works of the English comic author took place. The analysis is based both on published texts and archival materials: unpublished articles about Sterne, personal letters and documents of literary scholars of the 1930s.

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Speaker: Krishan Rajaratnam
Date: March 26, 2016
Title: Laurence Sterne and Soviet Literary Scholars of the 1930s
Abstract: We will explore the reception of Laurence Sterne's novels in Stalin's Russia in the 1930s. In that tragic time of executions, arrests and the suppression of free thought an interesting and paradoxical discussion about the personality and works of the English comic author took place. The analysis is based both on published texts and archival materials: unpublished articles about Sterne, personal letters and documents of literary scholars of the 1930s.

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Speaker: Anne Dranovski
Date: March 19, 2016
Title: Some Motivation for Geometric Representation Theory
Abstract: We'll reveal the why and how of the flag variety: where it comes from, and how it's used. On the how end, we'll recite a handful of fancy dualities, including, you guessed it, Howe duality. Examples may be ramped up from sl_2 to sl_3, though always over C.

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Speaker: Justin Martel
Date: March 5, 2016
Title: Virtual Cohomological Dimension and Borel-Serre Formula
Abstract: I look to describe some homological-duality problems related to the arithmetic groups SL(2,Z), Sp(4,Z), connectivity of their rational Tits buildings and Borel-Serre rational bordifications of the correspondant symmetric spaces.

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Speaker: Anton Izosimov
Date: February 27, 2016
Title: Integrable systems and Riemann surfaces
Abstract: I will give an informal review of the theory of integrable systems from the point of view of algebraic geometry. In particular, I will prove a classical result saying that integrable systems could be linearized on the Jacobian of the spectral curve. The talk will be based on simple (nevertheless,quite general) examples.

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Speaker: Özgür Esentepe
Date: January 30, 2016
Title: Some Motivation for Noncommutative Algebraic Geometry
Abstract: In the classical theory of algebraic geometry, there is a very nice correspondence between certain points on certain topological spaces and certain ideals of certain rings. Similarly, in representation theory, there is a very nice correspondence between certain ideals of certain rings and certain modules over those rings. Then, certain French mathematicians in certain decades of last century said that we should change our understanding of spaces and they introduced sheaves, and they established yet another correspondence between certain sheaves and certain of modules. This all happened in the commutative case. I will talk about what happens in noncommutative algebra and how certain modules act like points of certain so called noncommutative spaces.

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Speaker: Changho Han
Date: January 16, 2016
Title: Why moduli spaces
Abstract: Moduli spaces is a roughly 100 years old concept that changed the way mathematicians think. We will explore various significance of moduli spaces as families and will focus on examples. Examples will mainly be from algebraic geometry.

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Speaker: Eduard Duryev
Date: November 25, 2015
Title: Knots, Links and 3-manifolds
Abstract: This semester I was trying to get closer to such objects as knots, links and 3-manifolds. Various people were telling me interesting stories and I want to share some of them with you. My talk will be rather an amateur bus trip (if you know what I mean) by the colorful sights of topological wonders rather than a deep investigation of the topic. In particular, we will perform a dance with Seifert surfaces and make a surgery of the trefoil knot complement in the 3-sphere to extract its Alexander polynomial - a powerful invariant of knots and links. We will see how Thurston norm on the homologies of 3-manifolds, a younger sibling of Alexander polynomial, keeps record of all fibrations of this manifold over the circle.
And here are some questions you might want to bring to our trip:
Is there a foliation of a 3-sphere by smooth surfaces?
Why figure-8 knot complement is a hyperbolic 3-manifold?
Can you visualize a circle bundle over a torus with Chern class equal to 1?

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Speaker: Petr Pushkar
Date: November 14, 2015
Title: Borel-Weil-Bott Theorem and Weyl Character Formula
Abstract: I will present the one of the most basic interactions of Representation Theory and Geometry. There are many different ways to prove the Weyl character formula using just representation theory, I will show how to deduce it from the geometrical properties of the flag variety.

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Speaker: Dylan William Butson Esquire
Date: November 7, 2015
Title: Rational Homotopy Theory
Abstract: Some elements of Sullivan's approach to rational homotopy theory: homotopy theory in the categories of spaces, simplicial sets and differential graded algebras, the relationship between them, and in particular a practical method to compute the rational homotopy groups of spaces from their dgas of differential forms.

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Speaker: Louis-Philippe Thibault
Date: October 31, 2015
Title: McKay Correspondence
Abstract: The McKay correspondence describes a relationship between resolution of Kleinian singularities and representation theory of finite subgroups of SU(2, C). It gave rise to many interesting questions and results linking resolution of singularity, Auslander-Reiten theory and representation theory of algebras, as well as string theory, amoung others. In this talk, we will describe the correspondence and show some of its nice consequences. We will try to give an historical account, starting with some work of Plato and DaVinci.

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Speaker: Ivan Telpukhovskiy
Date: October 24, 2015
Title: On Moduli Space of Algebraic Curves
Abstract: I will give a talk on moduli space of algebraic curves. This is an important subject of study in algebraic geometry. I will start with examples in small dimensions, describe Deligne-Mumford compactification and maybe reach ELSV formula.

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Speaker: Leonid Monin
Date: xx
Title: Tait-Kneser Theorem
Abstract: xx

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Speaker: Gaurav Patil
Date: October 3, 2015
Title: Taniyama-Shimura Conjecture
Abstract: xx

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Speaker: Parker Glynn-Adey
Date: September 12, 2015
Title: q-Homotopy for Simplicial Complexes
Abstract: In the 1970s R.H. Atkin developed tools for studying connectivity properties of simplicial complexes in a series of strange papers. The resulting theory of q-connectedness was applied in many novel contexts but has since fallen in to disuse. The theory has been entirely neglected by topologists (perhaps because it was invented by a physicist and it is not a homeomorphism invariant). This talk will introduce the modern treatment of q-connectedness as a homotopy theory for combinatorial simplicial complexes. We'll apply it by computing some weird groups related to the Platonic solids and, time permitting, some buildings.

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Speaker: Vincent Gelinas
Date: July 12, 2015
Title: Cyclic Homology
Abstract: xx

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Speaker: Benjamin Briggs
Date: xx
Title: An Introduction to Derived Categories
Abstract: xx

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Speaker: Leonid Monin
Date: xx
Title: Tropics and Gromov-Witten Invariants
Abstract: xx

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Speaker: Leonid Monin
Date: xx
Title: Flat Surfaces
Abstract: xx

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Speaker: Leonid Monin
Date: xx
Title: On Bernstein-Kouchnirenko Theorem
Abstract: xx