Difference between revisions of "NTS/Abstracts"
(NTS/Abstracts moved to NTS/Abstracts Spring 2011) 

Line 1:  Line 1:  
−  #  +  == Anton Gershaschenko == 
+  
+  <center>  
+  { style="color:black; fontsize:100%" table border="2" cellpadding="10" width="700" cellspacing="20"  
+    
+   bgcolor="#DDDDDD" align="center" Title: Moduli of Representations of Unipotent Groups  
+    
+   bgcolor="#DDDDDD"  
+  Abstract: Representations of reductive groups are discretely parameterized, but unipotent groups can have nontrivial families of representations, so it makes sense try to construct and understand a moduli stack (or space) of representations of a given unipotent group. If you restrict to certain kinds of representations, it is possible to actually get your hands on the moduli stack and to construct a moduli space. I'll summarize the few things I know about the general case and then give you a tour of some interesting features that appear in small examples.  
+  }  
+  </center>  
+  
+  <br>  
+  
+  == Keerthi Madapusi ==  
+  
+  <center>  
+  { style="color:black; fontsize:100%" table border="2" cellpadding="10" width="700" cellspacing="20"  
+    
+   bgcolor="#DDDDDD" align="center" Title: A rationality property of Hodge cycles on abelian varieties, with an application to arithmetic compactiﬁcations of Shimura varieties  
+    
+   bgcolor="#DDDDDD"  
+  Abstract: Let A be an abelian variety over a number field E, and let v be a finite place of E where A has bad, split, semistable reduction. Then the toric part of the reduction of A at v gives rise to a partial integral structure on the (ladic, padic, deRham, log crystalline) cohomology group H^1(A_{E_v} ), arising essentially from the character group of the torus. Let η be a Hodge cycle on A; then one can ask if η is rational (in a precise sense) with respect to this new integral structure on the cohomology. This question was first considered by Andre ́.  
+  Using the theory of Shimura varieties and the FaltingsChai toroidal compactifications of the moduli of principally polarized abelian varieties, we convert this question into one of deciding if a certain subscheme of a torus embedding is again a torus embedding. In the situation where the MumfordTate group of A has a reductive model over Zp, for vp (this is the unramified situation), we employ a generalization of the methods introduced by Faltings and Kisin–initially used to construct smooth integral models of Shimura varieties–to answer this question positively.  
+  A byproduct of this rationality result is the construction of good toroidal compactifications of the integral models of Shimura varieties mentioned above. This was in fact the main motivation for considering, in the first place, the possibility of such a result. A formal consequence of the existence of these compactifications is the following result, which is a slightly weakened version of a conjecture of Yasuo Morita: Suppose the MumfordTate group G of A is anisotropic mod center, then, for any prime p such that G admits a reductive model over Zp, A has potentially good reduction at all finite places vp of E.  
+  The first part of the talk will be expository: we will introduce Hodge cycles on abelian varieties and their properties, so that we can state the problem at hand. Then, we will switch track and talk about the question on toric embeddings referenced above. After this, we will focus on the case where the reduction of A at v is a split torus. In this case, the theory is more combinatorial, but the key ideas for the general case are already visible. We will quickly sketch the properties that we need of Shimura varieties and the FaltingsChai compactification, and see how they can be used to reduce the problem to the one about toric embeddings. If time remains, we will say something about how to solve this latter problem using the FaltingsKisin method.  
+  }  
+  </center>  
+  
+  <br>  
+  
+  == Bei Zhang ==  
+  
+  <center>  
+  { style="color:black; fontsize:100%" table border="2" cellpadding="10" width="700" cellspacing="20"  
+    
+   bgcolor="#DDDDDD" align="center" Title: padic Lfunction of automorphic form of GL(2)  
+    
+   bgcolor="#DDDDDD"  
+  Abstract: Modular symbol is used to construct padic Lfunctions  
+  associated to a modular form. In this talk, I will explain how to  
+  generalize this powerful tool to the construction of padic Lfunctions  
+  attached to an automorphic representation on GL_{2}(A) where A is the ring  
+  of adeles over a number field. This is a joint work with Matthew Emerton.  
+  }  
+  </center>  
+  
+  <br>  
+  
+  == David Brown ==  
+  
+  <center>  
+  { style="color:black; fontsize:100%" table border="2" cellpadding="10" width="700" cellspacing="20"  
+    
+   bgcolor="#DDDDDD" align="center" Title: Explicit modular approaches to generalized Fermat equations  
+    
+   bgcolor="#DDDDDD"  
+  Abstract: TBA  
+  }  
+  </center>  
+  
+  
+  <br>  
+  
+  == Tony VárillyAlvarado ==  
+  
+  <center>  
+  { style="color:black; fontsize:100%" table border="2" cellpadding="10" width="700" cellspacing="20"  
+    
+   bgcolor="#DDDDDD" align="center" Title: TBA  
+    
+   bgcolor="#DDDDDD"  
+  Abstract: TBA  
+  }  
+  </center>  
+  
+  <br>  
+  == Wei Ho ==  
+  
+  <center>  
+  { style="color:black; fontsize:100%" table border="2" cellpadding="10" width="700" cellspacing="20"  
+    
+   bgcolor="#DDDDDD" align="center" Title: TBA  
+    
+   bgcolor="#DDDDDD"  
+  Abstract: TBA  
+  }  
+  </center>  
+  
+  <br>  
+  == Rob Rhoades ==  
+  
+  <center>  
+  { style="color:black; fontsize:100%" table border="2" cellpadding="10" width="700" cellspacing="20"  
+    
+   bgcolor="#DDDDDD" align="center" Title: TBA  
+    
+   bgcolor="#DDDDDD"  
+  Abstract: TBA  
+  }  
+  </center>  
+  
+  <br>  
+  == TBA ==  
+  
+  <center>  
+  { style="color:black; fontsize:100%" table border="2" cellpadding="10" width="700" cellspacing="20"  
+    
+   bgcolor="#DDDDDD" align="center" Title: TBA  
+    
+   bgcolor="#DDDDDD"  
+  Abstract: TBA  
+  }  
+  </center>  
+  
+  <br>  
+  == Chris Davis ==  
+  
+  <center>  
+  { style="color:black; fontsize:100%" table border="2" cellpadding="10" width="700" cellspacing="20"  
+    
+   bgcolor="#DDDDDD" align="center" Title: TBA  
+    
+   bgcolor="#DDDDDD"  
+  Abstract: TBA  
+  }  
+  </center>  
+  
+  <br>  
+  == Andrew Obus ==  
+  
+  <center>  
+  { style="color:black; fontsize:100%" table border="2" cellpadding="10" width="700" cellspacing="20"  
+    
+   bgcolor="#DDDDDD" align="center" Title: Cyclic Extensions and the Local Lifting Problem  
+    
+   bgcolor="#DDDDDD"  
+  Abstract: The "local lifting problem" asks: given a GGalois extension A/k[[t]], where k is algebraically closed of characteristic p, does there exist a GGalois extension A_R/R[[t]] that reduces to A/k[[t]], where R is a characteristic zero DVR with residue field k? (here a Galois extension is an extension of integrally closed rings that gives a Galois extension on fraction fields.) The Oort conjecture states that the local lifting problem should always have a solution for G cyclic. This is basic Kummer theory when p does not divide G, and has been proven when v_p(G) = 1 (Oort, Sekiguchi, Suwa) and when v_p(G) = 2 (Green, Matignon). We will first motivate the local lifting problem from geometry, and then we will show that it has a solution for a large family of cyclic extensions. This family includes all extensions where v_p(G) = 3 and many extensions where v_p(G) is arbitrarily high. This is joint work with Stefan Wewers.  
+  }  
+  </center>  
+  
+  <br>  
+  
+  == Bianca Viray ==  
+  
+  <center>  
+  { style="color:black; fontsize:100%" table border="2" cellpadding="10" width="700" cellspacing="20"  
+    
+   bgcolor="#DDDDDD" align="center" Title: Descent on elliptic surfaces and transcendental Brauer element  
+    
+   bgcolor="#DDDDDD"  
+  Abstract: Elements of the Brauer group of a variety X are hard to  
+  compute. Transcendental elements, i.e. those that are not in the  
+  kernel of the natural map Br X > Br \overline{X}, are notoriously  
+  difficult. Wittenberg and Ieronymou have developed methods to find  
+  explicit representatives of transcendental elements of an elliptic  
+  surface, in the case that the Jacobian fibration has rational  
+  2torsion. We use ideas from descent to develop techniques for general  
+  elliptic surfaces.  
+  }  
+  </center>  
+  
+  <br>  
+  
+  == Frank Thorne ==  
+  
+  <center>  
+  { style="color:black; fontsize:100%" table border="2" cellpadding="10" width="700" cellspacing="20"  
+    
+   bgcolor="#DDDDDD" align="center" Title: TBA  
+    
+   bgcolor="#DDDDDD"  
+  Abstract: TBA  
+  }  
+  </center>  
+  
+  <br>  
+  == Rafe Jones ==  
+  
+  <center>  
+  { style="color:black; fontsize:100%" table border="2" cellpadding="10" width="700" cellspacing="20"  
+    
+   bgcolor="#DDDDDD" align="center" Title: Galois theory of iterated quadratic rational functions  
+    
+   bgcolor="#DDDDDD"  
+  
+  Abstract: I'll describe recent work investigating the arboreal Galoisrepresentation attached to a degree2 rational function, focusing on the case where the function commuteswith a nontrivial Mobius transformation. In a sense this is a dynamical systems analogue to the padic Galois representation attached to an elliptic curve, with particular attention to the CM case. In joint work with Michelle Manes, we give criteria for the image of this representation to be large, and show thatthese criteria are often satisfied in the "CM" case.  
+  }  
+  </center>  
+  
+  <br>  
+  
+  == Jonathan Blackhurst ==  
+  
+  <center>  
+  { style="color:black; fontsize:100%" table border="2" cellpadding="10" width="700" cellspacing="20"  
+    
+   bgcolor="#DDDDDD" align="center" Title: Polynomials of the Bifurcation Points of the Logistic Map  
+    
+   bgcolor="#DDDDDD"  
+  Abstract: The logistic map f(r,x)=rx(1x) was originally studied by ecologists modeling the population of a species from one generation to the next. Here, the population of the next generation f(r,x) depends on the population x of the current generation and a parameter r called the probiotic potential. To find the longterm behavior of the population for a fixed r, we look at iterates of the critical point c=1/2 under the map f. If 0<r<2 the iterates approach zero, so the population goes extinct. If 2<r<3, the iterates approach a single nonzero value, and the population is in equilibrium. If 3<r<1+sqrt(6), the iterates oscillate between two values (even iterates approach one value, while odd iterates approach another) in a boomandbust cycle. As r increases, the population begins oscillating between four values, then eight, then sixteen, and this perioddoubling continues until an accumulation point (r=3.57, approximately) where the behavior of the population is no longer periodic. Even after the accumulation point, iterates of the critical point may begin again to exhibit periodic behavior. For example, at r=1+sqrt(8), they oscillate between three values. These values of r—2, 3, 1+sqrt(6), and 1+sqrt(8)—where qualitative behavior of the model changes are called bifurcation points. Little has been known about the polynomials these algebraic numbers satisfy. We find their degrees, show that their roots come in pairs whose mean is 1, put constraints on the size and prime factors of their constant coefficients, and record the number of real roots.  
+  }  
+  </center>  
+  
+  == Liang Xiao ==  
+  
+  <center>  
+  { style="color:black; fontsize:100%" table border="2" cellpadding="10" width="700" cellspacing="20"  
+    
+   bgcolor="#DDDDDD" align="center" Title: Computing logcharacteristic cycles using ramification theory  
+    
+   bgcolor="#DDDDDD"  
+  Abstract: There is an analogy among vector bundles with integrable  
+  connections, overconvergent Fisocrystals, and lisse ladic sheaves.  
+  Given one of the objects, the property of being clean says that the  
+  ramification is controlled by the ramification along all generic  
+  points of the ramified divisors. In this case, one expects that the  
+  Euler characteristics may be expressed in terms of (subsidiary) Swan  
+  conductors; and (in first two cases) the logcharacteristic cycles may  
+  be described in terms of refined Swan conductors. I will explain the  
+  proof of this in the vector bundle case and report on the recent  
+  progress on the overconvergent Fisocrystal case if time is permitted.  
+  }  
+  </center>  
+  
+  == Winnie Li ==  
+  
+  <center>  
+  { style="color:black; fontsize:100%" table border="2" cellpadding="10" width="700" cellspacing="20"  
+    
+   bgcolor="#DDDDDD" align="center" Title: Modularity of Low Degree Scholl Representations  
+    
+   bgcolor="#DDDDDD" To the space of ddimensional cusp forms of weight k > 2 for  
+  a noncongruence  
+  subgroup of SL(2, Z), Scholl has attached a family of 2ddimensional  
+  compatible ladic  
+  representations of the Galois group over Q. Since his construction is  
+  motivic, the associated  
+  Lfunctions of these representations are expected to agree with  
+  certain automorphic Lfunctions  
+  according to Langlands' philosophy. In this talk we shall survey  
+  recent progress on this topic.  
+  More precisely, we'll see that this is indeed the case when d=1. This  
+  also holds true when d=2,  
+  provided that the representation space admits quaternion  
+  multiplications. This is a joint work  
+  with Atkin, Liu and Long.  
+  
+  }  
+  </center>  
+  
+  
+  
+  == Avraham Eizenbud ==  
+  
+  <center>  
+  { style="color:black; fontsize:100%" table border="2" cellpadding="10" width="700" cellspacing="20"  
+    
+   bgcolor="#DDDDDD" align="center" Title: Multiplicity One Theorems  a Uniform Proof  
+    
+   bgcolor="#DDDDDD"  
+  
+  Abstract: Let F be a local field of characteristic 0.  
+  We consider distributions on GL(n+1,F) which are invariant under the adjoint action of  
+  GL(n,F). We prove that such distributions are invariant under  
+  transposition. This implies that an irreducible representation of  
+  GL(n+1,F), when restricted to GL(n,F) "decomposes" with multiplicity one.  
+  
+  
+  
+  Such property of a group and a subgroup is called strong Gelfand property.  
+  It is used in representation theory and automorphic forms. This property  
+  was introduced by Gelfand in the 50s for compact groups. However, for  
+  noncompact groups it is much more difficult to establish.  
+  
+  For our pair (GL(n+1,F),GL(n,F)) it was proven in 2007 in [AGRS] for  
+  nonArchimedean F, and in 2008 in [AG] and [SZ] for Archimedean F. In this  
+  lecture we will present a uniform for both cases.  
+  This proof is based on the above papers and an additional new tool. If time  
+  permits we will discuss similar theorems that hold for orthogonal and  
+  unitary groups.  
+  
+  [AG] A. Aizenbud, D. Gourevitch, Multiplicity one theorem for (GL(n+1,R),GL(n,R))", arXiv:0808.2729v1 [math.RT]  
+  
+  [AGRS] A. Aizenbud, D. Gourevitch, S. Rallis, G. Schiffmann, Multiplicity One Theorems, arXiv:0709.4215v1 [math.RT], to appear in the Annals of Mathematics.  
+  
+  
+  [SZ] B. Sun and C.B. Zhu, Multiplicity one theorems: the Archimedean case, preprint available at http://www.math.nus.edu.sg/~matzhucb/Multiplicity_One.pdf  
+  
+  }  
+  </center>  
+  
+  
+  
+  <br>  
+  
+  == Organizer contact information ==  
+  [http://www.math.wisc.edu/~brownda/ David Brown:]  
+  
+  [http://www.math.wisc.edu/~cais/ Bryden Cais:]  
+  
+  <br>  
+  
+  
+  
+    
+  Return to the [[NTSNumber Theory Seminar Page]]  
+  
+  Return to the [[AlgebraAlgebra Group Page]] 
Revision as of 17:32, 31 May 2011
Contents
Anton Gershaschenko
Title: Moduli of Representations of Unipotent Groups 
Abstract: Representations of reductive groups are discretely parameterized, but unipotent groups can have nontrivial families of representations, so it makes sense try to construct and understand a moduli stack (or space) of representations of a given unipotent group. If you restrict to certain kinds of representations, it is possible to actually get your hands on the moduli stack and to construct a moduli space. I'll summarize the few things I know about the general case and then give you a tour of some interesting features that appear in small examples. 
Keerthi Madapusi
Title: A rationality property of Hodge cycles on abelian varieties, with an application to arithmetic compactiﬁcations of Shimura varieties 
Abstract: Let A be an abelian variety over a number field E, and let v be a finite place of E where A has bad, split, semistable reduction. Then the toric part of the reduction of A at v gives rise to a partial integral structure on the (ladic, padic, deRham, log crystalline) cohomology group H^1(A_{E_v} ), arising essentially from the character group of the torus. Let η be a Hodge cycle on A; then one can ask if η is rational (in a precise sense) with respect to this new integral structure on the cohomology. This question was first considered by Andre ́. Using the theory of Shimura varieties and the FaltingsChai toroidal compactifications of the moduli of principally polarized abelian varieties, we convert this question into one of deciding if a certain subscheme of a torus embedding is again a torus embedding. In the situation where the MumfordTate group of A has a reductive model over Zp, for vp (this is the unramified situation), we employ a generalization of the methods introduced by Faltings and Kisin–initially used to construct smooth integral models of Shimura varieties–to answer this question positively. A byproduct of this rationality result is the construction of good toroidal compactifications of the integral models of Shimura varieties mentioned above. This was in fact the main motivation for considering, in the first place, the possibility of such a result. A formal consequence of the existence of these compactifications is the following result, which is a slightly weakened version of a conjecture of Yasuo Morita: Suppose the MumfordTate group G of A is anisotropic mod center, then, for any prime p such that G admits a reductive model over Zp, A has potentially good reduction at all finite places vp of E. The first part of the talk will be expository: we will introduce Hodge cycles on abelian varieties and their properties, so that we can state the problem at hand. Then, we will switch track and talk about the question on toric embeddings referenced above. After this, we will focus on the case where the reduction of A at v is a split torus. In this case, the theory is more combinatorial, but the key ideas for the general case are already visible. We will quickly sketch the properties that we need of Shimura varieties and the FaltingsChai compactification, and see how they can be used to reduce the problem to the one about toric embeddings. If time remains, we will say something about how to solve this latter problem using the FaltingsKisin method. 
Bei Zhang
Title: padic Lfunction of automorphic form of GL(2) 
Abstract: Modular symbol is used to construct padic Lfunctions associated to a modular form. In this talk, I will explain how to generalize this powerful tool to the construction of padic Lfunctions attached to an automorphic representation on GL_{2}(A) where A is the ring of adeles over a number field. This is a joint work with Matthew Emerton. 
David Brown
Title: Explicit modular approaches to generalized Fermat equations 
Abstract: TBA 
Tony VárillyAlvarado
Title: TBA 
Abstract: TBA 
Wei Ho
Title: TBA 
Abstract: TBA 
Rob Rhoades
Title: TBA 
Abstract: TBA 
TBA
Title: TBA 
Abstract: TBA 
Chris Davis
Title: TBA 
Abstract: TBA 
Andrew Obus
Title: Cyclic Extensions and the Local Lifting Problem 
Abstract: The "local lifting problem" asks: given a GGalois extension A/kt, where k is algebraically closed of characteristic p, does there exist a GGalois extension A_R/Rt that reduces to A/kt, where R is a characteristic zero DVR with residue field k? (here a Galois extension is an extension of integrally closed rings that gives a Galois extension on fraction fields.) The Oort conjecture states that the local lifting problem should always have a solution for G cyclic. This is basic Kummer theory when p does not divide G, and has been proven when v_p(G) = 1 (Oort, Sekiguchi, Suwa) and when v_p(G) = 2 (Green, Matignon). We will first motivate the local lifting problem from geometry, and then we will show that it has a solution for a large family of cyclic extensions. This family includes all extensions where v_p(G) = 3 and many extensions where v_p(G) is arbitrarily high. This is joint work with Stefan Wewers. 
Bianca Viray
Title: Descent on elliptic surfaces and transcendental Brauer element 
Abstract: Elements of the Brauer group of a variety X are hard to compute. Transcendental elements, i.e. those that are not in the kernel of the natural map Br X > Br \overline{X}, are notoriously difficult. Wittenberg and Ieronymou have developed methods to find explicit representatives of transcendental elements of an elliptic surface, in the case that the Jacobian fibration has rational 2torsion. We use ideas from descent to develop techniques for general elliptic surfaces. 
Frank Thorne
Title: TBA 
Abstract: TBA 
Rafe Jones
Title: Galois theory of iterated quadratic rational functions 
Abstract: I'll describe recent work investigating the arboreal Galoisrepresentation attached to a degree2 rational function, focusing on the case where the function commuteswith a nontrivial Mobius transformation. In a sense this is a dynamical systems analogue to the padic Galois representation attached to an elliptic curve, with particular attention to the CM case. In joint work with Michelle Manes, we give criteria for the image of this representation to be large, and show thatthese criteria are often satisfied in the "CM" case. 
Jonathan Blackhurst
Title: Polynomials of the Bifurcation Points of the Logistic Map 
Abstract: The logistic map f(r,x)=rx(1x) was originally studied by ecologists modeling the population of a species from one generation to the next. Here, the population of the next generation f(r,x) depends on the population x of the current generation and a parameter r called the probiotic potential. To find the longterm behavior of the population for a fixed r, we look at iterates of the critical point c=1/2 under the map f. If 0<r<2 the iterates approach zero, so the population goes extinct. If 2<r<3, the iterates approach a single nonzero value, and the population is in equilibrium. If 3<r<1+sqrt(6), the iterates oscillate between two values (even iterates approach one value, while odd iterates approach another) in a boomandbust cycle. As r increases, the population begins oscillating between four values, then eight, then sixteen, and this perioddoubling continues until an accumulation point (r=3.57, approximately) where the behavior of the population is no longer periodic. Even after the accumulation point, iterates of the critical point may begin again to exhibit periodic behavior. For example, at r=1+sqrt(8), they oscillate between three values. These values of r—2, 3, 1+sqrt(6), and 1+sqrt(8)—where qualitative behavior of the model changes are called bifurcation points. Little has been known about the polynomials these algebraic numbers satisfy. We find their degrees, show that their roots come in pairs whose mean is 1, put constraints on the size and prime factors of their constant coefficients, and record the number of real roots. 
Liang Xiao
Title: Computing logcharacteristic cycles using ramification theory 
Abstract: There is an analogy among vector bundles with integrable connections, overconvergent Fisocrystals, and lisse ladic sheaves. Given one of the objects, the property of being clean says that the ramification is controlled by the ramification along all generic points of the ramified divisors. In this case, one expects that the Euler characteristics may be expressed in terms of (subsidiary) Swan conductors; and (in first two cases) the logcharacteristic cycles may be described in terms of refined Swan conductors. I will explain the proof of this in the vector bundle case and report on the recent progress on the overconvergent Fisocrystal case if time is permitted. 
Winnie Li
Title: Modularity of Low Degree Scholl Representations 
To the space of ddimensional cusp forms of weight k > 2 for
a noncongruence subgroup of SL(2, Z), Scholl has attached a family of 2ddimensional compatible ladic representations of the Galois group over Q. Since his construction is motivic, the associated Lfunctions of these representations are expected to agree with certain automorphic Lfunctions according to Langlands' philosophy. In this talk we shall survey recent progress on this topic. More precisely, we'll see that this is indeed the case when d=1. This also holds true when d=2, provided that the representation space admits quaternion multiplications. This is a joint work with Atkin, Liu and Long. 
Avraham Eizenbud
Title: Multiplicity One Theorems  a Uniform Proof 
Abstract: Let F be a local field of characteristic 0. We consider distributions on GL(n+1,F) which are invariant under the adjoint action of GL(n,F). We prove that such distributions are invariant under transposition. This implies that an irreducible representation of GL(n+1,F), when restricted to GL(n,F) "decomposes" with multiplicity one.
Such property of a group and a subgroup is called strong Gelfand property. It is used in representation theory and automorphic forms. This property was introduced by Gelfand in the 50s for compact groups. However, for noncompact groups it is much more difficult to establish. For our pair (GL(n+1,F),GL(n,F)) it was proven in 2007 in [AGRS] for nonArchimedean F, and in 2008 in [AG] and [SZ] for Archimedean F. In this lecture we will present a uniform for both cases. This proof is based on the above papers and an additional new tool. If time permits we will discuss similar theorems that hold for orthogonal and unitary groups. [AG] A. Aizenbud, D. Gourevitch, Multiplicity one theorem for (GL(n+1,R),GL(n,R))", arXiv:0808.2729v1 [math.RT] [AGRS] A. Aizenbud, D. Gourevitch, S. Rallis, G. Schiffmann, Multiplicity One Theorems, arXiv:0709.4215v1 [math.RT], to appear in the Annals of Mathematics.

Organizer contact information
Return to the Number Theory Seminar Page
Return to the Algebra Group Page